Elements of Hydrodynamics

Nicolas Borghini

Version of September 2 2015

Nicolas BorghiniUniversitaumlt Bielefeld Fakultaumlt fuumlr PhysikHomepage httpwwwphysikuni-bielefeldde~borghiniEmail borghini at physikuni-bielefeldde

Foreword

The following pages were originally not designed to fall under your eyes They grew up fromhandwritten notes for myself listing the important points which I should not forget in the lectureroom As time went by more and more remarks or developments were added which is why Istarted to replace the growingly dirty sheets of paper by an electronic versionmdashthat could then alsobe easily uploaded on the web page of my lecture for the benefit() of the students

Again additional results calculations comments paragraphs or even whole chapters accumu-lated leading to the temporary outcome which you are reading now a not necessarily optimaloverall outline at times unfinished sentences not fully detailed proofs or calculationsmdashbecause themissing steps are obvious to memdash insufficient discussions of the physics of some resultsmdashwhichI hopefully provide in the classroommdash not-so-good-looking figures incomplete bibliography etcYou may also expect a few solecisms inconsistent notations and the usual unavoidable typos(lowast)

Eventually you will have to cope with the many idiosyncrasies in my writing as for instance myimmoderate use of footnotes dashes or parentheses quotation marks which are not considered asldquogood practicerdquo

In short the following chapters may barely be called ldquolecture notesrdquo they cannot replace atextbook(dagger) and the active participation in a course and in the corresponding tutorialexercisesessions

(lowast)Comments and corrections are welcome(dagger) which is one of several good reasons why you should think at least twice before printing a hard copy

Contents

Introductionbull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 1

I Basic notions on continuous media bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 2I1 Continuous medium a model for many-body systems 2

I11 Basic ideas and concepts 2

I12 General mathematical framework 4

I13 Local thermodynamic equilibrium 4

I2 Lagrangian description 7I21 Lagrangian coordinates 8

I22 Continuity assumptions 8

I23 Velocity and acceleration of a material point 8

I3 Eulerian description 9I31 Eulerian coordinates Velocity field 9

I32 Equivalence between the Eulerian and Lagrangian viewpoints 10

I33 Streamlines 10

I34 Material derivative 11

I4 Mechanical stress 13I41 Forces in a continuous medium 13

I42 Fluids 14

II Kinematics of a continuous medium bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull16II1 Generic motion of a continuous medium 16

II11 Local distribution of velocities in a continuous medium 17

II12 Rotation rate tensor and vorticity vector 18

II13 Strain rate tensor 19

II2 Classification of fluid flows 22II21 Geometrical criteria 22

II22 Kinematic criteria 22

II23 Physical criteria 23

Appendix to Chapter II bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull24IIA Deformations in a continuous medium 24

III Fundamental equations of non-relativistic fluid dynamics bull bull bull bull bull bull bull bull bull bull25III1 Reynolds transport theorem 25

III11 Closed system open system 25

III12 Material derivative of an extensive quantity 26

III2 Mass and particle number conservation continuity equation 28III21 Integral formulation 28

III22 Local formulation 29

III3 Momentum balance Euler and NavierndashStokes equations 29III31 Material derivative of momentum 30

III32 Perfect fluid Euler equation 30

III33 Newtonian fluid NavierndashStokes equation 34

III34 Higher-order dissipative fluid dynamics 38

III4 Energy conservation entropy balance 38III41 Energy and entropy conservation in perfect fluids 39

III42 Energy conservation in Newtonian fluids 40

III43 Entropy balance in Newtonian fluids 41

IV Non-relativistic flows of perfect fluids bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull44IV1 Hydrostatics of a perfect fluid 44

IV11 Incompressible fluid 45

IV12 Fluid at thermal equilibrium 45

IV13 Isentropic fluid 45

IV14 Archimedesrsquo principle 47

IV2 Steady inviscid flows 48IV21 Bernoulli equation 48

IV22 Applications of the Bernoulli equation 49

IV3 Vortex dynamics in perfect fluids 52IV31 Circulation of the flow velocity Kelvinrsquos theorem 52

IV32 Vorticity transport equation in perfect fluids 54

IV4 Potential flows 56IV41 Equations of motion in potential flows 56

IV42 Mathematical results on potential flows 57

IV43 Two-dimensional potential flows 60

V Waves in non-relativistic perfect fluids bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull70V1 Sound waves 70

V11 Sound waves in a uniform fluid at rest 71

V12 Sound waves on moving fluids 74

V13 Riemann problem Rarefaction waves 74

V2 Shock waves 75V21 Formation of a shock wave in a one-dimensional flow 75

V22 Jump equations at a surface of discontinuity 76

V3 Gravity waves 79V31 Linear sea surface waves 79

V32 Solitary waves 83

VI Non-relativistic dissipative flows bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull89VI1 Statics and steady laminar flows of a Newtonian fluid 89

VI11 Static Newtonian fluid 89

VI12 Plane Couette flow 90

VI13 Plane Poiseuille flow 91

VI14 HagenndashPoiseuille flow 92

VI2 Dynamical similarity 94VI21 Reynolds number 94

VI22 Other dimensionless numbers 95

VI3 Flows at small Reynolds number 96VI31 Physical relevance Equations of motion 96

VI32 Stokes flow past a sphere 97

VI4 Boundary layer 100VI41 Flow in the vicinity of a wall set impulsively in motion 100

VI42 Modeling of the flow inside the boundary layer 102

VI5 Vortex dynamics in Newtonian fluids 104VI51 Vorticity transport in Newtonian fluids 104

VI52 Diffusion of a rectilinear vortex 105

VI6 Absorption of sound waves 106

VII Turbulence in non-relativistic fluids bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 110VII1 Generalities on turbulence in fluids 110

VII11 Phenomenology of turbulence 110

VII12 Reynolds decomposition of the fluid dynamical fields 112

VII13 Dynamics of the mean flow 113

VII14 Necessity of a statistical approach 115

VII2 Model of the turbulent viscosity 116VII21 Turbulent viscosity 116

VII22 Mixing-length model 117

VII23 k-model 118

VII24 (k-ε)-model 118

VII3 Statistical description of turbulence 119VII31 Dynamics of the turbulent motion 119

VII32 Characteristic length scales of turbulence 120

VII33 The Kolmogorov theory (K41) of isotropic turbulence 122

VIII Convective heat transfer bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 125VIII1 Equations of convective heat transfer 125

VIII11 Basic equations of heat transfer 125

VIII12 Boussinesq approximation 127

VIII2 RayleighndashBeacutenard convection 128VIII21 Phenomenology of the RayleighndashBeacutenard convection 128

VIII22 Toy model for the RayleighndashBeacutenard instability 131

IX Fundamental equations of relativistic fluid dynamics bull bull bull bull bull bull bull bull bull bull bull 133IX1 Conservation laws 134

IX11 Particle number conservation 134

IX12 Energy-momentum conservation 136

IX2 Four-velocity of a fluid flow Local rest frame 137

IX3 Perfect relativistic fluid 139IX31 Particle four-current and energy-momentum tensor of a perfect fluid 139

IX32 Entropy in a perfect fluid 141

IX33 Non-relativistic limit 142

IX4 Dissipative relativistic fluids 144IX41 Dissipative currents 144

IX42 Local rest frames 147

IX43 General equations of motion 148

IX44 First order dissipative relativistic fluid dynamics 149

IX45 Second order dissipative relativistic fluid dynamics 151

Appendices to Chapter IX bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 153IXA Microscopic formulation of the hydrodynamical fields 153

IXA1 Particle number 4-current 153

IXA2 Energy-momentum tensor 154

IXB Relativistic kinematics 154

IXC Equations of state for relativistic fluids 154

X Flows of relativistic fluids bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 155X1 Relativistic fluids at rest 155

X2 One-dimensional relativistic flows 155X21 Landau flow 155

X22 Bjorken flow 155

Appendices bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 159

A Basic elements of thermodynamics bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 159

B Tensors on a vector space bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 160B1 Vectors one-forms and tensors 160

B11 Vectors 160

B12 One-forms 160

B13 Tensors 161

B14 Metric tensor 162

B2 Change of basis 164

C Tensor calculus bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 165C1 Covariant differentiation of tensor fields 165

C11 Covariant differentiation of vector fields 165

C12 Examples differentiation in Cartesian and in polar coordinates 167

C13 Covariant differentiation of general tensor fields 168

C14 Gradient divergence Laplacian 168

C2 Beginning of elements of an introduction to differential geometry 169

D Elements on holomorphic functions of a complex variable bull bull bull bull bull bull bull bull bull 170D1 Holomorphic functions 170

D11 Definitions 170

D12 Some properties 170

D2 Multivalued functions 171

D3 Series expansions 171D31 Taylor series 171

D32 Isolated singularities and Laurent series 171

D33 Singular points 171

D4 Conformal maps 172

Bibliography bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 173

Introduction

General introduction and outlineNotations conventions etc

General references(in alphabetical order)

bull Faber Fluid dynamics for physicists [1]

bull Guyon Hulin Petit amp Mitescu Physical hydrodynamics [2]

bull Landau amp Lifshitz Course of theoretical physics Vol 6 Fluid mechanics [3]= Landau amp Lifschitz Lehrbuch der theoretischen Physik Band VI Hydrodynamik [4]

bull Sommerfeld Lectures on theoretical physics Vol II Mechanics of deformable bodies [5]= Vorlesungen uumlber theoretische Physik Band II Mechanik der deformierbaren Medien [6]

CHAPTER I

Basic notions on continuous media

A system of many microscopic degrees of freedom is often more conveniently described as a materialbody that fills some region of space continuously rather than as a collection of discrete points(Sec I1) This theoretical approach which is especially suited to represent systems whose internaldeformations are relevant is an instance of physical modeling originally motivated by the agreementof its predictions with experimental observations Like every model that of a continuous mediumis valid only in some range of physical conditions in particular on macroscopic scales

Mathematically a classical continuous medium at a given instant is described as a volumemdashormore generally a manifoldmdashin usual Euclidean space The infinitesimal elements of this volumeconstitute the elementary ldquomaterial pointsrdquo which are entirely characterized by their position

To describe the time evolution of the physical system modeled as a continuous medium twoequivalent approaches are available The first one consists in following the trajectories of the materialpoints as time progresses (Sec I2) The physical picture of continuousness is then enforced byrequesting that the mapping between the position of a given point at some reference initial timeand its position at any later instant is continuous

The second point of view which will mostly be adopted in the remainder of these notes focuseson the change in the various physical quantities at a fixed position as time elapses (Sec I3)The reference for the medium evolution between successive instants t and t + dt is the ldquocurrentrdquoconfiguration of the material points ie at time t instead of their positions in the (far) past In thatdescription the spatial variables are no longer dynamical but only labels for the position at whichsome observable is considered Accordingly the dynamical quantities in the system are now time-dependent fields the desired continuousness of the medium translates into continuity conditions onthose fields

Eventually the mathematical object that models internal forces in a continuous medium iethe influence from neighboring material points on each other is shortly introduced (Sec I4) Thisallows the classification of deformable continuous media into two traditional large classes and inparticular the definition of fluids

I1 Continuous medium a model for many-body systemsIn this Section we first spell out a few arguments which lead to the introduction of the model ofa continuous medium (Sec I11) The basic ingredients of the mathematical implementation ofthe model are then presented and a few notions are defined (Sec I12) Eventually the physicalassumptions underlying the modeling are reexamined in greater detail and some more or less obviouslimitations of the continuous description are indicated (Sec I13)

I11 Basic ideas and concepts

The actual structure of matter at the microscopic scale is discrete and involves finite ldquoelemen-taryrdquo entities electrons atoms ions molecules which in the remainder of these notes will becollectively referred to as ldquoatomsrdquo Any macroscopic sample of matter contains a large amount ofthese atoms For instance the number density in an ideal gas under normal conditions is about

I1 Continuous medium a model for many-body systems 3

27times 1025 mminus3 so that one cubic millimeter still contains 27times 1016 atoms Similarly even thoughthe number density in the interstellar medium might be as low as about 102 mminus3 any volumerelevant for astrophysics ie with at least a kilometer-long linear size involves a large number ofatoms

Additionally these atoms are in constant chaotic motion with individual velocities of order102ndash103 mmiddotsminus1 for a system at thermal equilibrium at temperature T 300 K Given a mean freepath(i) of order 10minus7 m in a gas under normal conditions each atom undergoes 109ndash1010 times persecond ie its trajectory changes direction constantly when viewed with a macroscopic viewpoint

As is seen in Statistical Mechanics it is in general unnecessary to know the details of the motionof each atom in a macroscopic system as a matter of fact there emerge global characteristicsdefined as averages which can be predicted to a high degree of accuracy thanks to the large numberof degrees of freedom involved in their determination despite the chaoticity of the individual atomicbehaviors The macroscopic properties of systems at (global) thermodynamic equilibrium are thusentirely determined by a handful of collective variables either extensivemdashlike entropy internalenergy volume particle number total momentum mdash or intensivemdashas eg the respective densitiesof the various extensive variables temperature pressure chemical potential average velocity mdashwhere the latter take the same value throughout the system

When thermodynamic equilibrium does not hold globally in a system there is still the possi-bility that one may consider that it is valid locally ldquoat each pointrdquo in space In that situationmdashwhose underlying assumptions will be specified in greater detail in Sec I13mdashthe intensive thermo-dynamic variables characterizing the system macroscopically become fields which can vary frompoint to point More generally experience shows that it is fruitful to describe a large amount ofcharacteristicsmdashnot only thermodynamic but also of mechanical nature like forces and the dis-placements or deformations they inducemdashof macroscopic bodies as fields A ldquocontinuous mediumrdquois then intuitively a system described by such fields which should satisfy some (mathematical)continuity property with respect to the spatial variables that parameterize the representation of thephysical system as a geometrical quantity as will be better specified in Secs I2 and I3

Assuming the relevance of the model of a medium whose properties are described by continuousfields is often referred to as continuum hypothesis(ii)

The reader should keep in mind that the modeling of a given macroscopic system as a continuousmedium does not invalidate the existence of its underlying discrete atomic structure Specificphenomena will still directly probe the latter as eg X-ray scattering experiments That is themodel has limitations to its validity especially at small wavelengths or high frequencies whereldquosmallrdquo or ldquohighrdquo implies a comparison to some microscopic physical scale characteristic of thesystem under consideration Turning the reasoning the other way around the continuous-mediumpicture is often referred to as a long-wavelength low-frequency approximation to a more microscopicdescriptionmdashfrom which it can actually be shown to emerge in the corresponding limits

It is important to realize that the model itself is blind to its own limitations ie there isno a priori criterion within the mathematical continuous-medium description that signals thebreakdown of the relevance of the picture to actual physics In practice there might be hintsthat the equations of the continuous model are being applied in a regime where they should notas for instance if they yield negative values for a quantity which should be positive but suchoccurrences are not the general rule

Remarkslowast The model of a continuous model is not only applicablemdashand appliedmdashto obvious cases likegases liquids or (deformable) solids it may also be used to describe the behaviors of large crowdsfish schools car traffic provided the number of ldquoelementaryrdquo constituents is large and the systemis studied on a large enough scale(i)mittlere freie Weglaumlnge (ii)Kontinuumshypothese

4 Basic notions on continuous media

lowast Even if the continuous description is valid on ldquolong wavelengthsrdquo it remains obvious that anyphysical system viewed on a scale much larger than its spatial extent is to first approximation bestdescribed as pointlikeConsider for instance a molecular cloud of interstellar medium with a 10 parsec radius and about1010 H2 molecules per cubic meter For a star forming at its core it behaves a continuous medium1 kpc away however the inner degrees of the cloud are most likely already irrelevant and it is bestdescribed as a mere point

I12 General mathematical framework

Consider a non-relativistic classical macroscopic physical system Σ described by Newtonianphysics The positions of its individual atoms viewed as pointlike at a given instant tmdashwhich isthe same for all observersmdashare points in a three-dimensional Euclidean spaceE 3

In the description as a continuous medium the system Σ is represented by a geometrical manifoldinE 3 which for the sake of simplicity will be referred to as a ldquovolumerdquo and denoted by V Thebasic constituents of V are its infinitesimal elements d3V called material points(iii) or continuousmedium particles(iv)mdashwhich explains a posteriori our designating the discrete constituents of matteras ldquoatomsrdquomdash or in the specific case of the elementary subdivisions of a fluid fluid particles(v) Aswe shall state more explicitly in Sec I13 these infinitesimal elements are assumed to have thesame physical properties as a finite macroscopic piece

Associated with the physical picture attached to the notion of continuousness is the requirementthat neighboring material points in the medium remain close to each other throughout the systemevolution We shall see below how this picture is implemented in the mathematical description

Remark The volume V mdashwith the topology inherited fromE 3mdashneed not be simply connected Forinstance one may want to describe the flow of a river around a bridge pier the latter represents aphysical region which water cannot penetrate which is modeled as a hole throughout the volumeV occupied by fluid particles

To characterize the position of a given material point as well as some of the observables relativeto the physical system Σ one still needs to specify the reference frame in which the system isstudied corresponding to the point of view of a given observer and to choose a coordinate systemin that reference frame This choice allows one to define vectorsmdashlike position vectors velocitiesor forcesmdashand tensors

The basis vectors of the coordinate system will be designated as ~e1 ~e2 ~e3 while the componentsof a given vector will be denoted with upper (ldquocontravariantrdquo) indices as eg ~c = ci~ei where theEinstein summation convention over repeated upper and lower indices was used

Once the reference frame and coordinate system are determined the macroscopic state of thephysical system at time t is mapped onto a corresponding configuration(vi) κt of the mediumconsisting of the continuous set of the position vectors ~r = xi~ei of its constituting material pointsSince the volume occupied by the latter may also depends on time it will also be labeled by t Vt

To be able to formalize the necessary continuity conditions in the following Sections one alsointroduces a reference time t0mdashconveniently taken as the origin of the time axis t0 = 0mdashand thecorresponding reference configuration κ0 of the medium which occupies a volume V0 The genericposition vector of a material point in this reference configuration will be denoted as ~R = Xi~ei

Remark In so-called ldquoclassicalrdquo continuous media as have been introduced here the material pointsare entirely characterized by their position vector In particular they have no intrinsic angularmomentum

(iii)Materielle Punkte (iv)Mediumteilchen (v)Fluidteilchen (vi)Konfiguration

I13 Local thermodynamic equilibrium

In a more bottom-up approach to the modeling of a system Σ of discrete constituents as a con-tinuous medium one should first divide Σ (in thought) into small cells of fixedmdashyet not necessarilyuniversalmdashsize fulfilling two conditions

(i) each individual cell can meaningfully be treated as a thermodynamic system ie it must belarge enough that the relative fluctuation of the usual extensive thermodynamic quantitiescomputed for the content of the cell are negligible

(ii) the thermodynamic properties vary little over the cell scale ie cells cannot be too large sothat (approximate) homogeneity is ensured

The rationale behind these two requirements is illustrated by Fig I1 which represents schematicallyhow the value of a local macroscopic quantity eg a density depends on the resolution of theapparatus with which it is measured ie equivalently on the length scale on which it is defined If theapparatus probes too small a length scale so that the discrete degrees of freedom become relevantthe measured value strongly fluctuates from one observation to the next one as hinted at by thedisplayed envelope of possible results of measurements this is the issue addressed by condition (i)Simultaneously a small change in the measurement resolution even with the apparatus still centeredon the same point in the system can lead to a large variation in the measured value of the observablecorresponding to the erratic behavior of the curve at small scales shown in Fig I1 This fluctuatingpattern decreases with increasing size of the observation scale since this increase leads to a growthin the number of atoms inside the probed volume and thus a drop in the size of relative fluctuationsAt the other end of the curve one reaches a regime where the low resolution of the observationleads to encompassing domains with enough atoms to be rid of fluctuations yet with inhomogeneousmacroscopic properties in a single probed regionmdashin violation of condition (ii) As a result themeasured value of the density under consideration slowly evolves with the observation scale

In between these two domains of strong statistical fluctuations and slow macroscopic variationslies a regime where the value measured for an observable barely depends on the scale over which it is

observation scale

measuredlocalq

uantity

envelope of the setof possible values

strong variationson ldquoatomicrdquo scale

well-defined local value

macroscopic variationof the local quantity

Figure I1 ndash Typical variation of the measured value for a ldquolocalrdquo macroscopic observable asa function of the size scale over which it is determined

determined This represents the appropriate regime for meaningfully definingmdashand measuringmdashalocal density and more general local quantities

It is important to note that this intermediate ldquomesoscopicrdquo interval may not always exist Thereare physical systems in which strong macroscopic variations are already present in a range of scaleswhere microscopic fluctuations are still sizable For such systems one cannot find scale-independentlocal variables That is the proper definition of local quantities implicitly relies on the existenceof a clear separation of scales in the physical system under consideration which is what will beassumed in the remainder of these notes

Remark The smallest volume over which meaningful local quantities can be defined is sometimescalled representative volume element (vii) (RVE) or representative elementary volume

When conditions (i) and (ii) hold one may in particular define local thermodynamic variablescorresponding to the values taken in each intermediate-size cellmdashlabeled by its position ~rmdashby theusual extensive parameters internal energy number of atoms Since the separation betweencells is immaterial nothing prevents energy or matter from being transported from a cell to itsneighbors even if the global system is isolated Accordingly the local extensive variables in anygiven cell are actually time-dependent in the general case In addition it becomes important toadd linear momentummdashwith respect to some reference framemdashto the set of local extensive variablescharacterizing the content of a cell

The size of each cell is physically irrelevant as long as it satisfies the two key requirements thereis thus no meaningful local variable corresponding to volume Similarly the values of the extensivevariables in a given cell which are by definition proportional to the cell size are as arbitrary asthe latter They are thus conveniently replaced by the respective local densities internal energydensity e(t~r) number density n(t~r) linear momentum density ρ(t~r)~v(t~r) where ρ denotes themass density entropy density s(t~r)

Remark Rather than considering the densities of extensive quantities some authorsmdashin particularLandau amp Lifshitz [3 4]mdashprefer to work with specific quantities ie their respective amounts perunit mass instead of per unit volume The relation between densities and specific quantities istrivial denoting by x j resp x jm a generic local density resp specific amount for the same physicalquantity one has the identity

x j(t~r) = ρ(t~r) x jm(t~r) (I1)

in every cellmdashlabeled by ~rmdashand at every time t

Once the local extensive variables have been meaningfully defined one can develop the usualformalism of thermodynamics in each cell In particular one introduces the conjugate intensivevariables as eg local temperature T (t~r) and pressure P (t~r) The underlying important hypoth-esis is the assumption of a local thermodynamic equilibrium According to the latter the equation(s)of state of the system inside the small cell expressed with local thermodynamic quantities is thesame as for a macroscopic system in the actual thermodynamic limit of infinitely large volume andparticle number

Consider for instance a non-relativistic classical ideal gas its (mechanical) equation of statereads PV = NkBT with N the number of atoms which occupy a volume V at uniform pressure Pand temperature T while kB is the Boltzmann constant This is trivially recast as P = nkBT withn the number density of atoms The local thermodynamic equilibrium assumption then states thatunder non-uniform conditions of temperature and pressure the equation of state in a local cell atposition ~r is given by

P (t~r) = n(t~r)kBT (t~r) (I2)

at every time t

(vii)Repraumlsentatives Volumen-Element

I2 Lagrangian description 7

The last step towards the continuous-medium model is to promote ~r which till now was simplythe discrete label attached to a given cell to be a continuous variable taking its values in R3mdashor rather in the volume Vt attached to the system at the corresponding instant t Accordinglytaking into account the time-dependence of physical quantities the local variables in particular thethermodynamic parameters become fields on RtimesR3

The replacement of the fine-resolution description in which atoms are the relevant degrees offreedom by the lower-resolution model which assimilates small finite volumes of the former tostructureless points is called coarse graining(viii)

This is a quite generic procedure in theoretical physics whereby the finer degrees of freedom of amore fundamental description are smoothed awaymdashtechnically this is often done by performingaverages or integrals so that these degrees of freedom are ldquointegrated outrdquomdashand replaced bynovel effective variables in a theory with a more limited range of applicability but which ismore tractable for ldquolong-rangerdquo phenomena

Coming back to condition (ii) we already stated that it implicitly involves the existence of atleast one large length scale L over which the macroscopic physical properties of the system mayvary This scale can be a characteristic dimension of the system under consideration as eg thediameter of the tube in which a liquid is flowing In the case of periodic waves propagating inthe continuous medium L also corresponds to their wavelength More generally if G denotes amacroscopic physical quantity one may consider

L sim=

[∣∣~nablaG(t~r)∣∣

|G(t~r)|

]minus1

where ~nabla denotes the (spatial) gradientCondition (i) in particular implies that the typical size of the cells which are later coarse grained

should be significantly larger than the mean free path `mfp of atoms so that thermodynamic equi-librium holds in the local cells Since on the other hand this same typical size should be significantlysmaller than the scale L of macroscopic variations one deduces the condition

Kn equiv`mfp

L 1 (I4)

on the dimensionless Knudsen number Kn(a)

In air under normal conditions P = 105 Pa and T = 300 K the mean free path is `mfp asymp 01 micromIn the study of phenomena with variations on a characteristic scale L asymp 10 cm one finds Kn asymp 10minus6so that air can be meaningfully treated as a continuous gas

The opposite regime Kn gt 1 is that of a rarefied medium as for instance of the so-calledKnudsen gas in which the collisions between atoms are negligiblemdashand in particular insufficientto ensure thermal equilibrium as an ideal gas The flow of such systems is not well described byhydrodynamics but necessitates alternative descriptions like molecular dynamics in which thedegrees of freedom are explicitly atoms

I2 Lagrangian descriptionThe Lagrangian(b) perspective which generalizes the approach usually adopted in the description ofthe motion of a (few) point particle(s) focuses on the trajectories of the material points where thelatter are labeled by their position in the reference configuration Accordingly physical quantitiesare expressed as functions of time t and initial position vectors ~R and any continuity condition hasto be formulated with respect to these variables(viii)Vergroumlberung(a)M Knudsen 1871-1949 (b)J-L Lagrange 1736ndash1813

I21 Lagrangian coordinates

Consider a material point M in a continuous medium Given a reference frame R which allowsthe definition of its position vector at any time t one can follow its trajectory ~r(t) which afterhaving chosen a coordinate system is equivalently represented by the xi(t) for i = 1 2 3

Let ~R resp Xi denote the position resp coordinates of the material point M at t0 Thetrajectory obviously depends on this ldquoinitialrdquo position and ~r can thus be viewed as a function of tand ~R where the latter refers to the reference configuration κ0

~r = ~r(t ~R) (I5a)

with the consistency condition~r(t= t0 ~R) = ~R (I5b)

In the Lagrangian description also referred to as material description or particle descriptionthis point of view is generalized and the various physical quantities G characterizing a continuousmedium are viewed at any time as mathematical functions of the variables t and ~R

G = G(t ~R) (I6)

where the mapping Gmdashwhich as often in physics will be denoted with the same notation as thephysical quantity represented by its valuemdashis defined for every t on the initial volume V0 occupiedby the reference configuration κ0

Together with the time t the position vector ~Rmdashor equivalently its coordinates X1 X2 X3 ina given systemmdashare called Lagrangian coordinates

I22 Continuity assumptions

An important example of physical quantity function of t and ~R is simply the (vector) positionin the reference frame R of material points at time t ie ~r or equivalently its coordinates xi asgiven by relation (I5a) which thus relates the configurations κ0 and κt

More precisely ~r(t ~R) maps for every t the initial volume V0 onto Vt To implement mathe-matically the physical picture of continuity it will be assumed that the mapping ~r(t middot ) V0 rarr Vt

is also one-to-one for every tmdashie all in all bijectivemdash and that the function ~r and its inverse

~R = ~R(t~r) (I7)

are continuous with respect to both time and space variables This requirement in particularensures that neighboring points remain close to each other as time elapses It also preserves theconnectedness of volumes (closed) surfaces or curves along the evolution one may then definematerial domains ie connected sets of material points which are transported together in theevolution of the continuous medium

For the sake of simplicity it will be assumed that the mapping ~r and its inverse and moregenerally every mathematical function G representing a physical quantity is at least twice continu-ously differentiable (ie of class C 2) To be able to accommodate for important phenomena that arebetter modeled with discontinuities like shock waves in fluids (Sec V2) or ruptures in solidsmdashforinstance in the Earthrsquos crustmdash the C 2-character of functions under consideration may only holdpiecewise

I23 Velocity and acceleration of a material point

As mentioned above for a fixed reference position ~R the function t 7rarr ~r(t ~R) is the trajectory ofthe material point which passes through ~R at the reference time t0 As a consequence the velocityin the reference frame R of this same material point at time t is simply

~v(t ~R) =part~r(t ~R)

partt (I8)

I3 Eulerian description 9

Since the variable ~R is independent of t one could actually also write ~v(t ~R) = d~r(t ~R)dtIn turn the acceleration of the material point in R is given at time t by

~a(t ~R) =part~v(t ~R)

partt (I9)

Remark The trajectory (or pathline(ix)) of a material point can be visualized by tagging the pointat time t0 at its position ~R for instance with a fluorescent or radioactive marker and then imagingthe positions at later times t gt t0

If on the other hand one regularlymdashsay for every instant t0 le tprime le tmdashinjects some marker at afixed geometrical point P the resulting tagged curve at time t is the locus of the geometrical pointsoccupied by medium particles which passed through P in the past This locus is referred to asstreakline(x) Denoting by ~rP the position vector of point P the streakline is the set of geometricalpoints with position vectors

~r = ~r(t ~R(tprime~rP )

)for t0 le tprime le t (I10)

I3 Eulerian descriptionThe Lagrangian approach introduced in the previous Section is actually not commonly used in fluiddynamics at least not in its original form except for specific problems

One reason is that physical quantities at a given time are expressed in terms of a referenceconfiguration in the (far) past a small uncertainty on this initial condition may actually yieldafter a finite duration a large uncertainty on the present state of the system which is problematicOn the other hand this line of argument explains why the Lagrangian point of view is adoptedto investigate chaos in many-body systems

The more usual description is the so-called Eulerian(c) perspective in which the evolution betweeninstants t and t+ dt takes the system configuration at time t as a reference

I31 Eulerian coordinates Velocity field

In contrast to the ldquomaterialrdquo Lagrangian point of view which identifies the medium particles in areference configuration and follows them in their evolution in the Eulerian description the emphasisis placed on the geometrical points Thus the Eulerian coordinates are time t and a spatial vector~r where the latter does not label the position of a material point but rather that of a geometricalpoint Accordingly the physical quantities in the Eulerian specification are described by fields onspace-time

Thus the fundamental field that entirely determines the motion of a continuous medium in agiven reference frame R is the velocity field ~vt(t~r) The latter is defined such that it gives the valueof the Lagrangian velocity ~v [cf Eq (I8)] of a material point passing through ~r at time t

~v =~vt(t~r) forallt forall~r isin Vt (I11)

More generally the value taken at given time and position by a physical quantity G whetherattached to a material point or not is expressed as a mathematical function Gt of the same Eulerianvariables

G = Gt(t~r) forallt forall~r isin Vt (I12)

Note that the mappings (t ~R) 7rarr G(t ~R) in the Lagrangian approach and (t~r) 7rarr Gt(t~r) in theEulerian description are in general different For instance the domains in R3 over which their spatial(ix)Bahnlinie (x)Streichlinie(c)L Euler 1707ndash1783

variables take their values differ constant (V0) in the Lagrangian specification time-dependent (Vt)in the case of the Eulerian quantities Accordingly the latter will be denoted with a subscript t

I32 Equivalence between the Eulerian and Lagrangian viewpoints

Despite the different choices of variables the Lagrangian and Eulerian descriptions are fullyequivalent Accordingly the prevalence in practice of the one over the other is more a technicalissue than a conceptual one

Thus it is rather clear that the knowledge of the Lagrangian specification can be used to obtainthe Eulerian formulation at once using the mapping ~r 7rarr ~R(t~r) between present and referencepositions of a material point Thus the Eulerian velocity field can be expressed as

~vt(t~r) = ~v(t ~R(t~r)

) (I13a)

This identity in particular shows that ~vt automatically inherits the smoothness properties of ~v ifthe mapping (t ~R) 7rarr ~r(t ~R) and its inverse are piecewise C 2 (cf Sec I22) then ~vt is (at least)piecewise C 1 in both its variables

For a generic physical quantity the transition from the Lagrangian to the Eulerian point of viewsimilarly reads

Gt(t~r) = G(t ~R(t~r)

) (I13b)

Reciprocally given a (well-enough behaved) Eulerian velocity field~vt on a continuous mediumone can uniquely obtain the Lagrangian description of the medium motion by solving the initialvalue problem

part~r(t ~R)

partt=~vt

(t~r(t ~R)

)~r(t0 ~R) = ~R

(I14a)

where the second line represents the initial condition That is one actually reconstructs the pathlineof every material point of the continuous medium Introducing differential notations the abovesystem can also be rewritten as

d~r =~vt(t~r) dt with ~r(t0 ~R) = ~R (I14b)

Once the pathlines ~r(t ~R) are known one obtains the Lagrangian function G(t ~R) for a givenphysical quantity G by writing

G(t ~R) = Gt(t~r(t ~R)

) (I14c)

Since both Lagrangian and Eulerian descriptions are equivalent we shall from now on drop thesubscript t on the mathematical functions representing physical quantities in the Eulerian point ofview

I33 Streamlines

At a given time t the streamlines(xi) of the motion are defined as the field lines of ~vt That isthese are curves whose tangent is everywhere parallel to the instantaneous velocity field at the samegeometrical point

Let ~x(λ) denote a streamline parameterized by λ The definition can be formulated as

d~x(λ)

dλ= α(λ)~v

(t ~x(λ)

)(I15a)

with α(λ) a scalar function Equivalently denoting by d~x(λ) a differential line element tangent to

(xi)Stromlinien

the streamline one has the condition

d~xtimes~v(t ~x(λ)

)= ~0 (I15b)

Eventually introducing a Cartesian system of coordinates the equation for a streamline isconveniently rewritten as

dx1(λ)

v1(t ~x(λ)

) =dx2(λ)

v2(t ~x(λ)

) =dx3(λ)

v3(t ~x(λ)

) (I15c)

in a point where none of the component vi of the velocity field vanishesmdashif one of the vi is zerothen so is the corresponding dxi thanks to Eq (I15b)

Remark Since the velocity field ~v depends on the choice of reference frame this is also the case ofits streamlines at a given instant

Consider now a closed geometrical curve in the volume Vt occupied by the continuous mediumat time t The streamlines tangent to this curve form in the generic case a tube-like surface calledstream tube(xii)

Let us introduce two further definitions related to properties of the velocity field

bull If ~v(t~r) has at some t the same value in every geometrical point ~r of a (connected) domainD sub Vt then the velocity field is said to be uniform across DIn that case the streamlines are parallel to each other over D

bull If~v(t~r) only depends on the position not on time then the velocity field and the correspond-ing motion of the continuous medium are said to be steady or equivalently stationary In that case the streamlines coincide with the pathlines and the streaklines

Indeed one checks that Eq (I14b) for the pathlines in which the velocity becomes time-independent can then be recast (in a point where all vi are non-zero) as

v1(t~r)=

v2(t~r)=

v3(t~r)

where the variable t plays no role this is exactly the system (I15c) defining the streamlinesat time t The equivalence between pathlines and streaklines is also trivial

I34 Material derivative

Consider a material point M in a continuous medium described in a reference frame R Let ~rresp ~r + d~r denote its position vectors at successive instants t resp t + dt The velocity of M attime t resp t + dt is by definition equal to the value of the velocity field at that time and at therespective position namely~v(t~r) resp~v(t+ dt~r+ d~r) For small enough dt the displacement d~rof the material point between t and t+dt is simply related to its velocity at time t by d~r =~v(t~r) dt

Let d~v equiv~v(t + dt~r + d~r) minus~v(t~r) denote the change in the material point velocity Assumingthat ~v(t~r) is differentiable (cf Sec I32) and introducing for simplicity a system of Cartesiancoordinates a Taylor expansion to lowest order yields

d~v part~v(t~r)

parttdt+

part~v(t~r)

partx1dx1 +

part~v(t~r)

partx2dx2 +

part~v(t~r)

partx3dx3

up to terms of higher order in dt or d~r Introducing the differential operator

d~r middot ~nabla = dx1 part

partx1+ dx2 part

partx2+ dx3 part

partx3

(xii)Stromroumlhre

this can be recast in the more compact form

d~v part~v(t~r)

parttdt+

(d~r middot ~nabla

)~v(t~r) (I16)

In the second term on the right-hand side d~r can be replaced by~v(t~r) dt On the other handthe change in velocity of the material point between t and t + dt is simply the product of itsacceleration ~a(t) at time t by the size dt of the time interval at least to lowest order in dt Dividingboth sides of Eq (I16) by dt and taking the limit dtrarr 0 in particular in the ratio d~vdt yield

~a(t) =part~v(t~r)

partt+[~v(t~r) middot ~nabla

]~v(t~r) (I17)

That is the acceleration of the material point consists of two terms

bull the local accelerationpart~v

partt which follows from the non-stationarity of the velocity field

bull the convective acceleration(~v middot ~nabla

)~v due to the non-uniformity of the motion

More generally one finds by repeating the same derivation that the time derivative of a physicalquantity G attached to a material point or domain yet expressed in terms of Eulerian fields is thesum of a local (partGpartt) and a convective [(~v middot ~nabla)G ] part irrespective of the tensorial nature of G Accordingly one introduces the operator

Dtequiv part

partt+~v(t~r) middot ~nabla (I18)

called material derivative(xiii) or (between others) substantial derivative(xiv) derivative following themotion hydrodynamic derivative Relation (I17) can thus be recast as

~a(t) =D~v(t~r)

Dt (I19)

Remarks

lowast Equation (I17) shows that even in the case of a steady motion the acceleration of a materialpoint may be non-vanishing thanks to the convective part

lowast The material derivative (I18) is also often denoted (and referred to) as total derivative ddt

lowast One also finds in the literature the denomination convective derivative(xv) To the eyes and earsof the author of these lines that name has the drawback that it does not naturally evoke the localpart but only the convective one which comes from the fact that matter is being transportedldquoconveyedrdquo with a non-vanishing velocity field~v(t~r)

lowast The two terms in Eq (I18) actually ldquomergerdquo together when considering the motion of a materialpoint in Galilean space-time RtimesR3 As a matter of fact one easily shows that DDt is the (Lie)derivative along the world-line of the material point

The world-line element corresponding to the motion between t and t+dt goes from (t x1 x2 x3) to(t+dt x1 +v1 dt x2 +v2 dt x3 +v3 dt) The tangent vector to this world-line thus has components(1 v1 v2 v3) ie the derivative along the direction of this vector is partt + v1part1 + v2part2 + v3part3 withthe usual shorthand notation parti equiv partpartxi

(xiii)Materielle Ableitung (xiv)Substantielle Ableitung (xv)Konvektive Ableitung

I4 Mechanical stress 13

I4 Mechanical stress

I41 Forces in a continuous medium

Consider a closed material domain V inside the volume Vt occupied by a continuous mediumand let S denote the (geometric) surface enclosing V One distinguishes between two classes offorces acting on this domain

bull Volume or body forces(xvi) which act in each point of the bulk volume of VExamples are weight long-range electromagnetic forces or in non-inertial reference framesfictitious forces (Coriolis centrifugal)For such forces which tend to be proportional to the volume they act on it will later be moreconvenient to introduce the corresponding volumic force density

bull Surface or contact forces(xvii) which act on the surface S like friction which we now discussin further detail

d2S ~en

d2 ~Fs

Figure I2

Consider an infinitesimally small geometrical surface element d2S at point P Let d2 ~Fs denotethe surface force through d2S That is d2 ~Fs is the contact force due to the medium exterior to Vthat a ldquotestrdquo material surface coinciding with d2S would experience The vector

~Ts equivd2 ~Fsd2S

representing the surface density of contact forces is called (mechanical) stress vector (xviii) on d2SThe corresponding unit in the SI system is the Pascal with 1 Pa = 1Nmiddotmminus2

Purely geometrically the stress vector ~Ts on a given surface element d2S at a given point canbe decomposed into two components namely

bull a vector orthogonal to plane tangent in P to d2S the so-called normal stress(xix) when itis directed towards the interior resp exterior of the medium domain being acted on it alsoreferred to as compression(xx) resp tension(xxi)

bull a vector in the tangent plane in P called shear stress(xxii) and often denoted as ~τ

Despite the short notation adopted in Eq (I20) the stress vector depends not only on theposition of the geometrical point P where the infinitesimal surface element d2S lies but also on the(xvi)Volumenkraumlfte (xvii)Oberflaumlchenkraumlfte (xviii)Mechanischer Spannungsvektor (xix)Normalspannung(xx)Druckspannung (xxi)Zugsspannung (xxii)Scher- Tangential- oder Schubspannung

orientation of the surface Let ~en denote the normal unit vector to the surface element directedtowards the exterior of the volume V (cf Fig I2) and let ~r denote the position vector of P in agiven reference frame The relation between ~en and the stress vector ~Ts on d2S is then linear

~Ts = σσσ(~r) middot~en (I21a)

with σσσ(~r) a symmetric tensor of rank 2 the so-called (Cauchy(d)) stress tensor (xxiii)

In a given coordinate system relation (I21a) yields

T is =

3sumj=1

σσσij ejn (I21b)

with T is resp ejn the coordinates of the vectors ~Ts resp ~en and σσσij the(

)-components of the stress

tensor

While valid in the case of a three-dimensional position space equation (I21a) should actuallybe better formulated to become valid in arbitrary dimension Thus the unit-length ldquonormalvectorrdquo to a surface element at point P is rather a 1-form acting on the vectors of the tangentspace to the surface at P As such it should be represented as the transposed of a vector [(~en)T]which multiplies the stress tensor from the left

~Ts = (~en)T middotσσσ(~r) (I21c)

This shows that the Cauchy stress tensor is a(

)-tensor (a ldquobivectorrdquo) which maps 1-forms onto

vectors In terms of coordinates this gives using Einsteinrsquos summation convention

T js = eniσσσij (I21d)

which thanks to the symmetry of σσσ is equivalent to the relation given above

Remark The symmetry property of the Cauchy stress tensor is intimately linked to the assumptionthat the material points constituting the continuous medium have no intrinsic angular momentum

I42 Fluids

With the help of the notion of mechanical stress we may now introduce the definition of a fluid which is the class of continuous media whose motion is described by hydrodynamics

A fluid is a continuous medium that deforms itself as long as it is submitted to shear stresses

(I22)Turning this definition around one sees that in a fluid at restmdashor to be more accurate studied

in a reference frame with respect to which it is at restmdashthe mechanical stresses are necessarilynormal That is the stress tensor is in each point diagonal

More precisely for a locally isotropic fluidmdashwhich means that the material points are isotropicwhich is the case throughout these notesmdashthe stress

)-tensor is everywhere proportional to the

inverse metric tensor

σσσ(t~r) = minusP (t~r)gminus1(t~r) (I23)

with P (t~r) the hydrostatic pressure at position ~r at time t

Going back to relation (I21b) the stress vector will be parallel to the ldquounit normal vectorrdquo inany coordinate system if the square matrix of the

)-components σσσij is proportional to the

(xxiii)(Cauchyrsquoscher) Spannungstensor

(d)AL Cauchy 1789ndash1857

identity matrix ie σσσij prop δij where we have introduced the Kronecker symbol To obtain the(20

)-components σσσik one has to multiply σσσij by the component gjk of the inverse metric tensor

summing over k which precisely gives Eq (I23)

Remarks

lowast Definition (I22) as well as the two remarks hereafter rely on an intuitive picture of ldquodeforma-tionsrdquo in a continuous medium To support this picture with some mathematical background weshall introduce in Sec IIA an appropriate strain tensor which quantifies these deformations atleast as long as they remain small

lowast A deformable solid will also deform itself when submitted to shear stress However for a givenfixed amount of tangential stress the solid will after some time reach a new deformed equilibriumpositionmdashotherwise it is not a solid but a fluid

lowast The previous remark is actually a simplification valid on the typical time scale of human beings Thusmaterials which in our everyday experience are solidsmdashas for instance those forming the mantle of the Earthmdashwill behave on a longer time scale as fluidsmdashin the previous example on geological time scales Whethera given substance behaves as a fluid or a deformable solid is sometimes characterized by the dimensionlessDeborah number [7] which compares the typical time scale for the response of the substance to a mechanicalstress and the observation time

lowast Even nicer the fluid vs deformable solid behavior may actually depend on the intensity of theapplied shear stress ketchup

Bibliography for Chapter Ibull National Committee for Fluid Mechanics films amp film notes on Eulerian Lagrangian description

and on Flow visualization(1)

bull Faber [1] Chapter 11ndash13

bull Feynman [8 9] Chapter 31ndash6

bull Guyon et al [2] Chapter 11

bull Sedov [10] Chapters 1 amp 21ndash22

bull Sommerfeld [5 6] beginning of Chapter II5

(1)The visualization techniques have probably evolved since the 1960s yet pathlines streaklines or streamlines arestill defined in the same way

CHAPTER II

Kinematics of a continuous medium

The goal of fluid dynamics is to investigate the motion of fluids under consideration of the forcesat play as well as to study the mechanical stresses exerted by moving fluids on bodies with whichthey are in contact The description of the motion itself irrespective of the forces is the object ofkinematics

The possibilities for the motion of a deformable continuous medium in particular of a fluid arericher than for a mere point particle or a rigid body besides translations and global rotations adeformable medium may also rotate locally and undergo deformations The latter term actuallyencompasses two different yet non-exclusive possibilities namely either a change of shape or avariation of the volume All these various types of motion are encoded in the local properties ofthe velocity field at each instant (Sec II1) Generic fluid motions are then classified according toseveral criteria especially taking into account kinematics (Sec II2)

For the sake of reference the characterization of deformations themselves complementing thatof their rate of change is briefly presented in Sec IIA That formalism is not needed within fluiddynamics but rather for the study of deformable solids like elastic ones

II1 Generic motion of a continuous mediumLet ~v denote the velocity field in a continuous medium with respect to some reference frame RTo illustrate (some of) the possible motions that occur in a deformable body Fig II1 shows thepositions at successive instants t and t+δt of a small ldquomaterial vectorrdquo δ~(t) that is a continuous set

δ~(t)

δ~(t+ δt)

~v(t~r + δ~(t)

~v(t~r)

Figure II1 ndash Positions of a material line element δ~ at successive times t and t+ δt

II1 Generic motion of a continuous medium 17

of material points distributed along the (straight) line element stretching between two neighboringgeometrical points positions Let ~r and ~r+ δ~(t) denote the geometrical endpoints of this materialvector at time t

Thanks to the continuity of the mappings ~R 7rarr ~r(t~r) and its inverse ~r 7rarr ~R(t~r) the materialvector defined at instant t remains a connected set of material points as time evolves in particularat t + δt Assuming that both the initial length |δ~(t)| as well as δt are small enough the evolvedset at t + δt remains approximately along a straight line and constitutes a new material vectordenoted by δ~(t+dt) The position vectors of these endpoints simply follow from the initial positionsof the corresponding material points ~r resp ~r + δ~(t) to which should be added the respectivedisplacement vectors between t and t+δt namely the product by δt of the initial velocity ~v

resp~v(t~r + δ~(t)

) That is one finds

δ~(t+ δt) = δ~(t) +[~v(t~r + δ~(t)

)minus~v(t~r)]δt+O

(δt2) (II1)

Figure II1 already suggests that the motion of the material vector consists not only of a translationbut also of a rotation as well as an ldquoexpansionrdquomdashthe change in length of the vector

II11 Local distribution of velocities in a continuous medium

Considering first a fixed time t let~v(t~r) resp~v(t~r)+ δ~v be the velocity at the geometric pointsituated at position ~r resp at ~r + δ~r in R

Introducing for simplicity a system of Cartesian coordinates in R the Taylor expansion of thei-th component of the velocity fieldmdashwhich is at least piecewise C 1 in its variables see Sec I32mdashgives to first order

δvi 3sumj=1

partvi(t~r)

partxjδxj (II2a)

Introducing the(

)-tensor ~nabla~v~nabla~v~nabla~v(t~r) whose components in the coordinate system used here are the

partial derivatives partvi(t~r)partxj the above relation can be recast in the coordinate-independentform

δ~v ~nabla~v~nabla~v~nabla~v(t~r) middot δ~r (II2b)

Like every rank 2 tensor the velocity gradient tensor ~nabla~v~nabla~v~nabla~v(t~r) at time t and position ~r can bedecomposed into the sum of the symmetric and an antisymmetric part

~nabla~v~nabla~v~nabla~v(t~r) = DDD(t~r) +RRR(t~r) (II3a)

where one conventionally writes

DDD(t~r) equiv 1

(~nabla~v~nabla~v~nabla~v(t~r) +

[~nabla~v~nabla~v~nabla~v(t~r)

]T) RRR(t~r) equiv 1

(~nabla~v~nabla~v~nabla~v(t~r)minus

]T) (II3b)

with[~nabla~v~nabla~v~nabla~v(t~r)

]T the transposed tensor to ~nabla~v~nabla~v~nabla~v(t~r) These definitions are to be understood as followsUsing the same Cartesian coordinate system as above the components of the two tensors DDD RRRviewed for simplicity as

)-tensors respectively read

DDDij(t~r) =1

[partvi(t~r)

partxj+partvj(t~r)

partxi

] RRRij(t~r) =

[partvi(t~r)

partxjminus partvj(t~r)

partxi

] (II3c)

Note that here we have silently used the fact that for Cartesian coordinates the positionmdashsubscriptor exponentmdashof the index does not change the value of the component ie numerically vi = vi forevery i isin 1 2 3

Relations (II3c) clearly represent the desired symmetric and antisymmetric parts Howeverone sees that the definitions would not appear to fulfill their task if both indices were not both

18 Kinematics of a continuous medium

either up or down as eg

DDDij(t~r) =

[partvi(t~r)

partxj+partvj(t~r)

partxi

]in which the symmetry is no longer obvious The trick is to rewrite the previous identity as

DDDij(t~r) =

2δikδlj

[partvk(t~r)

partxl+partvl(t~r)

partxk

2gik(t~r)glj(t~r)

[partvk(t~r)

partxl+partvl(t~r)

partxk

where we have used the fact that the metric tensor of Cartesian coordinates coincides withthe Kronecker symbol To fully generalize to curvilinear coordinates the partial derivatives inthe rightmost term should be replaced by the covariant derivatives discussed in Appendix C1leading eventually to

DDDij(t~r) =

2gik(t~r)glj(t~r)

[dvk(t~r)

dvl(t~r)

](II4a)

RRRij(t~r) =1

2gik(t~r)glj(t~r)

[dvk(t~r)

dxlminus dvl(t~r)

](II4b)

With these new forms which are valid in any coordinate system the raising or lowering ofindices does not affect the visual symmetric or antisymmetric aspect of the tensor

Using the tensors DDD and RRR we just introduced whose physical meaning will be discussed atlength in Secs II12ndashII13 relation (II2b) can be recast as

~v(t~r + δ~r

)=~v(t~r)

+DDD(t~r) middot δ~r +RRR(t~r) middot δ~r +O(|δ~r|2

)(II5)

where everything is at constant time

Under consideration of relation (II5) with δ~r = δ~(t) Eq (II1) for the time evolution of thematerial line element becomes

δ~(t+ δt) = δ~(t) +[DDD(t~r) middot δ~(t) +RRR(t~r) middot δ~(t)

]δt+O

(δt2)

Subtracting δ~(t) from both sides dividing by δt and taking the limit δtrarr 0 one finds for the rateof change of the material vector which is here denoted by a dot

)middot

(t) = DDD(t~r) middot δ~(t) +RRR(t~r) middot δ~(t) (II6)

In the following two subsections we shall investigate the physical content of each of the tensorsRRR(t~r) and DDD(t~r)

II12 Rotation rate tensor and vorticity vector

The tensor RRR(t~r) defined by Eq (II3b) is called for reasons that will become clearer belowrotation rate tensor (xxiv)

By construction this tensor is antisymmetric Accordingly one can naturally associate with ita dual (pseudo)-vector ~Ω(t~r) such that for any vector ~V

RRR(t~r)middot ~V = ~Ω

(t~r)times ~V forall~V isin R3

In Cartesian coordinates the components of ~Ω(t~r) are related to those of the rotation rate tensorby

Ωi(t~r) equiv minus1

3sumjk=1

εijkRRRjk(t~r) (II7a)

(xxiv)Wirbeltensor

with εijk the totally antisymmetric Levi-Civita symbol Using the antisymmetry of RRR(t~r) this

equivalently reads

Ω1(t~r) equiv minusRRR23(t~r) Ω2(t~r) equiv minusRRR31(t~r) Ω3(t~r) equiv minusRRR12(t~r) (II7b)

Comparing with Eq (II3c) one finds

~Ω(t~r) =1

2~nablatimes~v(t~r) (II8)

Let us now rewrite relation (II6) with the help of the vector ~Ω(t~r) assuming that DDD(t~r)vanishes so as to isolate the effect of the remaining term Under this assumption the rate of changeof the material vector between two neighboring points reads

)middot

(t) = RRR(t~r) middot δ~(t) = ~Ω(t~r)times δ~(t) (II9)

The term on the right hand side is then exactly the rate of rotation of a vector ~(t) in the motionof a rigid body with instantaneous angular velocity ~Ω(t~r) Accordingly the pseudovector ~Ω(t~r) isreferred to as local angular velocity (xxv) This a posteriori justifies the denomination rotation ratetensor for the antisymmetric tensor RRR(t~r)

Remarkslowast Besides the local angular velocity ~Ω(t~r) one also defines the vorticity vector (xxvi) as the curl ofthe velocity field

~ω(t~r) equiv 2~Ω(t~r) = ~nablatimes~v(t~r) (II10)

In fluid mechanics the vorticity is actually more used than the local angular velocity

lowast The local angular velocity ~Ω(t~r) or equivalently the vorticity vector ~ω(t~r) define at fixed tdivergence-free (pseudo)vector fields since obviously ~nabla middot (~nablatimes~v) = 0 The corresponding field linesare called vorticity lines(xxvii) and are given by [cf Eq (I15)]

d~xtimes ~ω(t~r) = ~0 (II11a)

or equivalently in a point where none of the components of the vorticity vector vanishes

ω1(t~r)=

ω2(t~r)=

ω3(t~r) (II11b)

II13 Strain rate tensor

According to the previous subsection the local rotational motion of a material vector is governedby the (local and instantaneous) rotation rate tensor RRR(t~r) In turn the translational motion issimply the displacementmdashwhich must be described in an affine space not a vector onemdashof one ofthe endpoints of δ~ by an amount given by the product of velocity and length of time interval Thatis both components of the motion of a rigid body are already accounted for without invoking thesymmetric tensor DDD(t~r)

In other words the tensor DDD(t~r) characterizes the local deviation between the velocity fields ina deformable body in particular a fluid and in a rigid body rotating with angular velocity ~Ω(t~r)Accordingly it is called strain rate tensor or deformation rate tensor (xxviii)

As we shall now see the diagonal and off-diagonal components of DDD(t~r) actually describethe rates of change of different kinds of deformation For simplicity we assume throughout thissubsection that ~Ω(t~r) = ~0(xxv)Wirbelvektor (xxvi)Wirbligkeit (xxvii)Wirbellinien (xxviii)VerzerrungsgeschwindigkeitstensorDeformationsgeschwindigkeitstensor

II13 a

Diagonal components

We first assume that all off-diagonal terms in the strain rate tensor vanish DDDij(t~r) = 0 fori 6= j so as to see the meaning of the diagonal components

Going back to Eq (II1) let us simply project it along one of the axes of the coordinate systemsay along direction i Denoting the i-th component of δ~ as δ`i one thus finds

δ`i(t+ δt) = δ`i(t) +[vi(t~r + δ~(t)

)minus vi

(t~r)]δt+O

(δt2)

Taylor-expanding the term between square brackets to first order then yields

δ`i(t+ δt)minus δ`i(t) 3sumj=1

partvi(t~r)

partxjδ`j(t) δt

up to terms of higher order in |δ~(t)| or δt Since we have assumed that both ~Ω(t~r)mdashor equivalentlythe componentsRRRij(t~r) of the rotation rate tensormdashand the off-diagonalDDDij(t~r) with i 6= j vanishone checks that the partial derivative partvi(t~r)partxj vanishes for i 6= j That is there is only theterm j = i in the sum so that the equation simplifies to

δ`i(t+ δt)minus δ`i(t) partvi(t~r)

partxiδ`i(t) δt = DDDi

i(t~r) δ`i(t) δt

Thus the relative elongation of the i-th componentmdashremember that there is no local rotation sothat the change in δ`i is entirely due to a variation of the length of the material vectormdashin δt isgiven by

δì(t+ δt)minus δì(t)δì(t)

= DDDii(t~r) δt (II12)

This means that the diagonal component DDDii(t~r) represents the local rate of linear elongation in

direction i

Volume expansion rateInstead of considering a one-dimensional material vector one can study the evolution of a small

ldquomaterial parallelepipedrdquo of the continuous medium situated at t at position ~r with instantaneousside lengths δL1(t) δL2(t) δL3(t)mdashfor simplicity the coordinate axes are taken along the sides ofthe parallelepiped Accordingly its volume at time t is simply δV(t) = δL1(t) δL2(t) δL3(t)

Taking into account Eq (II12) for the relative elongation of each side length one finds that therelative change in volume between t and t+ δt is

δV(t+δt)minus δV(t)

δV(t)=δL1(t+δt)minus δL1(t)

δL1(t)+δL2(t+δt)minus δL2(t)

δL3(t)

=[DDD1

1(t~r) +DDD22(t~r) +DDD3

3(t~r)]δt

In the second line one recognizes the trace of the tensor DDD(t~r) which going back to the definitionof the latter is equal to the divergence of the velocity fluid

DDD11(t~r) +DDD2

2(t~r) +DDD33(t~r) =

partv1(t~r)

partx1+partv2(t~r)

partx2+partv3(t~r)

partx3= ~nabla middot ~v(t~r)

That is this divergence represents the local and instantaneous volume expansion rate of the conti-nuous medium Accordingly the flow of a fluid is referred to as incompressible in some region whenthe velocity field in that region is divergence free

incompressible flow hArr ~nabla middot ~v(t~r) = 0 (II13)

II13 b

Off-diagonal components

Let us now assume thatDDD12(t~r) and thereby automaticallyDDD21(t~r) is the only non-vanishingcomponent of the strain rate tensor To see the influence of that component we need to consider

v1 δt

(v1+δv1)δt

v2 δt

(v2+δv2)δtδα1

Figure II2 ndash Evolution of a material rectangle caught in the motion of a continuous mediumbetween times t (left) and t+ δt (right)

the time evolution of a different object than a material vector since anything that can affect thelattermdashtranslation rotation dilatationmdashhas already been described above

Accordingly we now look at the change between successive instants t and t+δt of an elementaryldquomaterial rectanglerdquo as pictured in Fig II2 We denote by ~v resp ~v + δ~v the velocity at time t atthe lower left resp upper right corner of the rectangle Taylor expansions give for the componentsof the shift δ~v

δv1 =partv1(t~r)

partx2δ`2 δv2 =

partv2(t~r)

partx1d`1

Figure II2 shows that what was at time t a right angle becomes an angle π2minusδα at t+dt whereδα = δα1 + δα2 In the limit of small δt both δα1 and δα2 will be small and thus approximatelyequal to their respective tangents Using the fact that the parallelogram still has the same areamdashsince the diagonal components of DDD vanishmdashthe projection of any side of the deformed rectangle attime t+ δt on its original direction at time t keeps approximately the same length up to correctionsof order δt One thus finds for the oriented angles

δα1 δv2 δt

δ`1and δα2 minus

δv1 δt

With the Taylor expansions given above this leads to

δα1 partv2(t~r)

partx1δt δα2 minus

partv1(t~r)

partx2δt

Gathering all pieces one finds

δt partv2(t~r)

partx1minus partv1(t~r)

partx2= 2DDD21(t~r) (II14)

Going to the limit δt rarr 0 one sees that the off-diagonal component DDD21(t~r) represents half thelocal velocity of the ldquoangular deformationrdquomdashthe shearmdasharound direction x3

Remark To separate the two physical effects present in the strain rate tensor it is sometimeswritten as the sum of a diagonal rate-of-expansion tensor proportional to the identity 111mdashwhich isin fact the

)-form of the metric tensor g of Cartesian coordinatesmdashand a traceless rate-of-shear

tensor SSSDDD(t~r) =

[~nabla middot ~v(t~r)

]111+SSS(t~r) (II15a)

withSSS(t~r) equiv 1

]T minus 2

) (II15b)

Component-wise and generalizing to curvilinear coordinates this reads

DDDij(t~r) =1

]gij(t~r) +SSSij(t~r) (II15c)

with [cf Eq (II4a)]

SSSij(t~r) equiv1

[gki(t~r)g

lj(t~r)

(dvk(t~r)

dvl(t~r)

)minus 2

]gij(t~r)

] (II15d)

Summary

Gathering the findings of this Section the most general motion of a material volume elementinside a continuous medium in particular in a fluid can be decomposed in four elements

bull a translation

bull a rotation with a local angular velocity ~Ω(t~r) given by Eq (II8)mdashie related to the anti-symmetric part RRR(t~r) of the velocity gradientmdashand equal to twice the (local) vorticity vector~ω(t~r)

bull a local dilatation or contraction in which the geometric form of the material volume elementremains unchanged whose rate is given by the divergence of the velocity field ~nabla middot ~v(t~r) ieencoded in the diagonal elements of the strain rate tensor DDD(t~r)

bull a change of shape (ldquodeformationrdquo) of the material volume element at constant volume con-trolled by the rate-of-shear tensor SSS(t~r) [Eqs (II15b)(II15d)] obtained by taking the trace-less symmetric part of the velocity gradient

II2 Classification of fluid flowsThe motion or flow (xxix) of a fluid can be characterized according to several criteria either purelygeometrical (Sec II21) kinematic (Sec II22) or of a more physical nature (Sec II23) ie takinginto account the physical properties of the flowing fluid

II21 Geometrical criteria

In the general case the quantities characterizing the properties of a fluid flow will depend ontime as well as on three spatial coordinates

For some more or less idealized models of actual flows it may turn out that only two spatialcoordinates play a role in which case one talks of a two-dimensional flow An example is the flow ofair around the wing of an airplane which in first approximation is ldquoinfinitelyrdquo long compared to itstransverse profile the (important) effects at the ends of the wing which introduce the dependenceon the spatial dimension along the wing may be left aside in a first approach then considered in asecond more detailed step

In some cases eg for fluid flows in pipes one may even assume that the properties only dependon a single spatial coordinate so that the flow is one-dimensional In that approximation thephysical local quantities are actually often replaced by their average value over the cross section ofthe pipe

On a different level one also distinguishes between internal und external fluid flows accordingto whether the fluid is enclosed inside solid wallsmdasheg in a pipemdashor flowing around a bodymdashegaround an airplane wing(xxix)Stroumlmung

II2 Classification of fluid flows 23

II22 Kinematic criteria

The notions of uniformmdashthat is independent of positionmdashand steadymdashindependent of timemdashmotions were already introduced at the end of Sec I33 Accordingly there are non-uniform andunsteady fluids flows

If the vorticity vector ~ω(t~r) vanishes at every point ~r of a flowing fluid then the correspondingmotion is referred to as an irrotational flow (xxx) or for reasons that will be clarified in Sec IV4potential flow The opposite case is that of a vortical or rotational flow (xxxi)

According to whether the flow velocity v is smaller or larger than the (local) speed of soundcs one talks of subsonic or supersonic motion(xxxii) corresponding respectively to a dimensionlessMach number (e)

Ma equiv v

cs(II16)

smaller or larger than 1 Note that the Mach number can a priori be defined and take differentvalues Ma(t~r) at every point in a flow

When the fluid flows in layers that do not mix with each other so that the streamlines remainparallel the flow is referred to as laminar In the opposite case the flow is turbulent

II23 Physical criteria

All fluids are compressible more or less according to the substance and its thermodynamicstate Nevertheless this compressibility is sometimes irrelevant for a given motion in which case itmay fruitful to consider that the fluid flow is incompressible which as seen in sect II13 a technicallymeans that its volume expansion rate vanishes ~nabla middot~v = 0 In the opposite case (~nabla middot~v 6= 0) the flowis said to be compressible It is however important to realize that the statement is more a kinematicone than really reflecting the thermodynamic compressibility of the fluid

In practice flows are compressible in regions where the fluid velocity is ldquolargerdquo namely wherethe Mach number (II16) is not much smaller than 1 ie roughly speaking Ma amp 02

In an analogous manner one speaks of viscous resp non-viscous flows to express the fact that thefluid under consideration is modeled as viscous resp inviscidmdashwhich leads to different equations ofmotionmdash irrespective of the fact that every fluid has a non-zero viscosity

Other thermodynamic criteria are also used to characterize possible fluid motions isothermalflowsmdashie in which the temperature is uniform and remains constantmdash isentropic flowsmdashie with-out production of entropymdash and so on

Bibliography for Chapter IIbull National Committee for Fluid Mechanics film amp film notes on Deformation of ContinuousMedia

bull Faber [1] Chapter 24

bull Guyon et al [2] Chapters 31 32

bull Sommerfeld [5 6] Chapter I

(xxx)wirbelfreie Stroumlmung (xxxi)Wirbelstroumlmung (xxxii)Unterschall- bzw Uumlberschallstroumlmung(e)E Mach 1838ndash1916

Appendix to Chapter II

IIA Deformations in a continuous mediumStrain tensor

CHAPTER III

Fundamental equations ofnon-relativistic fluid dynamics

Some of the most fundamental laws of physics are conservation equations for various quantitiesenergy momentum (electric) charge and so on When applying them to many-body systemsin particular to continuous media like moving fluids care must be taken to consider isolated andclosed systems to ensure their validity At the very least the amount of quantity exchanged withthe exterior of the systemmdashfor example the change in momentum per unit time due to externalforces as given by Newtonrsquos second law or the change in energy due to the mechanical work ofthese forcesmdashmust be quantifiable

When this is the case it is possible to re-express global conservation laws or more generallybalance equationsmdashgiven in terms of macroscopic quantities like total mass total energy totalmomentum etcmdashin a local form involving densities using the generic recipe provided by Reynoldsrsquotransport theorem (Sec III1) In the framework of a non-relativistic theory in which the massor equivalently the particle number of a closed system is conserved one may thus derive a generalcontinuity equation holding at every point of the continuous medium (Sec III2)

The same approach may be followed to derive equations expressing the time evolution of mo-mentum or energy under the influence of external forces acting at every point of the fluid In eithercase it is however necessary to account for the possibility that several physical phenomena maycontribute to the transport of momentum and energy Depending on whether or not and how everyform of transport is taken into account one has different fluid models leading to different equationsfor the local expressions of Newtonrsquos second law (Sec III3) or of energy balance (Sec III4)

III1 Reynolds transport theoremThe material derivative of a quantity was already introduced in Sec I34 in which its action ona local function of both time t and position vector ~r was given In this Section we shall derive aformula for the substantial derivative of an extensive physical quantity attached to a ldquomacroscopicrdquomaterial system This formula will in the remainder of the Chapter represent the key relation whichwill allow us to express the usual conservation laws of mechanics which hold for closed systems interms of Eulerian variables

III11 Closed system open system

Consider the motion of a continuous medium in particular a flowing fluid in a reference frameR Let S be an arbitrary closed geometrical surface which remains fixed in R This surface willhereafter be referred to as control surface and the geometrical volume V it encloses as controlvolume

Due to the macroscopic transport of matter in the flowing medium the fluid contained insidethe control surface S represents an open system which can exchange matter with its exterior astime elapses Let Σ be the closed system consisting of the material points that occupy the controlvolume V at some given time t At a shortly later time t+ δt the material system Σ has moved and

26 Fundamental equations of non-relativistic fluid dynamics

2+-boundary ofΣ at time t

boundary of Σ

at time t+ δt

streamlines

Figure III1 ndash Time evolution of a closed material system transported in the motion of acontinuous medium

it occupies a new volume in the reference frame On Fig III1 one can distinguish between threeregions in position space

bull (1) which is common to the successive positions of Σ at t ant t+ δt

bull (2minus) which was left behind by Σ between t and t+ δt

bull (2+) into which Σ penetrates between t and t+ δt

III12 Material derivative of an extensive quantity

Let G(t) be one of the extensive quantities that characterize a macroscopic physical propertyof the closed material system Σ To this extensive quantity one can associate at every point ~r thecorresponding intensive specific density g

m(t~r) defined as the local amount of G per unit mass of

matter Denoting by dG(t~r) resp dM(t~r) the amount of G resp the mass inside a small materialvolume at position ~r at time t one can write symbolically

(t~r) =dG(t~r)

dM(t~r) (III1)

where the notation with differentials was used to suggest that the identity holds in the limit of asmall material volume

For instance the linear momentum resp the kinetic energy of a mass dM of fluid moving withvelocity~v is d~P =~v dM resp dK =~v2 dM2 so that the associated specific density is d~PdM =~vresp dKdM =~v22

Remark These examples illustrate the fact that the tensorial naturemdashscalar vector tensor of higherrankmdashof the function associated with quantity G can be arbitrary

For a material system Σ occupying at time t a volume V bounded by the control surface SEq III1 leads to

G(t) =

(t~r) dM =

(t~r) ρ(t~r) d3~r (III2)

for the value of G of the system where in the second identity ρ(t~r) = dMd3~r is the local massdensity

Let us now assume that the material system Σ is moving as part of a larger flowing continuousmedium To find the substantial derivative DG(t)Dt of G(t) we shall first compute the variation

III1 Reynolds transport theorem 27

δG for the material system Σ between times t and t + δt where δ is assumed to be small At theend of the calculation we shall take the limit δtrarr 0

Going back to the regions (1) (2minus) (2+) defined in Fig III1 one can write

δG =(G1 + G2+

)t+δtminus(G1 + G2minus

= δG1 + δG2

where the various indices denote the respective spatial domains and instants and

δG1 equiv(G1

)t+δtminus(G1

)t δG2 equiv

)t+δtminus(G2minus

bull δG1 represents the variation of G inside region (1) due to the non-stationarity of the fluidflow In the limit δtrarr 0 this region (1) coincides with the control volume V to leading orderin δt one thus has

δG1 =dG1(t)

dtδt =

[ intV

(t~r) ρ(t~r) d3~r

]δt =

(t~r) ρ(t~r) d3~r]δt (III3)

where the first identity is a trivial Taylor expansion the second one replaces the volume ofregion (1) by V while the last identity follows from the independence of the control volumefrom time

bull δG2 represents the algebraic amount of G traversing between t and t+δt the control surface Seither leaving (region 2+) or entering (region 2minus) the control volume V where in the latter casethe amount is counted negatively This is precisely the fluxmdashin the mathematical acceptationof the termmdashthrough the surface S oriented towards the exterior of an appropriate fluxdensity for quantity G (2)

-|~v| δt

Let~v denote the velocity of the continuous medium at position ~rat time t The amount of quantity G that traverses in δt a surfaceelement d2S situated in ~r equals the amount inside an elementarycylinder with base d2S and height |~v| δt ie d3G = g

mρd3V with

d3V = |d2~S middot~v| δt where the vector d2~S is normal to the surfaceelement

Integrating over all surface elements all over the control surface the amount of quantity Gflowing through S thus reads

δG2 =

(t~r) ρ(t~r)~v(t~r)]middot d2~S δt (III4)

All in all Eqs (III3)ndash(III4) yield after dividing by δt and taking the limit δt rarr 0 the so-calledReynolds transport theorem(xxxiii)(f)

(t~r) ρ(t~r) d3~r]

(t~r) ρ(t~r)~v(t~r)]middot d2~S (III5)

The first term on the right hand side of this relation represents a local time derivative partGparttsimilar to the first term in Eq I18 In contrast the second term is of convective type ie directlycaused by the motion of matter and represents the transport of G

(2)This flux density can be read off Eq (III4) namely gm(t~r) ρ(t~r)~v(t~r)(xxxiii)Reynoldsrsquoscher Transportsatz(f)O Reynolds 1842ndash1912

Anticipating on the rest of the Chapter this theorem will help us as follows The ldquousualrdquo lawsof dynamics are valid for closed material systems Σ rather than for open ones Accordingly theselaws involve time derivatives ldquofollowing the system in its motionrdquo that is precisely the materialderivative DDt Reynoldsrsquo transport theorem (III5) expresses the latter for extensive quantitiesG(t) in terms of local densities attached to fixed spatial positions ie in Eulerian variables

Remarks

lowast The medium velocity ~v(t~r) entering Reynolds transport theorem (III5) is measured in thereference frame R in which the control surface S remains motionless

lowast Since relation (III5) is traditionally referred to as a theorem one may wonder what are itsassumptions Obviously the derivation of the result relies on the assumption that the specificdensity g

m(t~r) and the velocity field ~v(t~r) are both continuous and differentiable in agreement

with the generic hypotheses in Sec I22 Figure III1 actually also embodies the hidden butnecessary assumption that the motion is continuous which leads to the smooth evolution of theconnected system of material points which are together inside the control surface S at time t Againthis follows from suitable properties of~v

lowast Accordingly the Reynolds transport theorem (III5) does not hold if the velocity field or thespecific density g

m is discontinuous As was already mentioned in Sec I22 such discontinuities are

however necessary to account for some phenomena (shock waves boundary between two immisciblefluids ) In such cases it will be necessary to reformulate the transport theorem to take intoaccount the discontinuities

III2 Mass and particle number conservation continuity equationThe mass M and the particle number N of a closed non-relativistic system Σ remain constant inits motion

Dt= 0 (III6)

These conservation laws lead with the help of Reynoldsrsquo transport theorem to partial differentialequations for some of the local fields that characterize a fluid flow

III21 Integral formulation

For an arbitrary control volume V delimited by surface S the Reynolds transport theorem (III5)with G(t) = M to which corresponds the specific density g

m(t~r) = 1 reads

[ intVρ(t~r) d3~r

[ρ(t~r)~v(t~r)

]middot d2~S = 0 (III7)

That is the time derivative of the mass contained in V is the negative of the mass flow rate throughS In agreement with footnote 2 ρ(t~r)~v(t~r) is the mass flux density (xxxiv) while its integral isthe mass flow rate(xxxv)

Taking now G(t) = N the associated specific density is gm

(t~r) = NM Since the productof NM with the mass density ρ(t~r) is precisely the particle number density n(t~r) Reynoldsrsquotheorem (III5) leads to

[ intV

n(t~r) d3~r

[n(t~r)~v(t~r)

where n(t~r)~v(t~r) represents the particle number flux density (xxxvi)

(xxxiv)Massenstromdichte (xxxv)Massenstrom (xxxvi)Teilchenstromdichte

III3 Momentum balance Euler and NavierndashStokes equations 29

Equation (III7) resp (III8) consitutes the integral formulation of mass resp particle numberconservation

Remark In the case of a steady motion Eq (III7) shows that the net mass flow rate through anarbitrary closed geometrical surface S vanishes That is the entrance of some amount of fluid intoa (control) volume V must be compensated by the simultaneous departure of an equal mass fromthe volume

III22 Local formulation

Since the control volume V in Eq (III7) resp (III8) is time-independent the time derivativecan be exchanged with the integration over volume Besides the surface integral can be transformedwith the help of the Gauss theorem into a volume integral All in all this yieldsint

partρ(t~r)

partt+ ~nabla middot

[ρ(t~r)~v(t~r)

]d3~r = 0

resp intV

partn(t~r)

[n(t~r)~v(t~r)

]d3~r = 0

These identities hold for an arbitrary integration volume V Using the continuity of the respectiveintegrands one deduces the following so-called continuity equations

partρ(t~r)

[ρ(t~r)~v(t~r)

]= 0 (III9)

resppartn(t~r)

[n(t~r)~v(t~r)

]= 0 (III10)

Equation (III9) represents the first of five dynamical (partial differential) equations which governthe evolution of a non-relativistic fluid flow

Remarks

lowast The form of the continuity equation (III9) does not depend on the properties of the flowingmedium as for instance whether dissipative effects play a significant role or not This should becontrasted with the findings of the next two Sections

lowast In the case of a steady fluid flow Eq (III9) gives ~nabla middot[ρ(t~r)~v(t~r)

]= 0 ie

ρ(t~r) ~nabla middot ~v(t~r) +~v(t~r) middot ~nablaρ(t~r) = 0

Thus the stationary flow of a homogeneous fluid ie for which ρ(t~r) is position independent willbe incompressible [~nabla middot~v(t~r) = 0 cf Eq (II13)]

III3 Momentum balance Euler and NavierndashStokes equationsFor a closed system Σ with total linear momentum ~P with respect to a given reference frame RNewtonrsquos second law reads

D~P (t)

Dt= ~F (t) (III11)

with ~F the sum of the ldquoexternalrdquo forces acting on ΣThe left hand side of this equation can be transformed with the help of Reynoldsrsquo transport

theorem (III5) irrespective of any assumption on the fluid under consideration (Sec III31) Incontrast the forces acting on a fluid element more precisely the forces exerted by the neighboring

elements do depend on the properties of the fluid The two most widespread models used for fluidsare that of a perfect fluid which leads to the Euler equation (Sec III32) and of a Newtonian fluidfor which Newtonrsquos second law (III11) translates into the NavierndashStokes equation (Sec III33)Throughout this Section we use the shorter designation ldquomomentumrdquo instead of the more accurateldquolinear momentumrdquo

III31 Material derivative of momentum

As already noted shortly below Eq (III1) the specific density associated with the momentum~P (t) is simply the flow velocity ~v(t~r) Applying Reynoldsrsquo theorem (III5) for the momentum ofthe material system contained at time t inside a control volume V the material derivative on theleft hand side of Newtonrsquos law (III11) can be recast as

D~P (t)

[ intV~v(t~r) ρ(t~r) d3~r

∮S~v(t~r) ρ(t~r)~v(t~r) middot d2~S (III12)

Both terms on the right hand side can be transformed to yield more tractable expressions Onthe one hand since the control volume V is immobile in the reference frame R the time derivativecan be taken inside the integral Its action on ρ(t~r)~v(t~r) is then given by the usual product ruleOn the other hand one can show the identity∮

S~v(t~r) ρ(t~r)~v(t~r) middot d2~S =

minus~v(t~r)

partρ(t~r)

partt+ ρ(t~r)

[~v(t~r) middot ~nabla

]~v(t~r)

d3~r (III13)

All in all one thus obtains

D~P (t)

intVρ(t~r)

part~v(t~r)

]~v(t~r)

d3~r =

intVρ(t~r)

D~v(t~r)

Dtd3~r (III14)

Proof of relation (III13) let ~J(t) denote the vector defined by the surface integral on the lefthand side of that identity For the i-th component of this vector Gaussrsquo integral theorem gives

J i(t) =

[vi(t~r) ρ(t~r)~v(t~r)

]middot d2~S =

intV~nabla middot[vi(t~r) ρ(t~r)~v(t~r)

The action of the differential operator yields vi(t~r) ~nabla middot[ρ(t~r)~v(t~r)

]+ ρ(t~r)~v(t~r) middot ~nablavi(t~r)

the divergence in the first term can be expressed according to the continuity equation (III9) asthe negative of the time derivative of the mass density leading to

~nabla middot[vi(t~r) ρ(t~r)~v(t~r)

]= minusvi(t~r)

partρ(t~r)

partt+ ρ(t~r)

]vi(t~r)

This relation holds for all three components of ~J from where Eq (III13) follows

Remark The derivation of Eq (III14) relies on purely algebraic transformations either as encodedin Reynoldsrsquo transport theorem or when going from relation (III12) to (III14) That is it does notimply any modelmdashapart from that of a continuous mediummdashfor the fluid properties In particularwhether or not dissipative effects are important in the fluid did not play any role here

III32 Perfect fluid Euler equation

In this section we first introduce the notion or rather the model of a perfect fluid which isdefined by the choice of a specific ansatz for the stress tensor which encodes the contact forcesbetween neighboring fluid elements Using that ansatz and the results of the previous paragraphNewtonrsquos second law (III11) is shown to be equivalent to a local formulation the so-called Eulerequation Eventually the latter is recast in the generic form for a local conservation or balanceequation involving the time derivative of a local density and the divergence of the correspondingflux density

III32 a

Forces in a perfect fluid

The forces in a fluid were already discussed on a general level in Sec I41 Thus the total forceon the right hand side of Eq (III11) consist of volume and surface forces which can respectivelybe expressed as a volume or a surface integral

~F (t) =

intV~fV (t~r) d3~r +

∮S~Ts(t~r) d2S (III15)

where ~fV denotes the local density of body forces while ~Ts is the mechanical stress vector introducedin Eq (I20) The latter will now allow us to introduce various models of fluids

The first simplest model is that of a perfect fluid or ideal fluid

A perfect fluid is a fluid in which there are no shear stresses nor heat conduction (III16a)

Stated differently at every point of a perfect fluid the stress vector ~Ts on a (test) surface elementd2S moving with the fluid is normal to d2S irrespective of whether the fluid is at rest or in motionThat is introducing the normal unit vector~en(~r) to d2S oriented towards the exterior of the materialregion acted upon(3) one may write

~Ts(t~r) = minusP (t~r)~en(~r) (III16b)

with P (t~r) the pressure at position ~r Accordingly the mechanical stress tensor in a perfect fluidin a reference frame R which is moving with the fluid is given by

σσσ(t~r) = minusP (t~r)gminus1(t~r) (III16c)

with gminus1 the inverse metric tensor just like in a fluid at rest [Eq (I23)] In a given coordinatesystem in R the

)-components of σσσ thus simply read

σσσij(t~r) = minusP (t~r) gij(t~r) (III16d)

ie the(

)-components are σσσij(t~r) = minusP (t~r) δij

Using relation (III16b) the total surface forces in Eq (III15) can be transformed into a volumeintegral ∮

S~Ts(t~r) d2S = minus

P (t~r)~en(~r) d2S = minus∮S

P (t~r) d2~S = minusintV~nablaP (t~r) d3~r (III17)

where the last identity follows from a corollary of the usual divergence theorem

Remark Although this might not be intuitive at first the pressure P (t~r) entering Eqs (III16b)ndash(III16d) is actually the hydrostatic pressure already introduced in the definition of the mechanicalstress in a fluid at rest see Eq (I23) One heuristic justification is that the stresses are definedas the forces per unit area exerted by a piece of fluid situated on one side of a surface on the fluidsituated on the other side Even if the fluid is moving the two fluid elements on both sides of thesurfacemdashas well as the comoving test surfacemdashhave the same velocity(4) ie their relative velocityvanishes just like in a fluid at rest

III32 b

Euler equation

Gathering Eqs (III11) (III14) (III15) and (III17) yieldsintVρ(t~r)

part~v(t~r)

]~v(t~r)

d3~r =

[minus~nablaP (t~r) + ~fV (t~r)

(3)Cf the discussion between Eqs (I21a)ndash(I21c)(4) thanks to the usual continuity assumption this no longer holds at a discontinuity

Since this identity must hold irrespective of the control volume V the integrands on both sides mustbe equal That is the various fields they involve obey the Euler equation

ρ(t~r)

part~v(t~r)

]~v(t~r)

= minus~nablaP (t~r) + ~fV (t~r) (III18)

Remarks

lowast The term in curly brackets on the left hand side is exactly the acceleration (I17) of a materialpoint as in Newtonrsquos second law

lowast Due to the convective term (~v middot~nabla)~v the Euler equation is a non-linear partial differential equation

lowast Besides Newtonrsquos second law for linear momentum one could also think of investigating theconsequence of its analogue for angular momentum Since we have assumed that the materialpoints do not have any intrinsic spin the conservation of angular momentum apart from leading tothe necessary symmetry of the stress tensormdashwhich is realized in a perfect fluid see Eq (III16c)or (III16d) and will also hold in a Newtonian fluid see Eq (III26)mdashdoes not bring any newdynamical equation

III32 c

Boundary conditions

To fully formulate the mathematical problem representing a given fluid flow one must alsospecify boundary conditions for the various partial differential equations These conditions reflectthe geometry of the problem under consideration

bull Far from an obstacle or from walls one may specify a given pattern for the flow velocity fieldFor instance one may require that the flow be uniform ldquoat infinityrdquo as eg for the motion farfrom the rotating cylinder in Fig IV5 illustrating the geometry of the Magnus effect

bull At an obstacle in particular at a wall the component of velocity perpendicular to the obstacleshould vanish that is the fluid cannot penetrate the obstacle or wall which makes sense andwill be hereafter often referred to as impermeability condition In case the obstacle is itself inmotion one should consider the (normal component of the) relative velocity of the fluid withrespect to the obstacle

On the other hand the model of a perfect fluid in which there is by definition no friction doesnot specify the value of the tangential component of the fluid relative velocity at an obstacle

III32 d

Alternative forms of the Euler equation

In practice the volume forces acting on a fluid element are often proportional to its mass asare eg the gravitational Coriolis or centrifugal forces Therefore it is rather natural to introducethe corresponding force density per unit mass instead of per unit volume

~aV (t~r) equiv d~FV (t~r)

dM(t~r)=

~fV (t~r)

ρ(t~r)

With the help of this ldquospecific density of body forcesrdquo which has the dimension of an accelerationthe Euler equation (III18) can be recast as

D~v(t~r)

Dt= minus 1

ρ(t~r)~nablaP (t~r) + ~aV (t~r) (III19)

The interpretation of this form is quite straightforward the acceleration of a material point (lefthand side) is the sum of the acceleration due to the pressure forces and the acceleration due tovolume forces (right hand side)

Alternatively one may use the identity (in which the time and position variables have beenomitted for the sake of brevity)

~v times(~nablatimes~v

)= ~nabla

)minus(~v middot ~nabla

which can be proved either starting from the usual formula for the double cross productmdashwith asmall twist when applying the differential operator to a vector squaredmdashor by working componentby component Recognizing in the rightmost term the convective part of the Euler equation onecan rewrite the latter or equivalently Eq (III19) as

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

]minus ~v(t~r)times ~ω(t~r) = minus 1

where we have made use of the vorticity vector (II10) Note that the second term on the left handside of this equation involves the (gradient of the) kinetic energy per unit mass dKdM

In Sec IV21 we shall see yet another form of the Euler equation [Eq (IV8)] involving ther-modynamic functions other than the pressure

III32 e

The Euler equation as a balance equation

The Euler equation can be rewritten in the generic form for of a balance equation namely asthe identity of the sum of the time derivative of a density and the divergence of a flux density witha source termmdashwhich vanishes if the quantity under consideration is conserved Accordingly wefirst introduce two

Definitions One associates with the i-th component in a given coordinate system of the momentumof a material system its

bull density(xxxvii) ρ(t~r) vi(t~r) and (III21a)

bull flux density(xxxviii)(in direction j) TTTij(t~r) equiv P (t~r) gij(t~r) + ρ(t~r) vi(t~r) vj(t~r) (III21b)

with gij the components of the inverse metric tensor gminus1Physically TTTij represents the amount of momentum along ~ei transported per unit time through

a unit surface(5) perpendicular to the direction of ~ejmdashie transported in direction j That is it isthe i-th component of the force upon a test unit surface with normal unit vector ~ej

The first contribution to TTTij involving pressure is the transport due to the thermal randommotion of the atoms of the fluid On the other hand the second termmdashnamely the transportedmomentum density multiplied by the velocitymdasharises from the convective transport represented bythe macroscopic motion

Remarks

lowast As thermal motion is random and (statistically) isotropic it does not contribute to the momen-tum density ρ(t~r)~v(t~r) only to the momentum flux density

lowast In tensor notation the momentum flux density (III21b) viewed as a(

)-tensor is given by

TTT(t~r) = P (t~r)gminus1(t~r) + ρ(t~r)~v(t~r)otimes~v(t~r) for a perfect fluid (III22)

(5) which must be immobile in the reference frame in which the fluid has the velocity~v entering definition (III21b)

(xxxvii)Impulsdichte (xxxviii)Impulsstromdichte

lowast Given its physical meaning the momentum flux (density) tensor TTT is obviously related to theCauchy stress tensor σσσ More precisely TTT represents the forces exerted by a material point onits neighbors while σσσ stands for the stresses acting upon the material point due to its neighborsInvoking Newtonrsquos third lawmdashwhich in continuum mechanics is referred to as Cauchyrsquos fundamentallemmamdash these two tensors are simply opposite to each other

lowast Building on the previous remark the absence of shear stress defining a perfect fluid can bereformulated as a condition of the momentum flux tensor

A perfect fluid is a fluid at each point of which one can find a local velocity suchthat for an observer moving with that velocity the fluid is locally isotropicThe momentum flux tensor is thus diagonal in the observerrsquos reference frame

(III23)

Using definitions (III21) one easily checks that the Euler equation (III18) is equivalent to thebalance equations (for i = 1 2 3)

[ρ(t~r) vi(t~r)

3sumj=1

dTTTij(t~r)

dxj= f iV (t~r) (III24a)

with f iV the i-th component of the volume force density and d dxi the covariant derivatives (seeAppendix C1) that coincide with the partial derivatives in Cartesian coordinates

Proof For the sake of brevity the (t~r)-dependence of the various fields will not be specifiedOne finds

part(ρvi)

partt+

3sumj=1

dTTTij

dxj=partρ

parttvi + ρ

partvi

partt+

3sumj=1

gijdPdxj

3sumj=1

vid(ρvj)

3sumj=1

ρvjdvi

= vi[partρ

partt+ ~nabla middot (ρ~v)

[partvi

partt+ (~v middot ~nabla)vi

where we have usedsumj g

ijddxj = ddxi The first term between square brackets vanishesthanks to the continuity equation (III9) In turn the second term is precisely the i-th componentof the left member of the Euler equation (III18) ie it equals the i-th component of ~fV minusthe third term which represents the i-th component of ~nablaP

In tensor notation Eq (III24a) reads

[ρ(t~r)~v(t~r)

]+ ~nabla middotTTT(t~r) = ~fV (t~r) (III24b)

where we have used the symmetry of the momentum flux tensorTTT while the action of the divergenceon a

)-tensor is defined through its components which is to be read off Eq (III24a)

III33 Newtonian fluid NavierndashStokes equation

In a real moving fluid there are friction forces that contribute to the transport of momentumbetween neighboring fluid layers when the latter are in relative motion Accordingly the momentumflux-density tensor is no longer given by Eq (III21b) or (III22) but now contains extra termsinvolving derivatives of the flow velocity Accordingly the Euler equation must be replaced by analternative dynamical equation including the friction forces

III33 a

Momentum flux density in a Newtonian fluid

The momentum flux density (III21b) in a perfect fluid only contains two termsmdashone propor-tional to the components gij of the inverse metric tensor the other proportional to vi(t~r) vj(t~r)

Since the coefficients in front of these two terms could a priori depend on~v2 this represents the mostgeneral symmetric tensor of degree 2 which can be constructed with the help of the flow velocityonly

If the use of terms that depend on the spatial derivatives of the velocity field is also allowed thecomponents of the momentum flux-density tensor can be of the following form where for the sakeof brevity the variables t and ~r are omitted

TTTij = Pgij + ρvi vj +Advi

dxj dxk

)+ middot middot middot (III25)

with coefficients A B that depend on i j and on the fluid under consideration

This ansatz for TTTij as well as the form of the energy flux density involved in Eq (III35) belowcan be ldquojustifiedrdquo by starting from a microscopic kinetic theory of the fluid and writing thesolutions of the corresponding equation of motion as a specific expansionmdashwhich turns out tobe in powers of the Knudsen number (I4) This also explains why terms of the type vipartPpartxjor vipartTpartxj with T the temperature were not considered in Eq (III25)Despite these theoretical considerations in the end the actual justification for the choices ofmomentum or energy flux density is the agreement with the measured properties of fluids

As discussed in Sec I13 the description of a system of particles as a continuous mediumand in particular as a fluid in local thermodynamic equilibrium rests on the assumption that themacroscopic quantities of relevance for the medium vary slowly both in space and time Accordingly(spatial) gradients should be small the third and fourth terms in Eq (III25) should thus beon the one hand much smaller than the first two ones on the other hand much larger than therightmost term as well as those involving higher-order derivatives or of powers of the first derivativesNeglecting these smaller terms one obtains ldquofirst-order dissipative fluid dynamicsrdquo which describesthe motion of Newtonian fluidsmdashthis actually defines the latter

Using the necessary symmetry of TTTij the third and fourth terms in Eq (III25) can be rewrittenas the sum of a traceless symmetric contribution and a tensor proportional to the inverse metrictensor This leads to the momentum flux-density tensor

TTTij(t~r) = P (t~r) gij(t~r) + ρ(t~r)vi(t~r)vj(t~r)

minus η(t~r)

[dvi(t~r)

dvj(t~r)

dximinus 2

3gij(t~r)~nabla middot~v(t~r)

](III26a)

minus ζ(t~r)gij(t~r)~nabla middot~v(t~r)

In geometric formulation this reads

TTT(t~r) = P (t~r)gminus1(t~r) + ρ(t~r)~v(t~r)otimes~v(t~r) +πππ(t~r) (III26b)

where dissipative effects are encored in the viscous stress tensor (xxxix)

πππ(t~r) equiv minus2η(t~r)

[DDD(t~r)minus 1

[~nablamiddot~v(t~r)

]gminus1(t~r)

]minusζ(t~r)

]gminus1(t~r)

for a Newtonian fluid(III26c)

with DDD(t~r) the strain rate tensor discussed in Sec II13 Component-wise

πij(t~r) equiv minus2η(t~r)

[DDDij(t~r)minus 1

[~nabla middot~v(t~r)

]gij(t~r)

]minus ζ(t~r)

]gij(t~r) (III26d)

(xxxix)viskoser Spannungstensor

In terms of the traceless rate-of-shear tensor (II15b) or of its components (II15d) one may alter-natively write

πππ(t~r) equiv minus2η(t~r)SSS(t~r)minus ζ(t~r)[~nabla middot~v(t~r)

]gminus1(t~r) (III26e)

πij(t~r) equiv minus2η(t~r)SSSij(t~r)minus ζ(t~r)[~nabla middot~v(t~r)

]gij(t~r) (III26f)

This viscous stress tensor involves two novel characteristics of the medium so-called transportcoefficients namely

bull the (dynamical) shear viscosity(xl) η which multiplies the traceless symmetric part of thevelocity gradient tensor ie the conveniently termed rate-of-shear tensor

bull the bulk viscosity also called second viscosity (xli) ζ which multiplies the volume-expansionpart of the velocity gradient tensor ie the term proportional to ~nabla middot~v(t~r)

The two corresponding contributions represent a diffusive transport of momentum in the fluidmdashrepresenting a third type of transport besides the convective and thermal ones

Remarks

lowast In the case of a Newtonian fluid the viscosity coefficients η and ζ are independent of the flowvelocity However they still depend on the temperature and pressure of the fluid so that they arenot necessarily uniform and constant in a real flowing fluid

lowast In an incompressible flow ~nabla middot~v(t~r) = 0 the last contribution to the momentum flux den-sity (III26) drops out Thus the bulk viscosity ζ only plays a role in compressible fluid motions(6)

lowast Expression (III26c) or (III26d) of the viscous stress tensor assumes implicitly that the fluid is(locally) isotropic since the coefficients η ζ are independent of the directions i j

III33 b

Surface forces in a Newtonian fluid

The Cauchy stress tensor corresponding to the momentum flux density (III26) of a Newtonianfluid is

σσσ(t~r) = minusP (t~r)gminus1(t~r)minusπππ(t~r) (III27a)

that is using the form (III26e) of the viscous stress tensor

σσσ(t~r) = minusP (t~r)gminus1(t~r) + 2η(t~r)SSS(t~r) + ζ(t~r)[~nabla middot~v(t~r)

]gminus1(t~r) (III27b)

Component-wise this becomes

σij(t~r) =

minusP (t~r)+

[ζ(t~r)minus 2

3η(t~r)

]~nablamiddot~v(t~r)

gij(t~r)+η(t~r)

[dvi(t~r)

dvj(t~r)

] (III27c)

Accordingly the mechanical stress vector on an infinitesimally small surface element situated atpoint ~r with unit normal vector ~en(~r) reads

~Ts(t~r) = σσσ(t~r) middot~en(~r) =

3sumij=1

[minusP (t~r) +

(ζ(t~r)minus 2

3η(t~r)

)~nabla middot~v(t~r)

]gij(t~r)

+ η(t~r)

(dvi(t~r)

dvj(t~r)

)nj(~r)~ei(t~r) (III28)

with nj(~r) the coordinate of ~en(~r) along direction j One easily identifies the two components of

(6)As a consequence the bulk viscosity is often hard to measuremdashone has to devise a compressible flowmdashso that itis actually not so well known for many substances even well-studied ones [11]

(xl)Scherviskositaumlt (xli)Dehnviskositaumlt Volumenviskositaumlt zweite Viskositaumlt

this stress vector (cf Sec I41)

bull the term proportional tosumgijn

j~ei = ~en is the normal stress on the surface element Itconsists of the usual hydrostatic pressure term minusP ~en and a second one proportional to thelocal expansion rate ~nabla middot~v in the compressible motion of a Newtonianmdashand more generally adissipativemdashfluid the normal stress is thus not only given by minusP ~en but includes additionalcontributions that vanish in the static case

bull the remaining term is the tangential stress proportional to the shear viscosity η Accordinglythe value of the latter can be deduced from a measurement of the tangential force acting ona surface element see Sec VI12

As in sect III32 a the external contact forces acting on a fluid element delimited by a surface Scan easily be computed Invoking the Stokes theorem yields∮

P (t~r)~en(~r) d2S minus∮Sπππ(t~r) middot~en(~r) d2S

= minusint

~nablaP (t~r) d3V +

~nabla middotπππ(t~r) d3V

= minusint

~nablaP (t~r) d3V +

~fvisc(t~r) d3V (III29a)

with the local viscous friction force density

~fvisc(t~r) =3sum

η(t~r)

[dvi(t~r)

dvj(t~r)

]~ej(t~r)

+ ~nabla[ζ(t~r)minus 2

3η(t~r)

]~nabla middot~v(t~r)

(III29b)

III33 c

NavierndashStokes equation

Combining the viscous force (III29b) with the generic equations (III12) (III14) and (III15)the application of Newtonrsquos second law to a volume V of fluid leads to an identity between sums ofvolume integrals Since this relation holds for any volume V it translates into an identity betweenthe integrands namely

ρ(t~r)

part~v(t~r)

]~v(t~r)

= minus~nablaP (t~r) + ~fvisc(t~r) + ~fV (t~r) (III30a)

or component-wise

ρ(t~r)

partvi(t~r)

]vi(t~r)

=minusdP (t~r)

[ζ(t~r)minus 2

3η(t~r)

3sumj=1

η(t~r)

[dvi(t~r)

dvj(t~r)

]+[~fV (t~r)

]i(III30b)

for i = 1 2 3If the implicit dependence of the viscosity coefficients on time and position is negligible one

may pull η and ζ outside of the spatial derivatives As a result one obtains the (compressible)NavierndashStokes equation(g)(h)

ρ(t~r)

part~v(t~r)

partt+[~v(t~r) middot~nabla

]~v(t~r)

= minus~nablaP (t~r) + η4~v(t~r) +

)~nabla[~nablamiddot~v(t~r)

]+ ~fV (t~r)

(III31)(g)C-L Navier 1785ndash1836 (h)G G Stokes 1819ndash1903

with 4 = ~nabla2 the Laplacian This is a non-linear partial differential equation of second order whilethe Euler equation (III18) is of first order

The difference between the order of the equations is not a mere detail while the Euler equationlooks like the limit η ζ rarr 0 of the NavierndashStokes equation the corresponding is not necessarilytrue of their solutions This is for instance due to the fact that their respective boundaryconditions differ

In the case of an incompressible flow the local expansion rate in the NavierndashStokes equa-tion (III31) vanishes leading to the incompressible NavierndashStokes equation

part~v(t~r)

]~v(t~r) = minus1

ρ~nablaP (t~r) + ν4~v(t~r) (III32)

with ν equiv ηρ the kinematic shear viscosity

Remark The dimension of the dynamic viscosity coefficients η ζ is MLminus1Tminus1 and the correspondingunit in the SI system is the Poiseuille(i) abbreviated Pamiddots In contrast the kinematic viscosity hasdimension L2Tminus1 ie depends only on space and time hence its denomination

III33 d

Boundary conditions

At the interface between a viscous fluid in particular a Newtonian one and another bodymdashbe itan obstacle in the flow a wall containing the fluid or even a second viscous fluid which is immisciblewith the first onemdashthe relative velocity between the fluid and the body must vanish This holdsnot only for the normal component of the velocity (ldquoimpermeabilityrdquo condition) as in perfect fluidsbut also for the tangential one to account for the friction forces The latter requirement is oftenreferred to as no-slip condition

III34 Higher-order dissipative fluid dynamics

Instead of considering only the first spatial derivatives of the velocity field in the momentumflux-density tensor (III25) one may wish to also include the second derivatives or even higherones Such assumptions lead to partial differential equations of motion replacing the NavierndashStokesequation of increasing order Burnett equation super Burnett equation [12]

The domain of validity of such higher-order dissipative fluid models is a priori larger than that offirst-order fluid dynamics since it becomes possible to account for stronger gradients On the otherhand this is at the cost of introducing a large number of new parameters besides the transportcoefficients already present in Newtonian fluids In parallel the numerical implementation of themodel becomes more involved so that a macroscopic description does not necessarily represent thebest approach

III4 Energy conservation entropy balanceThe conservation of mass and Newtonrsquos second law for linear momentum lead to four partial dif-ferential equations one scalarmdashcontinuity equation (III9)mdashand one vectorialmdashEuler (III18) orNavierndashStokes (III31)mdash describing the coupled evolutions of five fields ρ(t~r) the three compo-nents of~v(t~r) and P (t~r)(7) To fully determine the latter a fifth equation is needed For this lastconstraint there are several possibilities

A first alternative is if some of the kinematic properties of the fluid flow are known a prioriThus requiring that the motion be steady or irrotational or incompressible might suffice to fully

(7)The density of volume forces ~fV or equivalently the corresponding potential energy per unit mass Φ which standfor gravity or inertial forces are given ldquofrom the outsiderdquo and not counted as a degree of freedom

(i)J-L-M Poiseuille 1797ndash1869

III4 Energy conservation entropy balance 39

constrain the fluid flow for the geometry under consideration we shall see several examples in thenext three Chapters

A second possibility which will also be illustrated in Chap IVndashVI is that of a thermodynamicconstraint isothermal flow isentropic flow For instance one sees in thermodynamics that inan adiabatic process for an ideal gas the pressure and volume of the latter obey the relationPV γ = constant where γ denotes the ratio of the heat capacities at constant pressure (CP ) andconstant volume (CV ) Since V is proportional to 1ρ this so-called ldquoadiabatic equation of staterdquoprovides the needed constraint relating pressure and mass density

Eventually one may argue that non-relativistic physics automatically implies a further conser-vation law besides those for mass and linear momentum namely energy conservation Thus usingthe reasoning adopted in Secs (III2) and (III3) the rate of change of the total energymdashinternalkinetic and potentialmdashof the matter inside a given volume equals the negative of the flow of energythrough the surface delimiting this volume In agreement with the first law of thermodynamics onemust take into account in the energy exchanged with the exterior of the volume not only the con-vective transport of internal kinetic and potential energies but also the exchange of the mechanicalwork of contact forces andmdashfor dissipative fluidsmdashof heat

III41 Energy and entropy conservation in perfect fluids

In non-dissipative non-relativistic fluids energy is either transported convectivelymdashas it accom-panies some flowing mass of fluidmdashor exchanged as mechanical work of the pressure forces betweenneighboring regions Mathematically this is expressed at the local level by the equation

2ρ(t~r)~v(t~r)2 + e(t~r) + ρ(t~r)Φ(t~r)

]+ ~nabla middot

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r) + ρ(t~r)Φ(t~r)

]~v(t~r)

(III33)

where e denotes the local density of internal energy and Φ the potential energy per unit mass ofvolume forcesmdashassumed to be conservativemdashsuch that the acceleration ~aV present in Eq (III19)equals minus~nablaΦ

Equation (III33) will not be proven heremdashwe shall see later in Sec IX33 that it emerges aslow-velocity limit of one of the equations of non-dissipative relativistic fluid dynamics It is howeverclearly of the usual form for a conservation equation involving

bull the total energy density consisting of the kinetic (12ρ~v

2) internal (e) and potential (ρΦ)energy densities and

bull the total energy flux density which involves the previous three forms of energy as well as thatexchanged as mechanical work of the pressure forces(8)

Remarks

lowast The presence of pressure in the flux density however not in the density is reminiscent of thesame property in definitions (III21)

lowast The assumption that the volume forces are conservative is of course not innocuous For instanceit does not hold for Coriolis forces which means that one must be careful when working in a rotatingreference frame(8)Remember that when a system with pressure P increases its volume by an amount dV it exerts a mechanical work

P dV ldquoprovidedrdquo to its exterior

lowast The careful reader will have noticed that energy conservation (III33) constitutes a fifth equationcomplementing the continuity and Euler equations (III9) and (III18) yet at the cost of introducinga new scalar field the energy density so that now a sixth equation is needed The latter is providedby the thermal equation of state of the fluid which relates its energy density mass density andpressure(9) In contrast to the other equations this equation of state is not ldquodynamicalrdquo ie forinstance it does not involve time or spatial derivatives but is purely algebraic

One can showmdashagain this will be done in the relativistic case (sect IX32) can also be seen as spe-cial case of the result obtained for Newtonian fluids in Sec III43mdashthat in a perfect non-dissipativefluid the relation (III33) expressing energy conservation locally together with thermodynamic re-lations lead to the local conservation of entropy expressed as

parts(t~r)

[s(t~r)~v(t~r)

]= 0 (III34)

where s(t~r) is the entropy density while s(t~r)~v(t~r) represents the entropy flux density Themotion of a perfect fluid is thus automatically isentropic

This equation together with a thermodynamic relation is sometimes more practical than theenergy conservation equation (III33) to which it is however totally equivalent

III42 Energy conservation in Newtonian fluids

In a real fluid there exist further mechanisms for transporting energy besides the convectivetransport due to the fluid motion namely diffusion either of momentum or of energy

bull On the one hand the viscous friction forces in the fluid which already lead to the transportof momentum between neighboring fluid layers moving with different velocities exert somework in the motion which induces a diffusive transport of energy This is accounted for by acontribution πππ middot~v to the energy flux densitymdashcomponent-wise a contribution

sumj π

ij vj to the

i-th component of the flux densitymdash with πππ the viscous stress tensor given in the case of aNewtonian fluid by Eq (III26c)

bull On the other hand there is also heat conduction from the regions with higher temperaturestowards those with lower temperatures This transport is described by the introduction inthe energy flux density of a heat current(xlii) ~Q(t~r) = minusκ(t~r)~nablaT (t~r)mdashin accordance withthe local formulation of Fourierrsquos law (j) see eg Sec 121 in Ref [2]mdash with κ the heatconductivity(xliii) of the fluid

Taking into account these additional contributions the local formulation of energy conservation ina Newtonian fluid in the absence of external volume forces reads

2ρ(t~r)~v(t~r)2 + e(t~r)

]+ ~nabla middot

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r)

]~v(t~r)

minus η(t~r)

[(~v(t~r) middot ~nabla

)~v(t~r) + ~nabla

(~v(t~r)2

)]minus[ζ(t~r)minus 2η(t~r)

][~nabla middot ~v(t~r)

]~v(t~r)minus κ(t~r)~nablaT (t~r)

(III35)

(9)This is where the assumption of local thermodynamic equilibrium (sect I13) plays a crucial role(xlii)Waumlrmestromvektor (xliii)Waumlrmeleitfaumlhigkeit(j)J B Fourier 1768ndash1830

If the three transport coefficients η ζ and κ vanish this equation simplifies to that for perfectfluids Eq (III33)

Remark The energy flux density can be read off Eq (III35) since it represents the term betweencurly brackets One can check that it can also be written as[

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r)

]~v(t~r)minus 2η(t~r)SSS(t~r) middot~v(t~r)

minus ζ(t~r)[~nabla middot~v(t~r)

]~v(t~r)minus κ(t~r)~nablaT (t~r) (III36)

with SSS(t~r) the traceless symmetric rate-of-shear tensor One recognizes the various physical sourcesof energy transport

III43 Entropy balance in Newtonian fluids

In a real fluid with viscous friction forces and heat conductivity one can expect a priori thatthe transformation of mechanical energy into heat will lead in general to an increase in entropyprovided a closed system is being considered

Consider a volume V of flowing Newtonian fluid delimited by a surface S at each point~r of whichthe boundary conditions~v(t~r) middot~en(~r) = 0 and ~Q(t~r) middot~en(~r) = 0 hold where ~en(~r) denotes the unitnormal vector to S at ~r Physically these boundary conditions mean than neither matter nor heatflows across the surface S so that the system inside S is closed and isolated To completely excludeenergy exchanges with the exterior of S it is also assumed that there are no volume forces acting onthe fluid inside volume V We shall investigate the implications of the continuity equation (III10)the NavierndashStokes equation (III31) and the energy conservation equation (III35) for the totalentropy S of the fluid inside V For the sake of brevity the variables t ~r will be omitted in theremainder of this Section

Starting with the energy conservation equation (III35) the contribution

2ρ~v2

)+ ~nabla middot

2ρ~v2

]in its first two lines can be replaced by

ρ~v middot part~v

partt+

partρ

partt~v2 +

[~nabla middot(ρ~v)]~v2 +

3sumi=1

ρvi(~v middot ~nabla

3sumi=1

[partvi

partt+(~v middot ~nabla

)vi] (III37a)

where the continuity equation (III9) was usedAs recalled in Appendix A the fundamental thermodynamic relation U = TS minus PV + microN gives onthe one hand e+ P = Ts+ micron which leads to~nablamiddot[(e+P )~v

]= T~nablamiddot

+micro~nablamiddot(n~v)

+~v middot(s~nablaT +n~nablamicro

)= T~nablamiddot

+~v middot~nablaP (III37b)

where the second identity follows from the GibbsndashDuhem relation dP = sdT + n dmicro On the otherhand it leads to de = T ds+microdn which under consideration of the continuity equation for particlenumber yields

partt= T

partt+ micro

partnpartt

= Tparts

parttminus micro~nabla middot

(n~v) (III37c)

With the help of relations (III37a)ndash(III37c) the energy conservation equation (III35) can berewritten as

3sumi=1

[partvi

+ Tparts

partt+ T~nabla middot

+~v middot ~nablaP =

3sumij=1

partxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot~v

3sumi=1

partxi

[ζ(~nabla middot~v

+ ~nabla middot(κ~nablaT

) (III37d)

Multiplying the i-th component of Eq (III30b) by vi gives

[partvi

+ vipartPpartxi

=3sumj=1

vipart

partxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot~v

)]+ vi

partxi(ζ~nabla middot~v

Subtracting this identity summed over i = 1 2 3 from Eq (III37d) yields

Tparts

= η3sum

partvipartxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot ~v

)+ ζ(~nabla middot~v

)2+ ~nabla middot

(κ~nablaT

) (III38)

On the right hand side of this equation one may use the identity

3sumij=1

(partvi

partxj+partvj

partximinus2

3gij~nablamiddot~v

)(partvipartxj

+partvjpartximinus2

3gij~nablamiddot~v

3sumij=1

(partvi

partxj+partvj

partximinus2

3gij~nablamiddot~v

)partvjpartxi

(III39a)

which follows from the fact that both symmetric terms partvipartxj and partvjpartx

i from the left membergive the same contribution while the term in gij yields a zero contribution since it multiplies atraceless termAdditionally one has

~nabla middot(κ~nablaT

)= T~nabla middot

(κ~nablaTT

(~nablaT)2 (III39b)

All in all Eqs (III38) and (III39) lead to

(s~v)minus ~nabla middot

(κ~nablaTT

3sumij=1

(partvi

partxj+partvj

partximinus 2

)(partvipartxj

+partvjpartximinus 2

3gij~nabla middot~v

(~nabla middot~v

)2+ κ

(~nablaT)2

T 2 (III40a)

This may still be recast in the slightly more compact form

parts(t~r)

[s(t~r)~v(t~r)minus κ(t~r)

~nablaT (t~r)

T (t~r)

2η(t~r)SSS(t~r) SSS(t~r) + ζ(t~r)

]2+ κ(t~r)

[~nablaT (t~r)

]2T (t~r)

(III40b)

with SSS SSS equiv SSSijSSSij the scalar obtained by doubly contracting the rate-of-shear tensor with itselfThis equation can then be integrated over the V occupied by the fluid

bull When computing the integral of the divergence term on the left hand side with the Stokestheorem it vanishes thanks to the boundary conditions imposed at the surface S

bull the remaining term in the left member is simply the time derivative dSdt of the total entropyof the closed system

bull if all three transport coefficients η ζ κ are positive then it is also the case of the three termson the right hand side

One thus findsdS

dtge 0 in agreement with the second law of thermodynamics

Remarkslowast The previous derivation may be seen as a proof that the transport coefficients must be positiveto ensure that the second law of thermodynamics holds

lowast If all three transport coefficients η ζ κ vanish ie in the case of a non-dissipative fluidEq (III40) simply reduces to the entropy conservation equation in perfect fluids (III34)

Bibliography for Chapter IIIbull Feynman [8 9] Chapter 40ndash2 amp 41ndash1 41ndash2

bull Guyon et al [2] Chapters 33 41ndash43 51 52

bull LandaundashLifshitz [3 4] Chapter I sect 12 amp sect 67 (perfect fluids) and Chapters II sect 1516 amp V sect 49(Newtonian fluids)

bull Flieszligbach [13] Chapter 32

CHAPTER IV

Non-relativistic flows of perfect fluids

In the previous Chapter we have introduced the coupled dynamical equations that govern the flowsof perfect fluids in the non-relativistic regime namely the continuity (III9) Euler (III18) andenergy conservation (III33) equations for the mass density ρ(t~r) fluid velocity~v(t~r) and pressureP (t~r) The present Chapter discusses solutions of that system of equations ie possible motions ofperfect fluids(10) obtained when using various assumptions to simplify the problem so as to renderthe equations tractable analytically

In the simplest possible case there is simply no motion at all in the fluid yet the volume forcesacting at each point still drive the behavior of the pressure and local mass density throughout themedium (Sec IV1) Steady flows in which there is by definition no real dynamics are also easilydealt with both the Euler and energy conservation equations yield the Bernoulli equation whichcan be further simplified by kinematic assumptions on the flow (Sec IV2)

Section IV3 deals with the dynamics of vortices ie of the vorticity vector field in the motionof a perfect fluid In such fluids in case the pressure only depends on the mass density there existsa quantity related to vorticity that remains conserved if the volume forces at play are conservative

The latter assumption is also necessary to define potential flows (Sec IV4) in which the furtherhypothesis of an incompressible motion leads to simplified equations of motion for which a numberof exact mathematical results are known especially in the case of two-dimensional flows

Throughout the Chapter it is assumed that the body forces in the fluid whose volume densitywas denoted by ~fV in Chapter III are conservative so that they derive from a potential Morespecifically anticipating the fact that these volume forces are proportional to the mass they actupon we introduce the potential energy per unit mass Φ such that

~fV (t~r) = minusρ(t~r)~nablaΦ(t~r) (IV1)

IV1 Hydrostatics of a perfect fluidThe simplest possibility is that of static solutions of the system of equations governing the dy-namics of perfect fluids namely those with ~v = ~0 everywheremdashin an appropriate global referenceframemdashand additionally partpartt = 0 Accordingly there is no strictly speaking fluid flow this isthe regime of hydrostatics for which the only(11) non-trivial equationmdashfollowing from the Eulerequation (III18)mdashreads

ρ(~r)~nablaP (~r) = minus~nablaΦ(~r) (IV2)

Throughout this Section we adopt a fixed system of Cartesian coordinates (x1 x2 x3) = (x y z)with the basis vector~e3 oriented vertically and pointing upwards In the following examples we shallconsider the case of fluids in a homogeneous gravity field leading to Φ(~r) = gz with g = 98 mmiddot sminus2(10) at least in an idealized world Yet the reader is encouraged to relate the results to observations of her everyday

lifemdashbeyond the few illustrative examples provided by the authormdash and to wonder how a small set of seeminglyldquosimplerdquo mathematical equations can describe a wide variety of physical phenomena

(11)This is true only in the case of perfect fluids for dissipative ones there emerge new possibilities see Sec VI11

IV1 Hydrostatics of a perfect fluid 45

Remark If the stationarity condition is relaxed the continuity equation still leads to partρpartt = 0ie to a time-independent mass density Whether time derivatives vanish or not makes no changein the Euler equation when~v = ~0 Eventually energy conservation requires that the internal energydensity emdashand thereby the pressuremdashfollow the same time evolution as the ldquoexternalrdquo potentialenergy Φ Thus there is a non-stationary hydrostatics but in which the time evolution decouplesfrom the spatial problem

IV11 Incompressible fluid

Consider first an incompressible fluidmdashor more correctly a fluid whose compressibility can asa first approximation be neglectedmdashwith constant uniform mass density ρThe fundamental equation of hydrostatics (IV2) then yields

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= minusρg

ie one recovers Pascalrsquos law(k)

P (~r) = P (z) = P 0 minus ρgz (IV3)

with P 0 the pressure at the reference point with altitude z = 0For instance the reader is probably aware that at a depth of 10 m under water (ρ = 103 kgmiddotmminus3)

the pressure isP (minus10 m) = P (0) + 103 middot g middot 10 asymp 2times 105 Pa

with P (0) asymp 105 Pa the typical atmospheric pressure at sea level

IV12 Fluid at thermal equilibrium

To depart from the assumption of incompressibility whose range of validity is quite limited letus instead consider a fluid at (global) thermal equilibrium ie with a uniform temperature T forinstance an ideal gas obeying the thermal equation of state PV = NkBT

Denoting by m the mass of a molecule of that gas the mass density is related to pressure andtemperature by ρ = mPkBT so that Eq (IV2) reads

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= minus mg

kBTP (~r)

ie one obtains the barotropic formula(xliv)

P (~r) = P (z) = P 0 exp

(minusmgzkBT

Invoking the equation of state one sees that the molecule number density n(~r) is also exponentiallydistributed in agreement with the Maxwell distribution of statistical mechanics since mgz is thepotential gravitational energy of a molecule at altitude z

Taking as example airmdashwhich is a fictive ideal gas with molar mass(12) NAmair = 29 g middotmolminus1mdashthe ratio kBTmairg equals 88 times 103 m for T = 300 K ie the pressure drops by a factor 2 forevery elevation gain of ca 6 km Obviously however assuming a constant temperature in the Earthatmosphere over such a length scale is unrealistic(12)NA denotes the Avogadro number

(xliv)barometrische Houmlhenformel

(k)B Pascal 1623ndash1662

46 Non-relativistic flows of perfect fluids

IV13 Isentropic fluid

Let us now assume that the entropy per particle is constant throughout the perfect fluid at restunder study sn = constant with s the entropy density and n the particle number density

We shall show in sect IX32 that the ratio sn is always conserved in the motion of a relativisticperfect fluid Taking the low-velocity limit one deduces the conservation of sn in a non-relativistic non-dissipative flow D(sn)Dt = 0 implying that sn is constant along pathlinesie in the stationary regime along streamlines Here we assume that sn is constant everywhere

Consider now the enthalpy H = U + PV of the fluid whose change in an infinitesimal processis the (exact) differential dH = T dS + V dP + microdN (13) In this relation micro denotes the chemicalpotential which will however play no further role as we assume that the number of molecules in thefluid is constant leading to dN = 0 Dividing by N thus gives

)= T d

where the first term on the right-hand side vanishes since SN is assumed to be constant Dividingnow by the mass of a molecule of the fluid one finds

ρdP (IV4)

where w denotes the enthalpy densityThis identity relates the change in enthalpy pro unit mass wρ to the change in pressure P in

an elementary isentropic process If one considers a fluid at local thermodynamic equilibrium inwhich wρ and P takes different values at different places the identity relates the difference in wρto that in P between two (neighboring) points Dividing by the distance between the two pointsand in the limit where this distance vanishes one derives an identity similar to (IV4) with gradientsinstead of differentials

Together with Eq (IV2) one thus obtains

~nabla[w(~r)

ρ(~r)+ Φ(~r)

]= ~0 (IV5)

that isw(z)

ρ(z)+ gz = constant

Taking as example an ideal diatomic gas its internal energy is U = 52NkBT resulting in the

enthalpy density

w = e+ P =5

2nkBT + nkBT =

That isw

m with m the mass of a molecule of gas Equation (IV5) then gives

dT (z)

dz= minus mg7

In the case of air the term on the right hand side equals 977 times 10minus3 K middotmminus1 = 977 K middot kmminus1ie the temperature drops by ca 10 degrees for an elevation gain of 1 km This represents a muchbetter modeling of the (lower) Earth atmosphere as the isothermal assumption of Sec IV12

Remarkslowast The International Standard Atmosphere (ISA)(14) model of the Earth atmosphere assumes a(piecewise) linear dependence of the temperature on the altitude The adopted value of the tem-(13)The reader in need of a short reminder on thermodynamics is referred to Appendix A(14)See eg httpsenwikipediaorgwikiInternational_Standard_Atmosphere

perature gradient in the troposphere is smaller than the above namely 65 K middot kmminus1 to take intoaccount the possible condensation of water vapor into droplets or even ice

lowast Coming back to the derivation of relation (IV5) if we had not assumed sn constant we wouldhave found

ρ(~r)~nablaP (~r) = ~nabla

[w(~r)

ρ(~r)

]minus T (~r) ~nabla

[s(~r)

ρ(~r)

] (IV6)

which we shall use in Sec IV21

IV14 Archimedesrsquo principle

Consider now a fluid or a system of several fluids at rest occupying some region of space LetS be a closed control surface inside that region as depicted in Fig IV1 (left) and V be the volumedelimited by S The fluid inside S will be denoted by Σ and that outside by Σprime

fluid 1

fluid 2 S

fluid 1

fluid 2 solid body

Figure IV1 ndash Gedankenexperiment to illustrate Archimedesrsquo principle

The system Σ is in mechanical equilibrium ie the sum of the gravity forces acting at each pointof the volume V and the pressure forces exerted at each point of S by the fluid Σprime must vanish

bull The gravity forces at each point result in a single force ~FG applied at the center of mass Gof Σ whose direction and magnitude are those of the weight of the system Σ

bull According to the equilibrium condition the resultant of the pressure forces must equal minus~FG∮S

P (~r) d2~S = minus~FG

If one now replaces the fluid system Σ by a (solid) body B while keeping the fluids Σprime outsideS in the same equilibrium state the mechanical stresses inside Σprime remain unchanged Thus theresultant of the contact forces exerted by Σprime on B is still given by ~F = minus~FG and still applies at thecenter of mass G of the fluid system Σ This constitutes the celebrated Archimedes principle

Any object wholly or partially immersed in a fluid is buoyed up by a force equalto the weight of the fluid displaced by the object (IV7)

In addition we have obtained the point of application of the resultant force (ldquobuoyancyrdquo(xlv)) fromthe fluid

Remark If the center of mass G of the ldquoremovedrdquo fluid system does not coincide with the center ofmass of the body B the latter will be submitted to a torque since ~F and its weight are applied attwo different points

(xlv)statischer Auftrieb

IV2 Steady inviscid flowsWe now turn to stationary solutions of the equations of motion for perfect fluids all partial timederivatives vanishmdashand accordingly we shall drop the t variablemdash yet the flow velocity~v(~r) may nowbe non-zero Under those conditions the equations (III18) and (III33) expressing the conservationsof momentum and energy collapse onto a single equation (Sec IV21) Some applications of thelatter in the particular case of an incompressible fluid are then presented (Sec IV22)

IV21 Bernoulli equation

Replacing in the Euler equation (III20) the pressure term with the help of relation (IV6) andthe acceleration due to volume forces by its expression in term of the potential energy per unitmass one finds

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

]minus ~v(t~r)times ~ω(t~r) = T (t~r) ~nabla

[s(t~r)

ρ(t~r)

]minus ~nabla

[w(t~r)

ρ(t~r)

]minus ~nablaΦ(t~r) (IV8)

which is rather more clumsy than the starting point (III20) due to the many thermodynamicquantities it involves on its right hand side

Gathering all gradient terms together one obtains

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

2+w(t~r)

ρ(t~r)+ Φ(t~r)

]= ~v(t~r)times ~ω(t~r) + T (t~r) ~nabla

[s(t~r)

ρ(t~r)

] (IV9)

In the stationary regime the first term on the left-hand side disappears(15)mdashand we now omit thetime variable from the equations

Let d~(~r) denote a vector tangential to the streamline at position ~r ie parallel to ~v(~r) Whenconsidering the scalar product of d~(~r) with Eq (IV9) both terms on the right hand side yield zeroFirst the mixed product d~(~r) middot [~v(~r) times ~ω(~r)] is zero for it involves two collinear vectors Secondd~(~r) middot ~nabla[s(~r)ρ(~r)] vanishes due to the conservation of sn in flows of perfect fluids which togetherwith the stationarity reads~v(~r) middot ~nabla[s(~r)n(~r)] = 0 where n is proportional to ρ

On the other hand d~(~r) middot ~nabla represents the derivative along the direction of d~ ie along thestreamline at ~r Thus the derivative of the term in squared brackets on the left hand side ofEq (IV9) vanishes along a streamline ie the term remains constant on a streamline

~v(~r)2

2+w(~r)

ρ(~r)+ Φ(~r) = constant along a streamline (IV10)

where the value of the constant depends on the streamline Relation (IV10) is referred as to theBernoulli equation(m)

In the stationary regime the energy conservation equation (III33) in which one recognizes theenthalpy density w(~r) = e(~r) + P (~r) in the flux term leads to the same relation (IV10)

The first term in Eq (III33) vanishes due to the stationarity condition leaving (we drop thevariables)

~nabla middot[(

ρ+ Φ

Applying the product rule to the left member one finds a first term proportional to ~nabla middot (ρ~v)mdashwhich vanishes thanks to the continuity equation (III9)mdash leaving only the other term whichis precisely ρ times the derivative along~v of the left hand side of the Bernoulli equation

(15)This yields a relation known as Croccorsquos theorem(xlvi)(l)

(xlvi)Croccos Wirbelsatz(l)L Crocco 1909ndash1986 (m)D Bernoulli 1700ndash1782

IV2 Steady inviscid flows 49

Bernoulli equation in particular cases

Coming back to Eq (IV9) if the steady flow is irrotational ie ~ω(~r) = ~0 everywhere andisentropic ie s(~r)n(~r) is uniform then the gradient on the left hand side vanishes That isthe constant in the Bernoulli equation (IV10) is independent of the streamline ie it is the sameeverywhere

In case the flow is incompressible ie ~nabla middot~v(~r) = 0 then the continuity equation shows that themass density ρ becomes uniform throughout the fluid One may then replace pull the factor 1ρinside the pressure gradient in the Euler equation (III20) Repeating then the same steps as belowEq (IV9) one finds that the Bernoulli equation now reads

In incompressible flows~v(~r)2

P (~r)

ρ+ Φ(~r) is constant along a streamline (IV11)

This is the form which we shall use in the applications hereafter

Can this form be reconciled with the other one (IV10) which is still what follows from theenergy conservation equation Subtracting one from the other one finds that the ratio e(~r)ρis constant along streamlines That is since ρ is uniform the internal energy density is con-stant along pathlinesmdashwhich coincide with streamlines in a steady flow Now thermodynamicsexpresses the differential de through ds and dn since both entropy and particle number areconserved along a pathline so is internal energy ie Eq (IV10) is compatible with Eq (IV11)

IV22 Applications of the Bernoulli equation

Throughout this Section we assume that the flow is incompressible ie the mass density isuniform and rely on Eq (IV11) Of course one may release this assumption in which case oneshould replace pressure by enthalpy density everywhere below(16)

IV22 a

Drainage of a vessel Torricellirsquos law

Consider a liquid contained in a vessel with a small hole at its bottom through which the liquidcan flow (Fig IV2)

Figure IV2

At points A and B which lie on the same streamline the pressurein the liquid equals the atmospheric pressure(17) PA = PB = P 0The Bernoulli equation (at constant pressure) then yields

2+ gzA =

2+ gzB

with zA resp zB the height of point A resp B ie

v2B = v2

A + 2gh

If the velocity at point A vanishes one finds Torricellirsquos law (xlvii)(n)

vB =radic

That is the speed of efflux is the same as that acquired by a body in free fall from the same heighth in the same gravity field(16)The author confesses that he has a better physical intuition of pressure than of enthalpy hence his parti pris(17)One can show that the pressure in the liquid at point B equals the atmospheric pressure provided the local

streamlines are parallel to each othermdashthat is the flow is laminar(xlvii)Torricellis Theorem(n)E Torricelli 1608ndash1647

Remark To be allowed to apply the Bernoulli equation one should first show that the liquid flowssteadily If the horizontal cross section of the vessel is much larger than the aperture of the holeand h large enough this holds to a good approximation

IV22 b

Venturi effect

Consider now the incompressible flow of a fluid in the geometry illustrated in Fig IV3 As weshall only be interested in the average velocity or pressure of the fluid across a cross section of thetube the flow is effectively one-dimensional

s-v1 -v2

Figure IV3

The conservation of the mass flow rate in the tube which represents the integral formulation ofthe continuity equation (III9) leads to ρSv1 = ρs v2 ie v2 = (Ss)v1 gt v1 with S resp s the areaof the tube cross section in its broad resp narrow sectionOn the other hand the Bernoulli equation at constant height and thus potential energy gives

All in all the pressure in the narrow section is thus smaller than in the broad section P 2 lt P 1which constitutes the Venturi effect (o)

Using mass conservation and the Bernoulli equation one can express v1 or v2 in terms of thetube cross section areas and the pressure difference For instance the mass flow rate reads

P 1minusP 2

s2minus 1

IV22 c

Pitot tube

Figure IV4 represents schematically the flow of a fluid around a Pitot tube(p) which is a deviceused to estimate a flow velocity through the measurement of a pressure difference Three streamlinesare shown starting far away from the Pitot tube where the flow is (approximately) uniform andhas the velocity~v which one wants to measure The flow is assumed to be incompressible

Obull bullIbull

OprimebullA

-manometer

-bullAprime

Figure IV4 ndash Flow around a Pitot tube

The Pitot tube consists of two long thin concentric tubes

bull Despite the presence of the hole at the end point I the flow does not penetrate in the innertube so that~vI = ~0 to a good approximation

(o)G B Venturi 1746ndash1822 (p)H Pitot 1695ndash1771

bull In the broader tube there is a hole at a point A which is far enough from I to ensure that thefluid flow in the vicinity of A is no longer perturbed by the extremity of the tube ~vA =~vAprime ~vwhere the second identity holds thanks to the thinness of the tube which thus perturbs theflow properties minimally In addition the pressure in the broader tube is uniform so thatP = PB

If one neglects the height differencesmdashwhich is a posteriori justified by the numerical values we shallfindmdashthe (incompressible) Bernoulli equation gives first

PO + ρ~v2

along the streamline OI andPOprime + ρ

2= PAprime + ρ

~v2Aprime

2along the streamline OprimeAprime Using POprime PO PAprime PA and ~vAprime ~v the latter identity leads toPO PA = PB One thus finds

PI minus PB = ρ~v2

so that a measurement of PI minus PB gives an estimate of |~v|

For instance in air (ρ sim 13 kg middotmminus3) a velocity of 100 m middot sminus1 results in a pressure difference of65times 103 Pa = 65times 10minus2 atm With a height difference h of a few centimeters between O and Aprimethe neglected term ρgh is of order 1 Pa

Remarks

lowast The flow of a fluid with velocity~v around a motionless Pitot tube is equivalent to the motion ofa Pitot tube with velocity minus~v in a fluid at rest Thus Pitot tubes are used to measure the speed ofairplanes

lowast Is the flow of air really incompressible at velocities of 100 m middot sminus1 or higher Not really since theMach number (II16) becomes larger than 03 In practice one thus rather uses the ldquocompressiblerdquoBernoulli equation (IV10) yet the basic principles presented above remain valid

IV22 d

Magnus effect

Consider an initially uniform and steady flow with velocity ~v0 One introduces in it a cylinderwhich rotates about its axis with angular velocity ~ωC perpendicular to the flow velocity (Fig IV5)

~v0~ωC

Figure IV5 ndash Fluid flow around a rotating cylinder

Intuitively one can expect that the cylinder will drag the neighboring fluid layers along in itsrotation(18) In that case the fluid velocity due to that rotation will add up to resp be subtractedfrom the initial flow velocity in the lower resp upper region close to the cylinder in Fig IV5(18)Strictly speaking this is not true in perfect fluids only in real fluids with friction Nevertheless the tangential

forces due to viscosity in the latter may be small enough that the Bernoulli equation remains approximately validas is assumed here

Invoking now the Bernoulli equationmdashin which the height difference between both sides of thecylinder is neglectedmdash the pressure will be larger above the cylinder than below it Accordinglythe cylinder will experience a resulting force directed downwardsmdashmore precisely it is proportionalto~v0 times ~ωCmdash which constitutes the Magnus effect (q)

IV3 Vortex dynamics in perfect fluidsWe now turn back to the case of an arbitrary flow ~v(t~r) still in the case of a perfect fluid Thevorticity vector field defined as the rotational curl of the flow velocity field was introduced inSec II12 together with the vorticity lines Modulo a few assumptions on the fluid equation ofstate and the volume forces one can show that vorticity is ldquofrozenrdquo in the flow of a perfect fluidin the sense that there the flux of vorticity across a material surface remains constant as the latteris being transported This behavior will be investigated and formulated both at the integral level(Sec IV31) and differentially (Sec IV32)

IV31 Circulation of the flow velocity Kelvinrsquos theorem

Definition Let ~γ(t λ) be a closed curve parametrized by a real number λ isin [0 1] which is beingswept along by the fluid in its motion The integral

Γ~γ(t) equiv∮~γ

~v(t ~γ(t λ)) middot d~ (IV12)

is called the circulation around the curve of the velocity field

Remark According to Stokesrsquo theorem(19) if the area bounded by the contour ~γ(t λ) is simplyconnected Γ~γ(t) equals the surface integralmdashthe ldquofluxrdquomdashof the vorticity field over every surfaceS~γ(t) delimited by ~γ

Γ~γ(t) =

intS~γ

[~nablatimes~v(t~r)

]middot d2~S =

intS~γ~ω(t~r) middot d2~S (IV13)

Stated differently the vorticity field is the flux density of the circulation of the velocityThis relationship between circulation and vorticity will be further exploited hereafter we shall

now establish and formulate results at the integral level namely for the circulation which will thenbe expressed at the differential level ie in terms of the vorticity in Sec IV32

Many results take a simpler form in a so-called barotropic fluid (xlviii) in which the pressure canbe expressed as function of only the mass density P = P (ρ) irrespective of whether the fluid isotherwise perfect or dissipative An example of such a result isKelvinrsquos circulation theorem(r)

In a perfect barotropic fluid with conservative volume forces the circulation ofthe flow velocity around a closed curve comoving with the fluid is conserved (IV14a)

Denoting by ~γ(t λ) the closed contour in the theorem

DΓ~γ(t)

Dt= 0 (IV14b)

This result is also sometimes called Thomsonrsquos theorem

(19)which in its classical form used here is also known as KelvinndashStokes theorem(xlviii)barotropes Fluid(q)G Magnus 1802ndash1870 (r)W Thomson Baron Kelvin 1824ndash1907

IV3 Vortex dynamics in perfect fluids 53

Proof For the sake of brevity the arguments of the fields are omitted in case it is not necessaryto specify them Differentiating definition (IV12) first gives

DΓ~γDt

part~γ(t λ)

partλmiddot~v(t ~γ(t λ)) dλ =

[part2~γ

partλ parttmiddot~v +

part~γ

partλmiddot(part~v

partt+sumi

part~v

partxipartγi

Since the contour ~γ(t λ) flows with the fluidpart~γ(t λ)

partt=~v(t ~γ(t λ)) which leads to

DΓ~γDt

part~v

partλmiddot ~v +

part~γ

partλmiddot[part~v

The first term in the curly brackets is clearly the derivative with respect to λ of ~v22 so thatits integral along a closed curve vanishes The second term involves the material derivative of~v as given by the Euler equation Using Eq (III19) with ~aV = minus~nablaΦ leads to

DΓ~γDt

(minus~nablaPρminus ~nablaΦ

)middot part~γpartλ

Again the circulation of the gradient ~nablaΦ around a closed contour vanishes leaving

DΓ~γ(t)

Dt= minus

∮~γ

~nablaP (t~r)

ρ(t~r)middot d~ (IV15)

which constitutes the general case of Kelvinrsquos circulation theorem for perfect fluids with conser-vative volume forcesTransforming the contour integral with Stokesrsquo theorem yields the surface integral of

~nablatimes(~nablaP

)=~nablatimes ~nablaP

ρ+~nablaP times ~nablaρ

ρ2=~nablaP times ~nablaρ

ρ2 (IV16)

In a barotropic fluid the rightmost term of this identity vanishes since ~nablaP and ~nablaρ are collinearwhich yields relation (IV14)

Remark Using relation (IV13) and the fact that the area S~γ(t) bounded by the curve ~γ at time tdefines a material surface which will be transported in the fluid motion Kelvinrsquos theorem (IV14)can be restated as

In a perfect barotropic fluid with conservative volume forces the flux of thevorticity across a material surface is conserved (IV17)

Kelvinrsquos theorem leads to two trivial corollaries namely

Helmholtzrsquos theorem(s)

In the flow of a perfect barotropic fluid with conservative volume forces thevorticity lines and vorticity tubes move with the fluid (IV18)

Similar to the definition of stream tubes in Sec I33 a vorticity tube is defined as the surfaceformed by the vorticity lines tangent to a given closed geometrical curveAnd in the case of vanishing vorticity ~ω = ~0 one has

Lagrangersquos theorem

In a perfect barotropic fluid with conservative volume forces if the flow isirrotational at a given instant t it remains irrotational at later times (IV19)

Kelvinrsquos circulation theorem (IV14) and its corollaries imply that vorticity cannot be creatednor destroyed in the flow of a perfect barotropic fluid with conservative volume forces However(s)H von Helmholtz 1821ndash1894

the more general form (IV15) already show that in a non-barotropic fluid there is a ldquosourcerdquo forvorticity which leads to the non-conservation of the circulation of the flow velocity Similarly non-conservative forcesmdashfor instance a Coriolis force in a rotating reference framemdashmay contribute anon-vanishing term in Eq (IV15) leading to a change in the circulation We shall see in Sec VI5that viscous stresses also affect the transport of vorticity in a fluid

IV32 Vorticity transport equation in perfect fluids

Consider the Euler equation (III20) in the case of conservative volume forces ~aV = minus~nablaΦTaking the rotational curl of both sides yields after some straightforward algebra

part~ω(t~r)

parttminus ~nablatimes

[~v(t~r)times ~ω(t~r)

]= minus

~nablaP (t~r)times ~nablaρ(t~r)

ρ(t~r)2 (IV20)

This relation can be further transformed using the identity (we omit the variables)

~nablatimes(~v times ~ω

)=(~ω middot ~nabla

(~nabla middot ~ω

)~v minus

(~v middot ~nabla

)~ω minus

(~nabla middot~v

Since the divergence of the vorticity field ~nabla middot ~ω(t~r) vanishes the previous two equations yield

part~ω(t~r)

]~ω(t~r)minus

[~ω(t~r) middot ~nabla

]~v(t~r) = minus

]~ω(t~r)minus

ρ(t~r)2

(IV21)While it is tempting to introduce the material derivative D~ωDt on the left hand side of thisequation for the first two terms we rather define the whole left member to be a new derivative

D~v ~ω(t~r)

Dtequiv part~ω(t~r)

]~ω(t~r)minus

]~v(t~r) (IV22a)

or equivalentlyD~v ~ω(t~r)

Dtequiv D~ω(t~r)

Dtminus[~ω(t~r) middot ~nabla

]~v(t~r) (IV22b)

We shall refer to D~v Dt as the comoving time derivative for reasons that will be explained at theend of this Section

Using this definition Eq (IV21) reads

D~v ~ω(t~r)

Dt= minus

]~ω(t~r)minus

ρ(t~r)2 (IV23)

In the particular of a barotropic fluidmdashrecall that we also assumed that it is ideal and only hasconservative volume forcesmdashthis becomes

D~v ~ω(t~r)

Dt= minus

]~ω(t~r) (IV24)

Thus the comoving time-derivative of the vorticity is parallel to itselfFrom Eq (IV24) one deduces at once that if ~ω(t~r) vanishes at some time t it remains zeromdash

which is the differential formulation of corollary (IV19)

Invoking the continuity equation (III9) the volume expansion rate ~nabla middot~v on the right hand sideof Eq (IV24) can be replaced by minus(1ρ)DρDt For scalar fields material derivative and comovingtime-derivative coincide which leads to the compact form

[~ω(t~r)

ρ(t~r)

]= ~0 (IV25)

for perfect barotropic fluids with conservative volume forces That is anticipating on the discussion

of the comoving time derivative hereafter ~ωρ evolves in the fluid flow in the same way as theseparation between two material neighboring points the ratio is ldquofrozenrdquo in the fluid evolution

Comoving time derivative

To understand the meaning of the comoving time derivative D~v Dt let us come back to Fig II1depicting the positions at successive times t and t+δt of a small material vector δ~(t) By definitionof the material derivative the change in δ~ between these two instantsmdashas given by the trajectoriesof the two material points which are at ~r resp ~r + δ~(t) at time tmdashis

δ~(t+δt)minus δ~(t) =Dδ~(t)

On the other hand displacing the origin of δ~(t+δt) to let it coincide with that of δ~(t) one sees

δ~(t)δ~(t+ δt)

~v(t~r

+ δ~ (t)) δt

~v(t~r)δt

[δ~(t)middot~nabla

]~v(t~r)δt

Figure IV6 ndash Positions of a material line element δ~ at successive times t and t+ δt

on Fig IV6 that this change equals

δ~(t+δt)minus δ~(t) =[δ~(t)middot~nabla

]~v(t~r)δt

Equating both results and dividing by δt one findsDδ~(t)

Dt=[δ~(t)middot~nabla

]~v(t~r) ie precisely

D~vδ~(t)Dt

= ~0 (IV26)

Thus the comoving time derivative of a material vector which moves with the fluid vanishes Inturn the comoving time derivative at a given instant t of an arbitrary vector measures its rate ofchange with respect to a material vector with which it coincides at time t

This interpretation suggestsmdashthis can be proven more rigorouslymdashwhat the action of the co-moving time derivative on a scalar field should be In that case D~v Dt should coincide withthe material derivative which already accounts for all changesmdashdue to non-stationarity and con-vective transportmdashaffecting material points in their motion This justifies a posteriori our usingD~v ρDt = DρDt above

More generally the comoving time derivative introduced in Eq (IV22a) may be rewritten as

( middot ) equiv part

partt( middot ) + L~v( middot ) (IV27)

where L~v denotes the Lie derivative along the velocity field ~v(~r) whose action on an arbitrary

vector field ~ω(~r) is precisely (time plays no role here)

L~v ~ω(~r) equiv[~v(~r) middot ~nabla

]~ω(~r)minus

[~ω(~r) middot ~nabla

]~v(~r)

while it operates on an arbitrary scalar field ρ(~r) according to

L~v ρ(~r) equiv[~v(~r) middot ~nabla

]ρ(~r)

More information on the Lie derivative including its operation on 1-forms or more generallyon(mn

)-tensorsmdashfrom which the action of the comoving time derivative followsmdash can be found

eg in Ref [14 Chap 31ndash35]

IV4 Potential flowsAccording to Lagrangersquos theorem (IV19) every flow of a perfect barotropic fluid with conservativevolume forces which is everywhere irrotational at a given instant remains irrotational at every time

Focusing accordingly on the incompressible and irrotational motion of an ideal fluid with con-servative volume forces which is also referred to as a potential flow (xlix) the dynamical equationscan be recast such that the main one is a linear partial differential equation for the velocity potential(Sec IV41) for which there exist mathematical results (Sec IV42) Two-dimensional potentialflows are especially interesting since one may then introduce a complex velocity potentialmdashandthe corresponding complex velocitymdash which is a holomorphic function (Sec IV43) This allowsone to use the full power of complex analysis so as to devise flows around obstacles with variousgeometries by combining ldquoelementaryrdquo solutions and deforming them

IV41 Equations of motion in potential flows

Using a known result from vector analysis a vector field whose curl vanishes everywhere on asimply connected domain of R3 can be written as the gradient of a scalar field Thus in the caseof an irrotational flow ~nablatimes~v(t~r) = ~0 the velocity field can be expressed as

~v(t~r) = minus~nablaϕ(t~r) (IV28)

with ϕ(t~r) the so-called velocity potential (l)

Remarks

lowast The minus sign in definition (IV28) is purely conventional While the choice adopted here isnot universal it has the advantage of being directly analogous to the convention in electrostatics( ~E = minus~nablaΦCoul) or Newtonian gravitation physics (~g = minus~nablaΦNewt)

lowast Since Lagrangersquos theorem does not hold in a dissipative fluid in which vorticity can be createdor annihilated (Sec VI5) the rationale behind the definition of the velocity potential disappears

Using the velocity potential (IV28) and the relation ~aV = minus~nablaΦ expressing that the volumeforces are conservative the Euler equation (III20) reads

minuspart~nablaϕ(t~r)

partt+ ~nabla

[~nablaϕ(t~r)

+ Φ(t~r)

= minus 1

ρ(t~r)~nablaP (t~r)

Assuming that the flow is also incompressible and thus ρ constant this becomes

minus part~nablaϕ(t~r)

partt+ ~nabla

[~nablaϕ(t~r)

+P (t~r)

ρ+ Φ(t~r)

= ~0 (IV29)

(xlix)Potentialstroumlmung (l)Geschwindigkeitspotential

IV4 Potential flows 57

or equivalently

minuspartϕ(t~r)

partt+

[~nablaϕ(t~r)

+P (t~r)

ρ+ Φ(t~r) = C(t) (IV30)

where C(t) denotes a function of time onlyEventually expressing the incompressibility condition [cf Eq (II13)] ~nablamiddot~v(t~r) = 0 leads to the

Laplace equation(t)

4ϕ(t~r) = 0 (IV31)

for the velocity potential ϕ

Equations (IV28) (IV30) and (IV31) are the three equations of motion governing potentialflows Since the Laplace equation is partial differential it is still necessary to specify the corre-sponding boundary conditions

In agreement with the discussion in sect III32 c there are two types of condition at walls orobstacles which are impermeable to the fluid and ldquoat infinityrdquomdashfor a flow in an unbounded domainof spacemdash where the fluid flow is generally assumed to be uniform Choosing a proper referenceframe R this uniform motion of the fluid may be turned into having a fluid at rest Denoting byS(t) the material surface associated with walls or obstacles which are assumed to be moving withvelocity ~vobs in R and by ~en(t~r) the unit normal vector to S(t) at a given point ~r the conditionof vanishing relative normal velocity reads

minus~en(t~r) middot ~nablaϕ(t~r) = ~en(t~r) middot ~vobs(t~r) on S(t) (IV32a)

In turn the condition of rest at infinity reads

ϕ(t~r) sim|~r|rarrinfin

K(t) (IV32b)

where the scalar function K(t) will in practice be given

Remarks

lowast Since the Laplace equation (IV31) is linearmdashthe non-linearity of the Euler equation is inEq (IV30) which is ldquotrivialrdquo once the spatial dependence of the velocity potential has beendeterminedmdash it will be possible to superpose the solutions of ldquosimplerdquo problems to obtain thesolution for a more complicated geometry

lowast In potential flows the dependences on time and space are somewhat separated The Laplaceequation (IV31) governs the spatial dependence of ϕ and thus~v meanwhile time enters the bound-ary conditions (IV32) thus is used to ldquonormalizerdquo the solution of the Laplace equation In turnwhen ϕ is known relation (IV30) gives the pressure field where the integration ldquoconstantrdquo C(t)will also be fixed by boundary conditions

IV42 Mathematical results on potential flows

The boundary value problem consisting of the Laplace differential equation (IV31) together withthe boundary conditions on normal derivatives (IV32) is called a Neumann problem(u) or boundaryvalue problem of the second kind For such problems results on the existence and unicity of solutionshave been established which we shall now state without further proof(20)

(20)The Laplace differential equation is dealt with in many textbooks as eg in Ref [15 Chapters 7ndash9] [16 Chapter 4]or [17 Chapter VII]

(t)P-S (de) Laplace 1749ndash1827 (u)C Neumann 1832ndash1925

IV42 a

Potential flows in simply connected regions

The simplest case is that of a potential flow on a simply connected domain D of space D maybe unbounded provided the condition at infinity is that the fluid be at rest Eq (IV32b)

On a simply connected domain the Neumann problem (IV31)ndash(IV32) for the velocitypotential admits a solution ϕ(t~r) which is unique up to an additive constantIn turn the flow velocity field ~v(t~r) given by relation (IV28) is unique

(IV33)

For a flow on a simply connected region the relation (IV28) between the flow velocity and itspotential is ldquoeasilyrdquo invertible fixing some reference position ~r0 in the domain one may write

ϕ(t~r) = ϕ(t~r0)minusint~γ

~v(t~rprime) middot d~(~rprime) (IV34)

where the line integral is taken along any path ~γ on D connecting the positions ~r0 and ~r

That the line integral only depends on the path extremities ~r0 ~r not on the path itself isclearly equivalent to Stokesrsquo theorem stating that the circulation of velocity along any closedcontour in the domain D is zeromdashit equals the flux of the vorticity which is everywhere zerothrough a surface delimited by the contour and entirely contained in D

Thus ϕ(t~r) is uniquely defined once the value ϕ(t~r0) which is the arbitrary additive constantmentioned above has been fixed

This reasoning no longer holds in a multiply connected domain as we now further discuss

IV42 b

Potential flows in doubly connected regions

As a matter of fact in a doubly (or a fortiori multiply) connected domain there are by definitionnon-contractible closed paths Considering for instance the domain D traversed by an infinitecylindermdashwhich is not part of the domainmdashof Fig IV7 the path going from ~r0 to ~r2 along ~γ0rarr2

then coming back to ~r0 along ~γ prime0rarr2

(21) cannot be continuously shrunk to a point without leaving D This opens the possibility that the line integral in relation (IV34) depend on the path connectingtwo points

bull~r0

bull~r1

bull~r2

-~γ0rarr1

-~γ prime

0rarr1

6~γ0rarr2

6~γ primeprime

0rarr2

6~γ prime

0rarr2

Figure IV7

In a doubly connected domain D there is only a single ldquoholerdquo that prevents closed paths frombeing homotopic to a point ie contractible Let Γ(t) denote the circulation at time t of thevelocity around a closed contour with a given ldquopositiverdquo orientation circling the hole once Oneeasily checksmdasheg invoking Stokesrsquo theoremmdashthat this circulation has the same value for all closed(21)More precisely if ~γ prime

0rarr2is parameterized by λ isin [0 1] when going from ~r0 to ~r2 a path from ~r2 to ~r0 with the

same geometric supportmdashwhich is what is meant by ldquocoming back along ~γ prime0rarr2

rdquomdashis λ 7rarr ~γ prime0rarr2

(1minus λ)

paths with the same orientation going only once around the hole since they can be continuouslydeformed into each other without leaving D Accordingly the ldquouniversalrdquo circulation Γ(t) is alsoreferred to as cyclic constant (li) of the flow

More generally the circulation at time t of the velocity around a closed curve circling the holen times and oriented in the positive resp negative direction is nΓ(t) resp minusnΓ(t)

Going back to the line integral in Eq IV34 its value will generally depend on the path ~γ from~r0 to ~rmdashor more precisely on the class defined by the number of loops around the hole of thepath Illustrating this idea on Fig IV7 while the line integral from ~r0 to ~r2 along the path ~γ0rarr2

will have a given value I the line integral along ~γ prime0rarr2

will differ by one (say positive) unit of Γ(t)and be equal to I+Γ(t) In turn the integral along ~γ primeprime

0rarr2 which makes one more negatively oriented

loop than ~γ0rarr2 around the cylinder takes the value I minus Γ(t)These preliminary discussions suggest that if the Neumann problem (IV31)ndash(IV32) for the

velocity potential on a doubly connected domain admits a solution ϕ(t~r) the latter will not bea scalar function in the usual sense but rather a multivalued function whose various values at agiven position ~r at a fixed time t differ by an integer factor of the cyclic constant Γ(t)

All in all the following result holds provided the cyclic constant Γ(t) is known ie if its valueat time t is part of the boundary conditions

On a doubly connected domain the Neumann problem (IV31)ndash(IV32) for the velocitypotential with given cyclic constant Γ(t) admits a solution ϕ(t~r) which is uniqueup to an additive constant The associated flow velocity field ~v(t~r) is unique

(IV35)

The above wording does not specify the nature of the solution ϕ(t~r)

bull if Γ(t) = 0 in which case the flow is said to be acyclic the velocity potential ϕ(t~r) is aunivalued function

bull if Γ(t) 6= 0 ie in a cyclic flow the velocity potential ϕ(t~r) is a multivalued function of itsspatial argument Yet as the difference between the various values at a given ~r is function oftime only the velocity field (IV28) remains uniquely defined

Remarks

lowast Inspecting Eq (IV30) one might fear that the pressure field P (t~r) be multivalued reflectingthe term partϕ(t~r)partt Actually however Eq (IV30) is a first integral of Eq (IV29) in which the~r-independent multiples of Γ(t) distinguishing the multiple values of ϕ(t~r) disappear when thegradient is taken That is the term partϕ(t~r)partt is to be taken with a grain of salt since in fact itdoes not contain Γ(t) or its time derivative

lowast In agreement with the first remark the reader should remember that the velocity potential ϕ(t~r)is just a useful auxiliary mathematical function(22) yet the physical quantity is the velocity itselfThus the possible multivaluedness of ϕ(t~r) is not a real physical problem

(22)Like its cousins gravitational potential ΦNewt electrostatic potential ΦCoul magnetic vector potential ~A (li)zyklische Konstante

IV43 Two-dimensional potential flows

We now focus on two-dimensional potential flows for which the velocity fieldmdashand all otherfieldsmdashonly depend on two coordinates The latter will either be Cartesian coordinates (x y) whichare naturally combined into a complex variable z = x+ iy or polar coordinates (r θ) Throughoutthis Section the time variable t will not be denoted apart from possibly influencing the boundaryconditions it plays no direct role in the determination of the velocity potential

IV43 a

Complex flow potential and complex flow velocity

Let us first introduce a few useful auxiliary functions which either simplify the description oftwo-dimensional potential flows or allow one to ldquogeneraterdquo such flows at will

Stream functionIrrespective of whether the motion is irrotational or not in an incompressible two-dimensional

flow one can define a unique (up to an additive constant) stream function(lii) ψ(x y) such that

vx(x y) = minuspartψ(x y)

party vy(x y) =

partψ(x y)

partx(IV36)

at every point (x y) Indeed when the above two relations hold the incompressibility criterion~nabla middot~v(x y) = 0 is fulfilled automatically

Remark As in the case of the relation between the flow velocity field and the corresponding potentialEq (IV28) the overall sign in the relation between~v(~r) and ψ(~r) is conventional Yet if one wishesto define the complex flow potential as in Eq (IV39) below the relative sign of ϕ(~r) and ψ(~r) isfixed

The stream function for a given planar fluid motion is such that the lines along which ψ(~r) isconstant are precisely the streamlines of the flow

Let d~x(λ) denote a differential line element of a curve ~x(λ) of constant ψ(~r) ie a curve alongwhich ~nablaψ = ~0 Then d~x(λ) middot ~nablaψ

(~x(λ)

)= 0 at every point on the line using relations (IV36)

one recovers Eq (I15b) characterizing a streamline

Stream functions are also defined in three-dimensional flows yet in that case two of them areneeded More precisely one can find two linearly independent functions ψ1(~r) ψ2(~r) suchthat the streamlines are the intersections of the surfaces of constant ψ1 and of constant ψ2That is they are such that the flow velocity obeys ~v(~r) prop ~nablaψ1(~r) times ~nablaψ2(~r) with an a prioriposition-dependent proportionality factormdashwhich can be taken identically equal to unity in anincompressible flow

Consider now a potential flow ie which is not only incompressible but also irrotational Forsuch a two-dimensional flow the condition of vanishing vorticity reads

ωz(x y) =partvy(x y)

partxminus partvx(x y)

party= 0

which under consideration of relations (IV36) gives

4ψ(x y) = 0 (IV37a)

at every point (x y) That is the stream function obeys the Laplace equationmdashjust like the velocitypotential ϕ(~r)

A difference with ϕ(~r) arises with respect to the boundary conditions At an obstacle or wallsmodeled by a ldquosurfacerdquo Smdashin the plane R2 this surface is rather a curvemdash the impermeabilitycondition implies that the velocity is tangential to S ie S coincides with a streamline

ψ(x y) = constant on S (IV37b)

(lii)Stromfunktion

For a flow on an unbounded domain the velocity is assumed to be uniform at infinity~v(x y)rarr~vinfinwhich is the case if

ψ(x y) sim|~r|rarrinfin

vyinfin xminus vxinfin y (IV37c)

with vxinfin vyinfin the components of~vinfin

The boundary conditions (IV37b)ndash(IV37c) on the stream function are thus dissimilar from thecorresponding conditions (IV32a)ndash(IV32b) on the velocity potential In particular the conditionat an obstacle involves the stream function itself instead of its derivative the Laplace differentialequation (IV37a) with conditions (IV37b)ndash(IV37c) represents a Dirichlet problem(v) or boundaryvalue problem of the first kind instead of a Neumann problem

Complex flow potentialIn the case of a two-dimensional potential flow both the velocity potential φ(x y) and the stream

function ψ(x y) are so-called harmonic functions ie they are solutions to the Laplace differentialequation see Eqs (IV31) and (IV37a) In addition gathering Eqs (IV28) and (IV36) one seesthat they satisfy at every point (x y) the identities

partφ(x y)

partx=partψ(x y)

[= minusvx(x y)

partφ(x y)

party= minuspartψ(x y)

[= minusvy(x y)

] (IV38)

The relations between the partial derivatives of φ and ψ are precisely the CauchyndashRiemann equationsobeyed by the corresponding derivatives of the real and imaginary parts of a holomorphic functionof a complex variable z = x + iy That is the identities (IV38) suggest the introduction of acomplex (flow) potential

φ(z) equiv ϕ(x y) + iψ(x y) with z = x+ iy (IV39)

which will automatically be holomorphic on the domain where the flow is defined The functions ϕand ψ are then said to be conjugate to each other In line with that notion the curves in the planealong which one of the functions is constant are the field lines of the other and reciprocally

Besides the complex potential φ(z) one also defines the corresponding complex velocity as thenegative of its derivative namely

w(z) equiv minusdφ(z)

dz= vx(x y)minus ivy(x y) (IV40)

where the second identity follows at once from the definition of φ and the relations between ϕ or ψand the flow velocity Like φ(z) the complex velocity w(z) is an analytic function of z

IV43 b

Elementary two-dimensional potential flows

As a converse to the above construction of the complex potential the real and imaginary partsof any analytic function of a complex variable are harmonic functions ie any analytical functionφ(z) defines a two-dimensional potential flow on its domain of definition Accordingly we nowinvestigate a few ldquobasicrdquo complex potentials and the flows they describe

Uniform flowThe simplest possibility is that of a linear complex potential

φ(z) = minusv eminusiαz with v isin R α isin R (IV41)

(v)P G (Lejeune-)Dirichlet 1805ndash1859

3333333

33333 33

33333333

333333

Figure IV8

Using for instance Eq (IV40) this trivially leads to a uniformvelocity field making an angle α with the x-direction

~v(x y) =(

cosα~ex + sinα~ey)v

as illustrated in Fig IV8 in which a few streamlines are dis-played to which the equipotential lines (not shown) of ϕ(x y)are perpendicular

Flow source or sinkAnother flow with ldquosimplerdquo streamlines is that defined by the complex potential(23)

φ(z) = minus Q2π

log(z minus z0) with Q isin R z0 isin C (IV42a)

The resulting complex flow velocity

w(z) =Q

2π(z minus z0)(IV42b)

has a simple pole at z = z0 Using polar coordinates (r θ) centered on that pole the flow velocityis purely radial

~v(r θ) =Q

2πr~er (IV42c)

as displayed in the left panel of Fig IV9 while the flow potential and the stream function are

ϕ(r θ) = minus Q2π

log r ψ(r θ) = minus Q2π

θ (IV42d)

By computing the flux of velocity through a closed curve circling the polemdasheg a circle centeredon z0 which is an equipotential of ϕmdash one finds that Q represents the mass flow rate through thatcurve If Q is positive there is a source of flow at z0 is Q is negative there is a sink there in whichthe fluid disappears

Figure IV9 ndash Streamlines (full) and equipotential lines (dashed) for a flow source (IV42c)(left) and a pointlike vortex (IV43b) (right)

(23)The reader unwilling to take the logarithm of a dimensionful quantitymdashto which she is entirely entitledmdashmaydivide zminus z0 resp r by a length in the potentials (IV42a) and (IV43a) resp (IV42d) and (IV43c) or write thedifference in Eq (IV45) as the logarithm of a quotient She will however quickly convince herself that this doesnot affect the velocities (IV42b) and (IV43b) nor the potential (IV44a)

Pointlike vortexThe ldquoconjugaterdquo flow to the previous one ie that for which ϕ and ψ are exchanged corresponds

to the complex potential(23)

φ(z) =iΓ

2πlog(z minus z0) with Γ isin R z0 isin C (IV43a)

Using as above polar coordinates (r θ) centered on z0 the flow velocity is purely tangential

~v(r θ) =Γ

2πr2~eθ (IV43b)

as shown in Fig IV9 (right) where the basis vector ~eθ is normalized to r cf Eq (C6) Thecomplex potential (IV43a) thus describes a vortex situated at z0

In turn the velocity potential and stream function read

ϕ(r θ) = minus Γ

2πθ ψ(r θ) =

2πlog r (IV43c)

to be compared with those for a flow source Eq (IV42d)

Remark When writing down the complex velocity potentials (IV42a) or (IV43a) we left aside theissue of the (logarithmic) branch point at z = z0mdashand we did not specify which branch of thelogarithm we consider Now either potential corresponds to a flow that is actually defined on adoubly connected region since the velocity diverges at z = z0 From the discussion in sect IV42 b onsuch domains the potential is a multivalued object yet this is irrelevant for the physical quantitiesnamely the velocity field which remains uniquely defined at each point This is precisely what isillustrated here by the different branches of the logarithm which differ by a constant multiple of2iπ that does not affect the derivative

Flow dipoleA further possible irrotational and incompressible two-dimensional flow is that defined by the

complex potentialφ(z) =

micro eiα

z minus z0with micro isin R α isin R z0 isin C (IV44a)

leading to the complex flow velocity

w(z) =micro eiα

(z minus z0)2 (IV44b)

Again both φ(z) and w(z) are singular at z0Using polar coordinates (r θ) centered on z0 the flow velocity reads

~v(r θ) =micro

r2cos(θ minus α)~er +

r3sin(θ minus α)~eθ (IV44c)

which shows that the angle α gives the overall orientation of the flow with respect to the x-directionSetting for simplicity α = 0 and coming back momentarily to Cartesian coordinates the flow

potential and stream function corresponding to Eq (IV44a) are

ϕ(x y) =microx

x2 + y2 ψ(x y) = minus microy

x2 + y2 (IV44d)

Thus the streamlines are the curves x2 + y2 = consttimes y ie they are circles centered on the y-axisand tangent to the x-axis as represented in Fig IV10 where everything is tilted by an angle α

One can check that the flow dipole (IV44a) is actually the superposition of a pair of infinitelyclose source and sink with the same mass flow rate in absolute value

φ(z) = limεrarr0

[log(z minus z0 + ε eminusiα

)minus log

(z minus z0 minus ε eminusiα

)] (IV45)

Figure IV10 ndash Streamlines for a flow dipole (IV44a) centered on the origin

This is clearly fully analogous to an electric dipole potential being the superposition of the potentialscreated by electric charges +q and minusqmdashand justifies the denomination ldquodipole flowrdquo

One can similarly define higher-order multipoles flow quadrupoles octupoles for whichthe order of the pole of the velocity at z0 increases (order 1 for a source or a sink order 2 for adipole order 3 for a quadrupole and so on)

Remarkslowast The complex flow potentials considered until nowmdashnamely those of uniform flows (IV41) sourcesor sinks (IV42a) pointlike vortices (IV43a) and dipoles (IV44a) or multipolesmdashand their super-positions are the only two-dimensional flows valid on an unbounded domain

As a matter of fact demanding that the flow velocity ~v(~r) be uniform at infinity and that thecomplex velocity w(z) be analytic except at a finite number of singularitiesmdashsay only one at z0to simplify the argumentationmdash then w(z) may be expressed as a superposition of integer powersof 1(z minus z0)

w(z) =

infinsump=0

aminusp(z minus z0)p

(IV46a)

since any positive power of (z minus z0) would be unbounded when |z| rarr infin Integrating over z seeEq (IV40) the allowed complex potentials are of the form

φ(z) = minusa0z minus aminus1 log(z minus z0) +

infinsump=1

p aminuspminus1

(z minus z0)p (IV46b)

lowast Conversely the reader can checkmdashby computing the integral of w(z) along a contour at infinitymdashthat the total mass flow rate and circulation of the velocity field for a given flow are respectivelythe real and imaginary parts of the residue aminus1 in the Laurent series of its complex velocity w(z)ie are entirely governed by the sourcesink term (IV42a) and vortex term (IV43a) in the complexpotential

lowast Eventually the singularities that arise in the flow velocity will in practice not be a problem sincethese points will not be part of the physical flow as we shall see on an example in sect IV43 c

Flow inside or around a cornerAs a last example consider the complex flow potential

φ(z) = A eminusiα(z minus z0)n with A isin R α isin R n ge 1

2 z0 isin C (IV47a)

Figure IV11 ndash Streamlines for the flow defined by potential (IV47a) with from top to bottomand from left to right n = 3 3

35 and 1

Except in the case n = 1 this potential cannot represent a flow on an unbounded domain sinceone easily checks that the velocity is unbounded as |z| goes to infinity The interest of this potentiallies rather the behavior in the vicinity of z = z0

As a matter writing down the flow potential and the stream function in a system of polarcoordinates centered on z0

ϕ(r θ) = Arn cos(nθ minus α) ψ(r θ) = Arn sin(nθ minus α) (IV47b)

shows that they both are (πn)-periodic functions of the polar angle θ Thus the flow on thedomain D delimited by the streamlines ψ(r α) and ψ(r α+πn) is isolated from the motion in theremainder of the complex plane One may therefore assume that there are walls along these twostreamlines and that the complex potential (IV47a) describes a flow between them

For n = 1 one recovers the uniform flow (IV41)mdashin which we are free to put a wall along anystreamline restricting the domain D to a half plane instead of the whole plane If n gt 1 πn issmaller than π and the domain D is comprised between a half-plane in that case the fluid motionis a flow inside a corner On the other hand for 1

2 le n lt 1 πn gt π so that the motion is a flowpast a corner

The streamlines for the flows obtained with six different values for n are displayed in Fig IV11namely two flows in corners with angles π3 and 2π3 a uniform flow in the upper half plane twoflows past corners with inner angles 2π3 and π3 and a flow past a flat plaque correspondingrespectively to n = 3 3

35 and 1

IV43 c

Two-dimensional flows past a cylinder

Thanks to the linearity of the Laplace differential equations one may add ldquoelementaryrdquo solutionsof the previous paragraph to obtain new solutions which describe possible two-dimensional flowsWe now present two examples which represent flows coming from infinity where they are uniformand falling on a cylindermdasheither immobile or rotating around its axis

Acyclic flowLet us superpose the complex potentials for a uniform flow (IV41) along the x-direction and a

flow dipole (IV44a) situated at the origin and making an angle α = π with the vector ~ex

φ(z) = minusvinfin

) (IV48a)

where the dipole strength micro was written as R2vinfin Adopting polar coordinates (r θ) this ansatz

Figure IV12 ndash Streamlines for the acyclic potential flow past a cylinder (IV48a)

leads to the velocity potential and stream function

ϕ(r θ) = minusvinfin

)cos θ ψ(r θ) = minusvinfin

(r minus R2

)sin θ (IV48b)

One sees that the circle r = R is a line of constant ψ ie a streamline This means that the flowoutside that circle is decoupled from that inside In particular one may assume that the space insidethe circle is filled by a solid obstacle a ldquocylinderrdquo(24) without changing the flow characteristics onR2 deprived from the disk r lt R The presence of this obstacle has the further advantage that itldquohidesrdquo the singularity of the potential or the resulting velocity at z = 0 by cleanly removing itfrom the domain over which the flow is defined This is illustrated together with the streamlinesfor this flow in Fig IV12

From the complex potential (IV48a) follows at once the complex velocity

w(z) = vinfin

(1minus R2

) (IV49a)

which in polar coordinates gives

~v(r θ) = vinfin

[(1minus R2

)cos θ~er minus

)sin θ

] (IV49b)

The latter is purely tangential for r = R in agreement with the fact that the cylinder surface is astreamline The flow velocity even fully vanishes at the points with r = R and θ = 0 or π whichare thus stagnation points(liii)

Assuming that the motion is stationary one can calculate the force exerted on the cylinder bythe flowing fluid Invoking the Bernoulli equation (IV11)mdashwhich holds since the flow is steadyand incompressiblemdashand using the absence of vorticity which leads to the constant being the samethroughout the flow one obtains

P (~r) +1

2ρ~v(~r)2 = Pinfin +

2ρv2infin

(24)The denomination is motivated by the fact that even though the flow characteristics depend on two spatialcoordinates only the actual flow might take in place in a three-dimensional space in which case the obstacle isan infinite circular cylinder

(liii)Staupunkte

where Pinfin denotes the pressure at infinity That is at each point on the surface of the cylinder

P (R θ) = Pinfin +1

2ρ[v2infin minus~v(R θ)2

]= Pinfin +

2ρv2infin(1minus 4 sin2 θ

where the second identity follows from Eq (IV49b) The resulting stress vector on the vector ata given θ is directed radially towards the cylinder center ~Ts(R θ) = minusP (R θ)~er(R θ) Integratingover θ isin [0 2π] the total force on the cylinder due to the flowing fluid simply vanishesmdashin conflictwith the intuitionmdash phenomenon which is known as drsquoAlembert paradox (w)

The intuition according to which the moving fluid should exert a force on the immobile obstacleis good What we find here is a failure of the perfect-fluid model which is in that case tooidealized by allowing the fluid to slip without friction along the obstacle

Cyclic flowTo the flow profile which was just considered we add a pointlike vortex (IV43a) situated at the

originφ(z) = minusvinfin

2πlog

R (IV50a)

where we have divided z by R in the logarithm to have a dimensionless argument although thisplays no role for the velocity Comparing with the acyclic flow which models fluid motion arounda motionless cylinder the complex potential may be seen as a model for the flow past a rotatingcylinder as in the case of the Magnus effect (sect IV22 d)

Adopting polar coordinates (r θ) the velocity potential and stream function read

)cos θ minus Γ

2πθ ψ(r θ) = minusvinfin

(r minus R2

)sin θ +

2πlog

R (IV50b)

so that the circle r = R remains a streamline delimiting a fixed obstacleThe resulting velocity field reads in complex form

w(z) = vinfin

(1minus R2

)minus iΓ

2πz (IV51a)

and in polar coordinates

~v(r θ) = vinfin

[(1minus R2

)cos θ~er minus

r2minus Γ

2πrvinfin

)sin θ

] (IV51b)

The latter is purely tangential for r = R in agreement with the fact that the cylinder surface is astreamline

One easily checks that when the strength of the vortex is not too large namely Γ le 4πRvinfin theflow has stagnations points on the surface of the cylindermdashtwo if the inequality holds in the strictsense a single degenerate point if Γ = 4πR vinfinmdash as illustrated in Fig IV13 If Γ gt 4πRvinfin theflow defined by the complex potential (IV50a) still has a stagnation point yet now away from thesurface of the rotating cylinder as exemplified in Fig IV14

In either case repeating the same calculation based on the Bernoulli equation as for the acyclicflow allows one to derive the force exerted by the fluid on the cylinder The resulting force no longervanishes but equals minusΓρvinfin~ey on a unit length of the cylinder where ρ is the mass density of thefluid and ~ey the unit basis vector in the y-direction This is in line with the arguments presented insect IV22 d

(w)J le Rond drsquoAlembert 1717ndash1783

Figure IV13 ndash Streamlines for the cyclic potential flow past a (rotating) cylinder (IV50a)with Γ(4πRvinfin) = 025 (left) or 1 (right)

Figure IV14 ndash Streamlines for the cyclic potential flow past a (rotating) cylinder (IV50a)with Γ(4πRvinfin) = 4

IV43 d

Conformal deformations of flows

A further possibility to build two-dimensional potential flows is to ldquodistortrdquo the elementarysolutions of sect IV43 b or linear combinations of these building blocks Such deformations mayhowever not be arbitrary since they must preserve the orthogonality at each point in the fluid ofthe streamline (with constant ψ) and the equipotential line (constant ϕ) passing through that pointBesides rotations and dilationsmdashwhich do not distort the profile of the solution and are actuallyalready taken into account in the solutions of sect IV43 bmdash the generic class of transformations ofthe (complex) plane that preserve angles locally is that of conformal maps

As recalled in Appendix D4 such conformal mappingsmdashbetween open subsets of the complexplanes of variables z and Zmdashare defined by any holomorphic function Z = f (z) whose derivative iseverywhere non-zero and by its inverse F If φ(z) denotes an arbitrary complex flow potential onthe z-plane then Φ(Z) equiv φ(F (Z)) is a flow potential on the Z-plane Applying the chain rule theassociated complex flow velocity is w(F (Z))F prime(Z) where F prime denotes the derivative of F

A first example is to consider the trivial uniform flow with potential φ(z) = Az and theconformal mapping z 7rarr Z = f (z) = z1n with n ge 1

2 The resulting complex flow potential on theZ-plane is Φ(Z) = minusAZn

Except in the trivial case n = 1 f (z) is singular at z = 0 where f prime vanishes so that the mappingis non-conformal cutting a half-line ending at z = 0 f maps the complex plane deprived from thishalf-line onto an angular sector delimited by half-lines making an angle πnmdashas already seen insect IV43 b

Joukowsky transformA more interesting set of conformally deformed fluid flows consists of those provided by the use

of the Joukowsky transform(x)

Z = f (z) = z +R2

z(IV52)

where RJ isin RThe mapping (IV52) is obviously holomorphic in the whole complex z-plane deprived of the

originmdashwhich a single polemdash and has 2 points z = plusmnRJ at which f prime vanishes These two singularpoints correspond in the Z-plane to algebraic branch points of the reciprocal function z = F (Z) atZ = plusmn2RJ To remove them one introduces a branch cut along the line segment |X| le 2RJ On theopen domain U consisting complex Z-plane deprived from that line segment F is holomorphic andconformal One checks that the cut line segment is precisely the image by f of the circle |z| = RJ inthe complex z-plane Thus f and F provide a bijective mapping between the exterior of the circle|z| = RJ in the z-plane and the domain U in the Z-plane

Another property of the Joukowsky transform is that the singular points z = plusmnRJ are zeros off prime of order 1 so that angles are locally multiplied by 2 That is every continuously differentiablecurve going through z = plusmnRJ is mapped by f on a curve through Z = plusmn2RJ with an angular pointie a discontinuous derivative there

Consider first the circle C (0 R) in the z-plane of radius R gt RJ centered on the origin it canbe parameterized as

C (0 R) =z = R eiϑ 0 le ϑ le 2π

Its image in the Z-plane by the Joukowsky transform (IV52) is the set of points such that

)cosϑ+ i

(Rminus

)sinϑ 0 le ϑ le 2π

that is the ellipse centered on the origin Z = 0 with semi-major resp semi-minor axis R + R2JR

resp RminusR2JR along theX- resp Y -direction Accordingly the flows past a circular cylinder studied

in sect IV43 c can be deformed by f into flows past elliptical cylinders where the angle between theellipse major axis and the flow velocity far from the cylinder may be chosen at will

Bibliography for Chapter IVbull National Committee for Fluid Mechanics film amp film notes on Vorticity

bull Faber [1] Chapters 17 28ndash29 41ndash412

bull Feynman [8 9] Chapter 40

bull Guyon et al [2] Chapters 53ndash54 61ndash63 65ndash66 amp 7-1ndash73

bull LandaundashLifshitz [3 4] Chapter I sect 3 5 8ndash11

bull Sommerfeld [5 6] Chapters II sect 67 and IV sect 1819

(x)N Eukovski = N E Zhukovsky 1847ndash1921

CHAPTER V

Waves in non-relativistic perfect fluids

A large class of solutions of the equations of motion (III9) (III18) and (III33) is that of wavesQuite generically this denomination designates ldquoperturbationsrdquo of some ldquounperturbedrdquo fluid motionwhich will also be referred to as background flow

In more mathematical terms the starting point is a set of fields ρ0(t~r)~v0(t~r)P 0(t~r) solvingthe equations of motion representing the background flow The wave then consists of a second setof fields δρ(t~r) δ~v(t~r) δP (t~r) which are added the background ones such that the resultingfields

ρ(t~r) = ρ0(t~r) + δρ(t~r) (V1a)

P (t~r) = P 0(t~r) + δP (t~r) (V1b)

~v(t~r) =~v0(t~r) + δ~v(t~r) (V1c)

are solutions to the equations of motion

Different kinds of perturbationsmdashtriggered by some source which will not be specified hereafterand is thus to be seen as an initial conditionmdashcan be considered leading to different phenomena

A first distinction with which the reader is probably already familiar is that between travelingwaves which propagate and standing waves which do not Mathematically in the former case thepropagating quantity does not depend on space and time independently but rather on a combinationlike (in a one-dimensional case) x minus cϕt some propagation speed In contrast in standing wavesthe space and time dependence of the ldquopropagatingrdquo quantity factorize Hereafter we shall mostlymention traveling waves

Another difference is that between ldquosmallrdquo and ldquolargerdquo perturbations or in more technical termsbetween linear and nonlinear waves In the former case which is that of sound waves (Sec V1) orthe simplest gravity-controlled surface waves in liquids (Sec V31) the partial differential equationgoverning the propagation of the wave is linearmdashwhich means that nonlinear terms have beenneglected Quite obviously nonlinearities of the dynamical equationsmdashas eg the Euler equationmdashare the main feature of nonlinear waves as for instance shock waves (V2) or solitons (Sec V32)

V1 Sound wavesBy definition the phenomenon which in everyday life is referred to as ldquosoundrdquo consists of smalladiabatic pressure perturbations around a background flow where adiabatic actually means thatthe entropy remains constant In the presence of such a wave each point in the fluid undergoesalternative compression and rarefaction processes That is these waves are by construction (partsof) a compressible flow

We shall first consider sound waves on a uniform perfect fluid at rest (Sec V11)What then Doppler effect Riemann problem

V1 Sound waves 71

V11 Sound waves in a uniform fluid at rest

Neglecting the influence of gravity a trivial solution of the dynamical equations of perfect fluidsis that with uniform and time independent mass density ρ0 and pressure P 0 with a vanishing flowvelocity ~v0 = ~0 Assuming in addition that the particle number N0 in the fluid is conserved itsentropy has a fixed value S0 These conditions will represent the background flow we consider here

Inserting the values of the various fields in relations (V1) a perturbation of this backgroundflow reads

ρ(t~r) = ρ0 + δρ(t~r) (V2a)

P (t~r) = P 0 + δP (t~r) (V2b)

~v(t~r) = ~0 + δ~v(t~r) (V2c)

The necessary ldquosmallnessrdquo of perturbations means for the mass density and pressure terms

|δρ(t~r)| ρ0 |δP (t~r)| P 0 (V2d)

Regarding the velocity the background flow does not explicitly specify a reference scale with whichthe perturbation should be compared As we shall see below the reference scale is actually implicitlycontained in the equation(s) of state of the fluid under consideration and the condition of smallperturbation reads

|δ~v(t~r)| cs (V2e)

with cs the speed of sound in the fluid

Inserting the fields (V2) in the equations of motion (III9) and (III18) and taking into accountthe uniformity and stationarity of the background flow one finds

partδρ(t~r)

partt+ ρ0

~nabla middot δ~v(t~r) + ~nabla middot[δρ(t~r) δ~v(t~r)

]= 0 (V3a)

[ρ0 + δρ(t~r)

]partδ~v(t~r)

partt+[δ~v(t~r) middot ~nabla

]δ~v(t~r)

+ ~nablaδP (t~r) = 0 (V3b)

The required smallness of the perturbations will help us simplify these equations in that weshall only keep the leading-order terms in an expansion in which we consider ρ0 P 0 as zeroth-orderquantities while δρ(t~r) δP (t~r) and δ~v(t~r) are small quantities of first orderAccordingly the third term in the continuity equation is presumably much smaller than the othertwo and may be left aside in a first approximation Similarly the contribution of δρ(t~r) and theconvective term within the curly brackets on the left hand side of Eq (V3b) may be dropped Theequations describing the coupled evolutions of δρ(t~r) δP (t~r) and δ~v(t~r) are thus linearized

partδρ(t~r)

partt+ ρ0

~nabla middot δ~v(t~r) = 0 (V4a)

ρ0partδ~v(t~r)

partt+ ~nablaδP (t~r) = 0 (V4b)

To have a closed system of equations we still need a further relation between the perturbationsThis will be provided by thermodynamics ie by the implicit assumption that the fluid at rest iseverywhere in a state in which its pressure P is function of mass density ρ (local) entropy S and(local) particle number N ie that there exists a unique relation P = P (ρ SN) which is valid ateach point in the fluid and at every time Expanding this relation around the (thermodynamic)point corresponding to the background flow namely P 0 = P (ρ0 S0 N0) one may write

P(ρ0 + δρ S0 + δSN0 + δN

)= P 0 +

(partPpartρ

(partPpartS

(partPpartN

where the derivatives are taken at the point (ρ0 S0 N0) Here we wish to consider isentropic

72 Waves in non-relativistic perfect fluids

perturbations at constant particle number ie δS and δN vanish leaving

(partPpartρ

For this derivative we introduce the notation

c2s equiv

(partPpartρ

where both sides actually depend on ρ0 S0 and N0 One may then express δP as function of δρand replace ~nablaδP (t~r) by c2

s~nablaδρ(t~r) in Eq (V4b)

The resulting equations for δρ(t~r) and δ~v(t~r) are linear first order partial differential equa-tions Thanks to the linearity their solutions form a vector spacemdashat least as long as no initialcondition has been specified One may for instance express the solutions as Fourier transforms iesuperpositions of plane waves Accordingly we test the ansatz

δρ(t~r) = δρ(ω~k) eminusiωt+i~kmiddot~r δ~v(t~r) = δ~v(ω~k) eminusiωt+i~kmiddot~r (V6)

with respective amplitudes δρ δ~v that a priori depend on ω and ~k and are determined by the initialconditions for the problem In turn ω and ~k are not necessarily independent from each other

With this ansatz Eqs (V4) become

minusiωδρ(ω~k) + iρ0~k middot δ~v(ω~k) = 0 (V7a)

minusiωρ0 δ~v(ω~k) + ic2s~k δρ(ω~k) = 0 (V7b)

From the second equation the amplitude δ~v(ω~k) is proportional to ~k in particular it lies alongthe same direction That is the inner product ~k middot δ~v simply equals the product of the norms of thetwo vectors

Omitting from now on the (ω~k)-dependence of the amplitudes the inner product of Eq (V7b)with ~kmdashwhich does not lead to any loss of informationmdashallows one to recast the system as(

minusω ρ0

c2s~k 2 minusωρ0

)(δρ

~k middot δ~v

A first trivial solution to this system is δρ = 0 δ~v = ~0 ie the absence of any perturbation Inorder for non-trivial solutions to exist the determinant (ω2 minus c2

s~k 2)ρ0 of the system should vanish

This leads at once to the dispersion relation

ω = plusmncs|~k| (V8)

Denoting by ~e~k the unit vector in the direction of ~k the perturbations δρ(t~r) and δ~v(t~r) definedby Eq (V6) as well as δP (t~r) = c2

s δρ(t~r) are all functions of cstplusmn~r middot~e~k These are thus travelingwaves(liv) that propagate with the phase velocity ω(~k)|~k| = cs which is independent of ~k Thatis cs is the speed of sound For instance for air at T = 300 K the speed of sound is cs = 347 m middot sminus1

Air is a diatomic ideal gas ie it has pressure P = NkBTV and internal energy U = 52NkBT

This then gives c2s =

(partPpartρ

= minus V 2

(partPpartV

= minus V 2

[minusNkBT

V 2+NkB

(partT

(liv)fortschreitende Wellen

V1 Sound waves 73

The thermodynamic relation dU = T dS minus P dV + microdN yields at constant entropy and particlenumber

P = minus(partU

= minus5

(partT

ie NkB

(partT

= minus2P5

= minus2

leading to c2s =7

mair with mair = 29NA g middotmolminus1

Remarkslowast Taking the real parts of the complex quantities in the harmonic waves (V6) so as to obtainreal-valued δρ δP and δ~v one sees that these will be alternatively positive and negative and inaveragemdashover a duration much longer than a period 2πωmdashzero This in particular means thatthe successive compression and condensation (δP gt 0 δρ gt 0) or depression and rarefaction(lv)

(δP lt 0 δρ lt 0) processes do not lead to a resulting transport of matter

lowast A single harmonic wave (V6) is a traveling wave Yet if the governing equation or systemsof equations is linear or has been linearized as was done here the superposition of harmonicwaves is a valid solution In particular the superposition of two harmonic traveling waves withequal frequencies ω opposite waves vectors ~kmdashwhich is allowed by the dispersion relation (V8)mdashand equal amplitudes leads to a standing wave in which the dependence on time and space isproportional to eiωt cos(~k middot~r)

Coming back to Eq (V7b) the proportionality of δ~v(ω~k) and ~k means that the sound wavesin a fluid are longitudinalmdashin contrast to electromagnetic waves in vacuum which are transversalwaves

The nonexistence of transversal waves in fluids reflects the absence of forces that would actagainst shear deformations so as to restore some equilibrium shapemdashshear viscous effects cannotplay that roleIn contrast there can be transversal sound waves in elastic solids as eg the so-called S-modes(shear modes) in geophysics

The inner product of Eq (V7b) with ~k together with the dispersion relation (V8) and thecollinearity of δ~v and ~k leads to the relation

∣∣~k∣∣∣∣δ~v∣∣ = c2s

∣∣~k∣∣δρ hArr∣∣δ~v∣∣cs

for the amplitudes of the perturbations This justifies condition (V2e) which is then consistentwith (V2d) Similarly inserting the ansatz (V6) in Eq (V3b) the terms within curly bracketsbecome minusiω δ~v + i

(~k middot δ~v

)δ~v again neglecting the second with respect to the first is equivalent to

requesting∣∣δ~v∣∣ cs

Remark Going back to Eqs (V4) the difference of the time derivative of the first one and thedivergence of the second onemdashin which ~nablaP has been replaced by c2

s~nablaρmdashleads to the known wave

equation(25)

part2ρ(t~r)

partt2minus c2

s4ρ(t~r) = 0 (V9a)

If the flowmdashincluding the background flow on which the sound wave develops in case ~v0 is nottrivial as it is heremdashis irrotational so that one may write ~v(t~r) = minus~nablaϕ(t~r) then the velocitypotential ϕ also obeys the same equation

part2ϕ(t~r)

partt2minus c2

s4ϕ(t~r) = 0

(25)This traditional denomination is totally out of place in a chapter in which there are several types of waves each ofwhich has its own governing ldquowave equationrdquo Yet historically due to its role for electromagnetic or sound wavesit is the archetypal wave equation while the equations governing other types of waves often have a specific name

(lv)Verduumlnnung

V12 Sound waves on moving fluids

V13 Riemann problem Rarefaction waves

V2 Shock waves 75

V2 Shock wavesWhen the amplitude of the perturbations considered in Sec (V1) cannot be viewed as small asfor instance if |δ~v| cs does not hold then the linearization of the equations of motion (V3) is nolonger licit and the nonlinear terms play a role

A possibility is then that at a finite time t in the evolution of the fluid a discontinuity in someof the fields may appear referred to as shock wave(lvi) How this may arise will be discussed in thecase of a one-dimensional problem (Sec (V21)) At a discontinuity the differential formulation ofthe conservation laws derived in Chap III no longer holds and it becomes necessary to study theconservation of mass momentum and energy across the surface of discontinuity associated with theshock wave (Sec V22)

V21 Formation of a shock wave in a one-dimensional flow

As in Sec (V11) we consider the propagation of an adiabatic perturbation of a background fluidat rest neglecting the influence of gravity or other external volume forces In the one-dimensionalcase the dynamical equations (V3) read

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (V10a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

]+partδP (t x)

partx= 0 (V10b)

The variation of the pressure δP (t x) can again be expressed in terms of the variation in the massdensity δρ(t x) by invoking a Taylor expansion [cf the paragraph between Eqs (V4) and (V5)]Since the perturbation of the background ldquoflowrdquo is no longer small the thermodynamic state aroundwhich this Taylor expansion is performed is not necessarily that corresponding to the unperturbedfluid but rather an arbitrary state so that

δP (t x) cs(ρ)2δρ(t x)

where the speed of sound is that in the perturbed flow When differentiating this identity thederivative of δρ(t x) with respect to x is also the derivative of ρ(t x) since the unperturbed fluidstate is uniform Accordingly one may recast Eqs (V10) as

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (V11a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

]+ cs(ρ)2partρ(t x)

partx= 0 (V11b)

which constitutes a system of two coupled partial differential equations for the two unknown fieldsρ(t x) and δv(t x) = v(t x)

To tackle these equations one may assume that the mass density and the flow velocity haveparallel dependences on time and spacemdashas suggested by the fact that this property holds in thelinearized case of sound waves in which both ρ(t~r) and ~v(t~r) propagate with the same phase(cs|~k|t + ~k middot ~r) Thus the dependence of v on t and x is replaced with a functional dependencev(ρ(t x)

) with the known value v(ρ0) = 0 corresponding to the unperturbed fluid at rest Accord-

ingly the partial derivatives of the flow velocity with respect to t resp x become

partv(t x)

partt=

dv(ρ)

partρ(t x)

parttresp

partv(t x)

partx=

dv(ρ)

partρ(t x)

The latter identities may then be inserted in Eqs (V11) If one further multiplies Eq (V11a) by(lvi)Stoszligwelle

ρ(t x) dv(ρ)dρ and then subtracts Eq (V11b) from the result there comesρ2

[dv(ρ)

minus cs(ρ)2

partρ(t x)

partx= 0

that is discarding the trivial solution of a uniform mass density

dv(ρ)

dρ= plusmncs(ρ)

ρ (V12)

Under the simultaneous replacements v rarr minusv x rarr minusx cs rarr minuscs equations (V11)-(V12)remain invariant Accordingly one may restrict the discussion of Eq (V12) to the case with a+ signmdashthe minus case amounts to considering a wave propagating in the opposite direction with theopposite velocity The flow velocity is then formally given by

v(ρ) =

int ρ

cs(ρprime)

ρprimedρprime

where we used v(ρ0) = 0 while Eq (V11b) can be rewritten as

partρ(t x)

partt+[v(ρ(t x)

(ρ(t x)

)]partρ(t x)

partx= 0 (V13)

Assuming that the mass density perturbation propagates as a traveling wave ie making theansatz δρ(t x) prop f(xminuscwt) in Eq (V13) then its phase velocity cw will be given by cw = cs(ρ)+vInvoking Eq (V12) then shows that dv(ρ)dρ gt 0 so that cw grows with increasing mass densitythe denser regions in the fluid will propagate faster than the rarefied ones and possibly catch upwith themmdashin case the latter where ldquoin frontrdquo of the propagating perturbationmdashas illustrated inFig V1 In particular there may arise after a finite amount of time a discontinuity of the functionρ(t x) at a given point x0 The (propagating) point where this discontinuity takes place representsthe front of a shock wave

t4 gt t3

t3 gt t2

t2 gt t1

t1 gt t0

Figure V1 ndash Schematic representation of the evolution in time of the spatial distribution ofdense and rarefied regions leading to a shock wave

V2 Shock waves 77

V22 Jump equations at a surface of discontinuity

To characterize the properties of a flow in the region of a shock wave one needs first to specifythe behavior of the physical quantities of relevance at the discontinuity which is the object of thisSection Generalizing the finding of the previous Section in a one-dimensional setup in which thediscontinuity arises at a single (traveling) point in the three-dimensional case there will be a wholesurface of discontinuity (lvii) that propagates in the unperturbed background fluidFor the sake of brevity the dependence on t and ~r of the various fields of interest will be omitted

To describe the physics at the front of the shock wave we adopt a comoving reference frame Rwhich moves with the surface of discontinuity and in this reference frame we consider a system ofCartesian coordinates (x1 x2 x3) with the basis vector ~e1 perpendicular to the propagating surfaceThe region in front resp behind the surface will be denoted by (+) resp (minus) that is the fluid inwhich the shock waves propagates flows from the (+)- into the (minus)-region

The jump(lviii) of a local physical quantity g(~r) across the surface of discontinuity is defined as[[g]]equiv g

+minus gminus (V14)

where g+

resp gminus denotes the limiting value of g as x1 rarr 0+ resp x1 rarr 0minus In case such alocal quantity is actually continuous at the surface of discontinuity then its jump across the surfacevanishes

At a surface of discontinuity Sd the flux densities of mass momentum and energy across thesurface ie along the x1-direction must be continuous so that mass momentum and energy remainlocally conserved These requirements are expressed by the jump equations(lix)[[

ρ v1]]

= 0 (V15a)[[TTTi1]]

= 0 foralli = 1 2 3 (V15b)[[(1

2ρ~v2 + e+ P

]]= 0 (V15c)

where the momentum flux density tensor has components TTTij = P gij + ρ vi vj [see Eq (III21b)]with gij = δij in the case of Cartesian coordinates

The continuity of the mass flux density across the surface of discontinuity (V15a) can be recastas

(ρv1)minus= (ρv1)+ equiv j1 (V16)

A first trivial solution arises if there is no flow of matter across surface Sd ie if (v1)+ = (v1)minus = 0In that case Eq (V15c) is automatically satisfied Condition (V15b) for i = 1 becomes

= 0ie the pressure is the same on both sides of Sd Eventually Eq (V15b) with i = 2 or 3 holdsautomatically All in all there is no condition on the behavior of ρ v2 or v3 across the surface ofdiscontinuitymdashwhich means that these quantities may be continuous or not in the latter case withan arbitrary jump

If j1 does not vanish that is if matter does flow across Sd then the jump equation for thecomponent TTT21 = ρv2v1 resp TTT31 = ρv3v1 leads to

[[v2]]

= 0 resp[[

= 0 ie the component v2

resp v3 is continuous across the surface of discontinuity

(v2)minus= (v2)+ resp (v3)minus= (v3)+ (V17)

In turn rewriting the jump equation for TTT11 = P + ρ(v1)2 with the help of j1 yields

Pminusminus P + = j1[(v1)+minus (v1)minus

ρ+minus 1

ρminus

) (V18)

(lvii)Unstetigkeitsflaumlche (lviii)Sprung (lix)Sprunggleichungen

Thus if ρ+lt ρminus ie if the fluid is denser in the (minus)-region ldquobehindrdquo the shock frontmdashas is suggestedby Fig V1 yet still needs to be provedmdash then Pminusgt P + while relation (V16) yields (v1)+gt (v1)minusConversely ρ+ gt ρminus leads to Pminus lt P + and (v1)+ lt (v1)minus One can show that the former caseactually holds

Combining Eqs (V16) and (V18) yields[(v1)+

]2=j21

=Pminusminus P +

ρminusminus ρ+

ρminusρ+

=Pminusminus P +

ρminusminus ρ+

ρminusρ+

and similarly [(v1)minus

Pminusminus P +

ρminusminus ρ+

ρminus

If the jumps in pressure and mass density are small one can show that their ratio is approximatelythe derivative partPpartρ here at constant entropy and particle number ie[

]2 (partPpartρ

ρminusρ+

=ρminusρ+c2s

[(v1)minus

]2 ρ+

ρminusc2s

With ρminus gt ρ+ comes (v1)+ gt cs resp (v1)minus lt cs in front of resp behind the shock wave(26) Theformer identity means that an observer comoving with the surface of discontinuity sees in front afluid flowing with a supersonic velocity that is going temporarily back to a reference frame boundto the unperturbed fluid the shock wave moves with a supersonic velocity

Invoking the continuity across Sd of the product ρ v1 and of the components v2 v3 parallel tothe surface of discontinuity the jump equation (V15c) for the energy flux density simplifies to[[

2(v1)2 +

e+ Pρ

]]=j21

minus 1

ρ2minus

)+e+ + P +

ρ+minus eminus + Pminus

ρminus= 0

Expressing j21 with the help of Eq (V18) one finds

Pminus minus P +

ρminus

)=wminusρminusminus w+

ρ+(V19a)

with w = e+ P the enthalpy density or equivalently

Pminus + P +

ρ+minus 1

ρminus

ρ+minus eminusρminus (V19b)

Either of these equations represents a relation between the thermodynamic quantities on both sidesof the surface of discontinuity and define in the space of the thermodynamic states of the fluida so-called shock adiabatic curve also referred to as dynamical adiabatic curve(lx) or Hugoniot(y)

adiabatic curve or Rankine(z)ndashHugoniot relation

More generally Eqs (V16)ndash(V19) relate the dynamical fields on both sides of the surface ofdiscontinuity associated with a shock wave and constitute the practical realization of the continuityconditions encoded in the jump equations (V15)

(26)Here we are being a little sloppy one should consider the right (x1 rarr 0+) and left (x1 rarr 0minus) derivativescorresponding respectively to the (+) and (minus)-regions and thus find the associated speeds of sound (cs)+ and(cs)minus instead of a single cs

(lx)dynamische Adiabate(y)P H Hugoniot 1851ndash1887 (z)W J M Rankine 1820ndash1872

V3 Gravity waves 79

V3 Gravity wavesIn this Section we investigate waves that are ldquodrivenrdquo by gravity in the sense that the latter is themain force that acts to bring back the perturbed fluid to its unperturbed ldquobackgroundrdquo state Suchperturbations are generically referred to as gravity waves(lxi)

A first example is that of small perturbations at the free surface of a liquid originally at restmdashtheldquowavesrdquo of everyday language In that case some external source as eg wind or an earthquakeleads to a local rise of the fluid above its equilibrium level gravity then acts against this riseand tends to bring back the liquid to its equilibrium position In case the elevation caused bythe perturbation is small compared to the sea depth as well as in comparison to the perturbationwavelength one has linear sea surface waves (Sec V31) Another interesting case arises in shallowwater for perturbations whose horizontal extent is much larger than their vertical size in whichcase one may find so-called solitary waves (Sec V32)

Throughout this Section the flowsmdashcomprised of a background fluid at rest and the travelingperturbationmdashare supposed to be two-dimensional with the x-direction along the propagationdirection and the z-direction along the vertical oriented upwards so that the acceleration due togravity is ~g = minusg~ez The origin z = 0 is taken at the bottom of the sea ocean which for the sakeof simplicity is assumed to be flat

V31 Linear sea surface waves

A surface wave is a perturbation of the altitudemdashwith respect to the sea bottommdashof the freesurface of the sea which is displaced by an amount δh(t x) from its equilibrium position h0 wherethe latter corresponds to a fluid at rest with a horizontal free surface These variations in theposition of the free surface signal the motion of the sea water ie a flow with a corresponding flowvelocity throughout the sea~v(t x z)

We shall model this motion as vorticity-free which allows us to introduce a velocity potentialϕ(t x z) and assume that the mass density ρ of the sea water remains constant and uniform iewe neglect its compressibility The sea is supposed to occupy an unbounded region of space whichis a valid assumption if one is far from any coast

Equations of motion and boundary conditions

Under the assumptions listed above the equations of motion read [cf Eq (IV30) and (IV31)]

minus partϕ(t x z)

partt+

[~nablaϕ(t x z)

+P (t x z)

ρ+ gz = constant (V20a)

where gz is the potential energy per unit mass of water and[part2

partx2+part2

partz2

]ϕ(t x z) = 0 (V20b)

To fully specify the problem boundary conditions are still needed As in the generic case forpotential flow (Sec IV4) these will be Neumann boundary conditions involving the derivative ofthe velocity potential

bull At the bottom of the sea the water can have no vertical motion corresponding to the usualimpermeability condition that is

vz(z=0) = minuspartϕpartz

∣∣∣∣z=0

= 0 (V21a)

(lxi)Schwerewellen

bull At the free surface of the sea the vertical component vz of the flow velocity equals the velocityof the surface ie it equals the rate of change of the position of the (material) surface

minuspartϕ(t x z)

∣∣∣∣z=h0+δh(tx)

=Dδh(t x)

UsingD

Dt=part

partt+ vx

partx=part

parttminus partϕ

partx this gives[

partϕ(t x z)

partz+partδh(t x)

parttminus partδh(t x)

partϕ(t x z)

]z=h0+δh(tx)

= 0 (V21b)

bull At the free surface of the sea the pressure on the water sidemdashright below the surfacemdashisdirectly related to that just above the surface The latter is assumed to be constant and equalat some value P 0 which represents for instance the atmospheric pressure ldquoat sea levelrdquo Asa first approximationmdashwhose physical content will be discussed in the remark at the end ofthis paragraphmdash the pressure is equals on both sides of the sea surface

P(t x z=h0+δh(t x)

)= P 0 (V21c)

Expressing the pressure with the help of Eq (V20a) this condition may be recast as[minus partϕ(t x z)

partt+

[~nablaϕ(t x z)

]z=h0+δh(tx)

+ gδh(t x) = minusP 0

ρminus gh0 + constant (V21d)

where the whole right hand side of the equation represents a new constant

Hereafter we look for solutions consisting of a velocity potential ϕ(t x z) and a surface profileδh(t x) as determined by Eqs (V20) with conditions (V21)

Remark The assumption of an identical pressure on both sides of an interfacemdasheither between twoimmiscible liquids or between a liquid and a gas as heremdashis generally not warranted unless theinterface happens to be flat If there is the least curvature the surface tension associated with theinterface will lead to a larger pressure inside the concavity of the interface Neglecting this effectmdashwhich we shall consider again in Sec V32mdashis valid only if the typical radius of curvature of theinterface which as we shall see below is the wavelength of the surface waves is ldquolargerdquo especiallywith respect to the deformation scale δh

Harmonic wave assumption

Since the domain on which the wave propagates is unbounded a natural ansatz for the solutionof the Laplace equation (V20b) is that of a harmonic wave

ϕ(t x z) = f(z) cos(kxminus ωt) (V22)

propagating in the x-direction with a depth-dependent amplitude f(z) Inserting this form in theLaplace equation yields the linear ordinary differential equation

d2f(z)

dz2minus k2f(z) = 0

whose obvious solution is f(z) = a1 ekz + a2 eminuskz with a1 and a2 two real constantsThe boundary condition (V21a) at the sea bottom z = 0 gives a1 = a2 ie

ϕ(t x z) = C cosh(kz) cos(kxminus ωt) (V23)

with C a real constant

To make further progress with the equations of the system and in particular to determine theprofile of the free surface further assumptions are needed so as to obtain simpler equations We shallnow present a first such simplification leading to linear waves In Sec V32 another simplificationmdashof a more complicated started pointmdashwill be considered which gives rise to (analytically tractable)nonlinear waves

V3 Gravity waves 81

Linear waves

As in the case of sound waves we now assume that the perturbations are ldquosmallrdquo so as to beable to linearize the equations of motion and those expressing boundary conditions Thus we shallassume that the quadratic term (~nablaϕ)2 is much smaller than |partϕpartt| and that the displacement δhof the free surface from its rest position is much smaller than the equilibrium sea depth h0

To fix ideas the ldquoswell wavesrdquo observed far from any coast on the Earth oceans or seas have atypical wavelength λ of about 100 m and an amplitude δh0 of 10 m or lessmdashthe shorter thewavelength the smaller the amplitudemdash while the typical seaocean depth h0 is 1ndash5 km

The assumption (~nablaϕ)2 |partϕpartt| can on the one hand be made in Eq (V20a) leading to

minus partϕ(t x z)

partt+

P (t x z)

ρ+ gz =

ρ+ gh0 (V24)

in which the right member represents the zeroth order while the left member also contains firstorder terms which must cancel each other for the identity to hold On the other hand taking alsointo account the assumption |δh(t x)| h0 the boundary conditions (V21b) and (V21d) at thefree surface of the sea can be rewritten as

partϕ(t x z)

∣∣∣∣z=h0

+partδh(t x)

partt= 0 (V25a)

andminus partϕ(t x z)

∣∣∣∣z=h0

+ gδh(t x) = constant (V25b)

respectively Together with the Laplace differential equation (V20b) and the boundary conditionat the sea bottom (V21a) the two equations (V25) constitute the basis of the Airy(aa) linear wavetheory

Combining the latter two equations yields at once the condition[part2ϕ(t x z)

partt2+ g

partϕ(t x z)

Using the velocity potential (V23) this relation reads

minusω2C cosh(kh0) cos(kxminus ωt) + gkC sinh(kh0) cos(kxminus ωt) = 0

resulting in the dispersion relationω2 = gk tanh(kh0) (V26)

This relation becomes even simpler in two limiting cases

bull When kh0 1 or equivalently h0 λ where λ = 2πk denotes the wavelength whichrepresents the case of gravity waves at the surface of deep sea(27) then tanh(kh0) 1 Inthat case the dispersion relation simplifies to ω2 = gk the phase and group velocity of thetraveling waves are

cϕ =ω

radicg

k and cg =

dω(k)

radicg

respectively both independent from the sea depth h0(27)The sea may not be ldquotoo deeprdquo otherwise the assumed uniformity of the water mass density along the vertical

direction in the unperturbed state does not hold With λ 100 m the inverse wave number is kminus1 15 m sothat h0 = 100 m already represents a deep ocean in comparison the typical scale on which non-uniformities inthe mass density are relevant is rather 1 km

(aa)G B Airy 1801ndash1892

bull For kh0 1 ie in the case of a shallow sea with h0 λ the approximation tanh(kh0) kh0

leads to the dispersion relation ω2 = gh0k2 ie to phase and group velocities

cϕ = cg =radicgh0

independent from the wavelength λ signaling the absence of dispersive behavior

This phase velocity decreases with decreasing water depth h0 Accordingly this might lead toan accumulation similar to the case of a shock wave in Sec V2 whose description howeverrequires that one take into account the nonlinear terms in the equations which have beendiscarded here In particular we have explicitly assumed |δh(t x)| h0 in order to linearizethe problem so that considering the limiting case h0 rarr 0 is questionable

In addition a temptation when investigating the small-depth behavior h0 rarr 0 is clearlyto describe the breaking of waves as they come to shore Yet the harmonic ansatz (V23)assumes that the Laplace equation is considered on a horizontally unbounded domain iefar from any coast so again the dispersion relation (V26) may actually no longer be valid

The boundary condition (V25b) provides us directly with the shape of the free surface of thesea namely

δh(t x) =1

partϕ(t x z)

∣∣∣∣z=h0

gcosh(kh0) sin(kxminus ωt) equiv δh0 sin(kxminus ωt)

with δh0 equiv (ωCg) cosh(kh0) the amplitude of the wave which must remain much smaller than h0The profile of the surface waves of Airyrsquos linear theorymdashor rather its cross sectionmdashis thus a simplesinusoidal curve

This shape automatically suggests a generalization which is a first step towards taking intoaccount nonlinearities such that the free surface profile is sum of (a few) harmonics sin(kxminusωt)sin 2(kxminusωt) sin 3(kxminusωt) The approach leading to such a systematically expanded profilewhich relies on a perturbative expansion to deal with the (still small) nonlinearities is that ofthe Stokes waves

The gradient of the potential (V23) yields (the components of) the flow velocity

vx(t x z) =kg

cosh(kz)

cosh(kh0)δh0 sin(kxminus ωt)

vz(t x z) = minuskgω

sinh(kz)

cosh(kh0)δh0 cos(kxminus ωt)

Integrating these functions with respect to time leads to the two functions

x(t) = x0 +kgδh0

cosh(kz)

cosh(kh0)cos(kxminus ωt) = x0 +

δh0 cosh(kz)

sinh(kh0)cos(kxminus ωt)

z(t) = z0 +kgδh0

sinh(kz)

cosh(kh0)sin(kxminus ωt) = z0 +

δh0 sinh(kz)

sinh(kh0)sin(kxminus ωt)

with x0 and z0 two integration constants Choosing x0 x and z0 z if δh0 kminus1 these functionsrepresent the components of the trajectory (pathline) of a fluid particle that is at time t in the vicinityof the point with coordinates (x z) and whose velocity at that time is thus approximately the flowvelocity~v(t x z) Since

[x(t)minus x0]2

cosh2(kz)+

[z(t)minus z0]2

sinh2(kz)=

[kgδh0

ω2 cosh(kh0)

sinh(kh0)

this trajectory is an ellipse whose major and minor axes decrease with increasing depth h0 minus zIn the deep sea case kh0 1 one can use the approximations sinh(kz) cosh(kz) ekz2 for1 kz kh0 which shows that the pathlines close to the sea surface are approximately circles

V3 Gravity waves 83

Eventually the pressure distribution in the sea follows from Eq (V24) in which one uses thevelocity potential (V23) resulting in

P (t x z) = P 0 + ρg(h0 minus z) + ρpartϕ(t x z)

partt= P 0 + ρg

[h0 minus z + δh0

cosh(kz)

cosh(kh0)sin(kxminus ωt)

The contribution P 0 + ρg(h0 minus z) is the usual hydrostatic one corresponding to the unperturbedsea while the effect of the surface wave is proportional to its amplitude δh0 and decreases withincreasing depth

V32 Solitary waves

We now want to go beyond the linear limit considered in sect V31 c for waves at the free surfaceof a liquid in a gravity field To that extent we shall take a few steps back and first rewritethe dynamical equations of motion and the associated boundary conditions in a dimensionlessform (sect V32 a) This formulation involves two independent parameters and we shall focus onthe limiting case where both are smallmdashyet non-vanishingmdashand obey a given parametric relationIn that situation the equation governing the shape of the free surface is the Kortewegndashde Vriesequation which in particular describes solitary waves (sect V32 c)(28)

Dimensionless form of the equations of motion

As in sect V31 c the equations governing the dynamics of gravity waves at the surface of the seaare on the one hand the incompressibility condition

~nabla middot~v(t~r) = 0 (V27a)

and on the other hand the Euler equationpart~v(t~r)

]~v(t~r) = minus1

ρ~nablaP (t~r)minus g~ez (V27b)

The boundary conditions (V21) they obey are the absence of vertical velocity at the sea bottom

vz(t x z=0) = 0 (V27c)

the identity of the sea vertical velocity with the rate of change of the surface altitude h0 + δh(t x)

vz(t x z=h0+δh(t x)

)=partδh(t x)

partt+ vx(t~r)

partδh(t x)

partx (V27d)

and finally the existence of a uniform pressure at that free surface

P(t x z=h0+δh(t x)

)= P 0

In the sea at rest the pressure field is given by the hydrostatic formula

P st(t x z) = P 0 minus ρg(h0 minus z)

Defining the ldquodynamical pressurerdquo in the sea water as P dyn equiv P minusP st one finds first that the righthand side of the Euler equation (V27b) can be replaced by minus(1ρ)~nablaP dyn and secondly that theboundary condition at the free surface becomes

(t x z=h0+δh(t x)

)= ρgδh(t x) (V27e)

Let us now recast Eqs (V27) in a dimensionless form For that extent we introduce twocharacteristic lengths Lc for long-wavelength motions along x or z and δhc for the amplitude ofthe surface deformation for durations we define a scale tc which will later be related to Lc withthe help of a typical velocity With these scales we can construct dimensionless variables

tlowast equiv t

tc xlowast equiv x

Lc zlowast equiv z

(28)This Section follows closely the Appendix A of Ref [18]

and fieldsδhlowast equiv δh

δhc vlowastx equiv

vxδhctc

vlowastz equivvz

δhctc P lowast equiv

ρ δhcLct2c

Considering the latter as functions of the reduced variables tlowast xlowast zlowast one can rewrite theequations (V27a)ndash(V27e) The incompressibility thus becomes

partvlowastxpartxlowast

+partvlowastzpartzlowast

= 0 (V28a)

and the Euler equation projected successively on the x and z directions

partvlowastxparttlowast

(vlowastxpartvlowastxpartxlowast

+ vlowastzpartvlowastxpartzlowast

)= minuspartP lowast

partxlowast (V28b)

andpartvlowastzparttlowast

(vlowastxpartvlowastzpartxlowast

+ vlowastzpartvlowastzpartzlowast

partzlowast (V28c)

where we have introduced the dimensionless parameter ε equiv δhcLc In turn the various boundaryconditions are

vlowastz = 0 at zlowast = 0 (V28d)

at the sea bottom and at the free surface

vlowastz =partδhlowast

parttlowast+ εvlowastx

partδhlowast

partxlowastat zlowast = δ + εδhlowast (V28e)

with δ equiv h0Lc and

P lowast =gt2cLc

δhlowast at zlowast = δ + εδhlowast

Introducing the further dimensionless number

Fr equivradicLcg

the latter condition becomes

P lowast =1

Fr2 δhlowast at zlowast = δ + εδhlowast (V28f)

Inspecting these equations one sees that the parameter ε controls the size of nonlinearitiesmdashcfEqs (V28b) (V28c) and (V28e)mdash while δ measures the depth of the sea in comparison to thetypical wavelength Lc Both parameters are a priori independent δ is given by the physical setupwe want to describe while ε quantifies the amount of nonlinearity we include in the description

To make progress we shall from now on focus on gravity waves on shallow water ie assumeδ 1 In addition we shall only consider small nonlinearities ε 1 To write down expansionsin a consistent manner we shall assume that the two small parameters are not of the same orderbut rather that they obey ε sim δ2 Calculations will be considered up to order O(δ3) or equivalentlyO(δε)

For the sake of brevity we now drop the subscript lowast from the dimensionless variables and fields

Velocity potential

If the flow is irrotational partvxpartz = partvzpartx so that one may transform Eq (V28b) into

partvxpartt

(vxpartvxpartx

+ vzpartvzpartx

partδh

partx= 0 (V29)

In addition one may introduce a velocity potential ϕ(t x z) such that ~v = minus~nablaϕ With the latter

V3 Gravity waves 85

the incompressibility condition (V28a) becomes the Laplace equation

part2ϕ

partx2+part2ϕ

partz2= 0 (V30)

The solution for the velocity potential will be written as an infinite series in z

ϕ(t x z) =infinsumn=0

znϕn(t x) (V31)

with unknown functions ϕn(t x) Substituting this ansatz in the Laplace equation (V30) gives aftersome straightforward algebra

infinsumn=0

zn[part2ϕn(t x)

partx2+ (n+ 1)(n+ 2)ϕn+2(t x)

In order for this identity to hold for arbitrary zmdashat least for the values relevant for the flowmdasheach coefficient should individually vanish ie the ϕn should obey the recursion relation

ϕn+2(t x) = minus 1

(n+ 1)(n+ 2)

part2ϕn(t x)

partx2for n isin N (V32)

It is thus only necessary to determine ϕ0 and ϕ1 to know the whole seriesThe boundary condition (V28d) at the bottom reads partϕ(t x z = 0)partz = 0 for all t and x

which implies ϕ1(t x) = 0 so that all ϕ2n+1 identically vanish As a consequence ansatz (V31)with the recursion relation (V32) give

ϕ(t x z) = ϕ0(t x)minus z2

part2ϕ0(t x)

partx2+z4

part4ϕ0(t x)

partx4+

Differentiating with respect to x or z yields the components of the velocity~v = minus~nablaϕ

vx(t x z) = minuspartϕ0(t x)

partx+z2

part3ϕ0(t x)

partx3minus z4

part5ϕ0(t x)

partx5+

vz(t x z) = zpart2ϕ0(t x)

partx2minus z3

part4ϕ0(t x)

partx4+

Introducing the notation u(t x) equiv minuspartϕ0(t x)partx and anticipating that the maximal value of zrelevant for the problem is of order δ these components may be expressed as

vx(t x z) = u(t x)minus z2

part2u(t x)

partx2+ o(δ3) (V33a)

vz(t x z) = minusz partu(t x)

partx+z3

part3u(t x)

partx3+ o(δ3) (V33b)

where the omitted terms are beyond O(δ3)

Linear waves rediscoveredIf we momentarily set ε = 0mdashwhich amounts to linearizing the equations of motion and boundary

conditionsmdash consistency requires that we consider equations up to order δ at most That is wekeep only the first terms from Eqs (V33) at the surface at z δ they become

vx(t x z=δ) u(t x) vz(t x z=δ) minusδ partu(t x)

partx (V34a)

while the boundary condition (V28e) simplifies to

vz(t x z=δ) =partδh(t x)

partt= δ

partφ(t x)

partt (V34b)

where we have introduced φ(t x) equiv δh(t x)δ

Meanwhile Eq (V29) with ε = 0 reads

partvx(t x)

partt+

partφ(t x)

partx= 0 (V34c)

Together Eqs (V34a)ndash(V34c) yield after some straightforward manipulations the equation

part2u(t x)

partt2minus δ

part2u(t x)

partx2= 0 (V35)

ie a linear equation describing waves with the dimensionless phase velocityradicδFr =

radicgh0(Lctc)

Since the scaling factor of x resp t is Lc resp tc the corresponding dimensionful phase velocity iscϕ =

radicgh0 as was already found in sect V31 c for waves on shallow sea

Until now the scaling factor tc was independent from Lc Choosing tc equiv Lcradicgh0 ie the unit

in which times are measured the factor δFr2 equals 1 leading to the simpler-looking equation

partvx(t x z)

partt+ ε

[vx(t x z)

partvx(t x z)

partx+ vz(t x z)

partvz(t x z)

]+partφ(t x)

partx= 0 (V36)

instead of Eq (V29)

Non-linear waves on shallow water

Taking now ε 6= 0 and investigating the equations up to order O(δ3) O(δε) Eqs (V33) at thefree surface at z = δ(1 + εφ) become

vx(t x z=δ(1 + εφ)

)= u(t x)minus δ2

part2u(t x)

partx2 (V37a)

vz(t x z=δ(1 + εφ)

)= minusδ

[1 + εφ(t x)

]partu(t x)

partx+δ3

part3u(t x)

partx3 (V37b)

Inserting these velocity components in (V36) while retaining only the relevant orders yields

partu(t x)

parttminus δ2

part3u(t x)

partt partx2+ εu(t x)

partu(t x)

partx+partφ(t x)

partx= 0 (V38)

On the other hand the velocity components are also related by the boundary condition (V28e)which reads

partφ(t x)

partt+ δεvx

(t x z=δ(1 + εφ)

)partφ(t x)

Substituting Eq (V37a) resp (V37b) in the right resp left member yields

partφ(t x)

partt+ εu(t x)

partφ(t x)

partx+[1 + εφ(t x)

]partu(t x)

partxminus δ2

part3u(t x)

partx3= 0 (V39)

To leading order in δ and ε the system of nonlinear partial differential equations (V38)ndash(V39)simplifies to the linear system

partu(t x)

partt+partφ(t x)

partx= 0

partφ(t x)

partt+partu(t x)

partx= 0

which admits the solution u(t x) = φ(t x) under the condition

partu(t x)

partt+partu(t x)

partx= 0 (V40)

which describes a traveling wave with (dimensionless) velocity 1 u(t x) = u(xminust) We again recoverthe linear sea surface waves which we have already encountered twice

V3 Gravity waves 87

Going to next-to-leading order O(δ2) O(ε) we look for solutions in the form

u(t x) = φ(t x) + εu(ε)(t x) + δ2u(δ)(t x) (V41)

with φ u(ε) u(δ) functions that obey condition (V40) up to terms of order ε or δ2 Inserting thisansatz in Eqs (V38)ndash(V39) yields the system

partφ

partt+partφ

partx+ ε

partu(ε)

partx+ δ2partu(δ)

partx+ 2εφ

partφ

partxminus δ2

part3φ

partx3= 0

partφ

partt+partφ

partx+ ε

partu(ε)

partt+ δ2partu(δ)

partt+ εφ

partφ

partxminus δ2

part3φ

partx2 partt= 0

where for the sake of brevity the (t x)-dependence of the functions was not written Subtractingboth equations and using condition (V40) to relate the time and space derivatives of φ u(ε) andu(δ) one finds

[partu(ε)(t x)

partx+

2φ(t x)

partφ(t x)

]+ δ2

[partu(δ)(t x)

partxminus 1

part3φ(t x)

partx3

Since the two small parameters ε and δ are independent each term between square brackets in thisidentity must identically vanish Straightforward integrations then yield

u(ε)(t x) = minus1

4φ(t x) + C(ε)(t) u(δ)(t x) =

part2φ(t x)

partx2+ C(δ)(t)

with C(ε) C(δ) two functions of time onlyThese functions can then be substituted in the ansatz (V41) Inserting the latter in Eq (V39)

yields an equation involving the unknown function φ only namely

partφ(t x)

partt+partφ(t x)

partx+

2εφ(t x)

partφ(t x)

partx+

6δ2 part

3φ(t x)

partx3= 0 (V42)

The first two terms only are those of the linear-wave equation of motion (V40) Since the ε andδ nonlinear corrections also obey the same condition it is fruitful to perform a change of variablesfrom (t x) to (τ ξ) with τ equiv t ξ equiv xminus t Equation (V42) then becomes

partφ(τ ξ)

partτ+

2εφ(τ ξ)

partφ(τ ξ)

partξ+

6δ2 part

3φ(τ ξ)

partξ3= 0 (V43)

which is the Kortewegndashde Vries equation(ab)(ac)

Remark By rescaling the variables τ and ξ to a new set (τ ξ) one can actually absorb the pa-rameters ε δ which were introduced in the derivation Accordingly the more standard form of theKortewegndashde Vries (KdV) equation is

partφ(τ ξ)

partτ+ 6φ(τ ξ)

partφ(τ ξ)

partξ+part3φ(τ ξ)

partξ3= 0 (V44)

Solitary wavesThe Kortewegndashde Vries equation admits many different solutions Among those there is the class

of solitary waves or solitons which describe signals that propagate without changing their shape

(ab)D Korteweg 1848ndash1941 (ac)G de Vries 1866ndash1934

A specific subclass of solitons of the KdV equation of special interest in fluid dynamics consistsof those which at each given instant vanish at (spatial) infinity As solutions of the normalizedequation (V44) they read

φ(τ ξ) =φ0

cosh2[radicφ02 (ξminus 2φ0τ)

] (V45a)

with φ0 the amplitude of the wave Note that φ0 must be nonnegative which means that thesesolutions describe bumps above the mean sea levelmdashwhich is the only instance of such solitary waveobserved experimentally Going back first to the variables (τ ξ) then to the dimensionless variables(tlowast xlowast) and eventually to the dimensionful variables (t x) and field δh the soliton solution reads

δh(t x) =δhmax

radic3δhmax

[xminusradicgh0

δhmax

] (V45b)

with δhmax the maximum amplitude of the solitary wave This solution represented in Fig V2has a few properties that can be read directly off its expression and differ from those of linear seasurface waves namely

bull the propagation velocity csoliton of the solitonmdashwhich is the factor in front of tmdashis larger thanfor linear waves

bull the velocity csoliton increases with the amplitude δhmax of the soliton

bull the width of the soliton decreases with its amplitude

δhmax=1 t = t0

δhmax= 025 t = t0

δhmax=1 t = t1 gt t0

δhmax= 025 t = t1

δh(t x)

Figure V2 ndash Profile of the soliton solution (V45)

Bibliography for Chapter Vbull National Committee for Fluid Mechanics film amp film notes on Waves in Fluids

bull LandaundashLifshitz [3 4] Chapters I sect 12 VIII sect 64ndash65 IX sect 84ndash85 and X sect 99

bull Sommerfeld [5 6] Chapters III sect 13 V sect 23 24 amp 26 and VII sect 37

CHAPTER VI

Non-relativistic dissipative flows

The dynamics of Newtonian fluids is entirely governed by a relatively simple set of equationsnamely the continuity equation (III9) the NavierndashStokes equation (III31) andmdashwhen phenomenarelated with temperature gradients become relevantmdashthe energy conservation equation (III35) Asin the case of perfect fluids there are a priori more unknown dynamical fields than equations sothat an additional relation has to be provided either a kinematic constraint or an equation ofstate In this Chapter and the next two ones a number of simple solutions of these equationsare presented together with big classes of phenomena that are accounted in various more or lesssimplified situations

With the exception of the static-fluid case in which the only novelty with respect to the hydro-statics of perfect fluids is precisely the possible transport of energy by heat conduction (Sec VI11)the motions of interest in the present Chapter are mostly laminar flows in which viscous effects playan important role while heat transport is negligible Thus the role of the no-slip condition at aboundary of the fluid is illustrated with a few chosen examples of stationary motions within idealizedgeometrical setups (Sec VI1)

By introducing flow-specific characteristic length and velocity scales the NavierndashStokes equa-tion can be rewritten in a form involving only dimensionless variables and fields together withparametersmdashlike for instance the Reynolds number These parameters quantify the relative impor-tance of the several physical effects likely to play a role in a motion (Sec VI2)

According to the value of the dimensionless numbers entering the dynamical equations thelatter may possibly be simplified This leads to simpler equations with limited domain of validityyet which become more easily tractable as exemplified by the case of flows in which shear viscouseffects predominate over the influence of inertia (Sec VI3) Another simplified set of equations canbe derived to describe the fluid motion in the thin layer close to a boundary of the flow in whichthe influence of this boundary plays a significant role (Sec VI4)

Eventually the viscosity-induced modifications to the dynamics of vorticity (Sec VI5) and tothe propagation of sound waves (Sec VI6) are presented

VI1 Statics and steady laminar flows of a Newtonian fluidIn this Section we first write down the equations governing the statics of a Newtonian fluid(Sec VI11) then we investigate a few idealized stationary laminar fluid motions in which thevelocity field is entirely driven by the no-slip condition at boundaries (Secs VI12ndashVI14)

VI11 Static Newtonian fluid

Consider a motionless [~v(t~r) = ~0] Newtonian fluid in an external gravitational potential Φ(~r)mdashor more generally submitted to conservative volume forces such that

90 Non-relativistic dissipative flows

The three coupled equations (III9) (III31) and (III35) respectively simplify to

partρ(t~r)

partt= 0 (VI1a)

from where follows the time independence of the mass density ρ(t~r)

~nablaP (t~r) = minusρ(t~r)~nablaΦ(t~r) (VI1b)

similar to the fundamental equation (IV2) governing the hydrostatics of a perfect fluid and

parte(t~r)

partt= ~nabla middot

[κ(t~r)~nablaT (t~r)

] (VI1c)

which describes the transport of energy without macroscopic fluid motion ie non-convectivelythanks to heat conduction

VI12 Plane Couette flow

In the example of this Section and the next two ones (Secs VI13ndashVI14) we consider steadyincompressible laminar flows in absence of significant volume forces Since the mass density ρ isfixed thus known only four equations are needed to determine the flow velocity~v(~r) and pressureP (~r) the simplest possibility being to use the continuity and NavierndashStokes equations In thestationary and incompressible regime these become

~nabla middot~v(~r) = 0 (VI2a)[~v(~r) middot ~nabla

]~v(~r) = minus1

ρ~nablaP (~r) + ν4~v(~r) (VI2b)

with ν the kinematic shear viscosity assumed to be the same throughout the fluid

The so-called (plane) Couette flow(ad) is in its idealized version the motion of a viscous fluidbetween two infinitely extended plane plates as represented in Fig VI1 where the lower plate isat rest while the upper one moves in its own plane with a constant velocity ~u It will be assumed

Figure VI1 ndash Setup of the plane Couette flow

that the same pressure Pinfin holds rdquoat infinityrdquo in any directionAs the flow is assumed to be laminar the geometry of the problem is invariant under arbitrary

translations in the (x z)-plane This is automatically taken into account by the ansatz~v(~r) = v(y)~exfor the flow velocity Inserting this form in Eqs (VI2) yields

partv(y)

partx= 0 (VI3a)

v(y)partv(y)

partx~ex = minus1

ρ~nablaP (~r) + ν

d2v(y)

dy2~ex (VI3b)

With the ansatz for ~v(~r) the first equation is automatically fulfilled while the term on theleft hand side of the second equation vanishes Projecting the latter on the y and z directionsthus yields partP (~r)party = 0mdashexpressing the assumed absence of sizable effects from gravitymdashand(ad)M Couette 1858ndash1943

VI1 Statics and steady laminar flows of a Newtonian fluid 91

partP (~r)partz = 0mdashsince the problem is independent of z Along the x direction one finds

partP (~r)

partx= η

d2v(y)

dy2 (VI4)

Since the right member of this equation is independent of x and z a straightforward integration givesP (~r) = α(y)x+ β(y) where the functions α β only depend on y These functions are determinedby the boundary conditions since P (x=minusinfin) = P (x=infin) = Pinfin then α(y) = 0 β(y) = Pinfin andEq (VI4) simplifies to

d2v(y)

dy2= 0

This yields v(y) = γy + δ with γ and δ two integration constants which are again fixed by theboundary conditions At each plate the relative velocity of the fluid with respect to the plate mustvanish

v(y=0) = 0 v(y=h) = |~u|

leading to δ = 0 and γ = |~u|h All in all the velocity thus depends linearly on y

~v(~r) =y

h~u for 0 le y le h

Consider now a surface element d2S The contact force d2 ~Fs exerted on it by the fluid followsfrom the Cauchy stress tensor whose Cartesian components (III27c) here read

σij(~r) = minusP (~r)δij + η

[partvi(~r)

partxj+partvj(~r)

partxi

minusPinfin η |~u|h 0

η |~u|h minusPinfin 0

0 0 minusPinfin

The force per unit surface on the motionless plate at y = 0 corresponding to a unit normal vector~en(~r) = ~ey is

d2 ~Fs(~r)

d2S= ~Ts(~r) =

[ 3sumij=1

σij(~r)~ei otimes~ej]middot~ey =

3sumij=1

σij(~r)(~ej middot~ey

)~ei =

η |~u|h

minusPinfin0

Due to the friction exerted by the fluid the lower plate is dragged by the flow in the (positive) xdirection

Remark The tangential stress on the lower plate is η~uh proportional to the shear viscositymeasuring the tangential stress with known |~u| and h yields a measurement of η In practicethis measurement rather involves the more realistic cylindrical analog to the above plane flow theso-called CouettendashTaylor flow (ae)

VI13 Plane Poiseuille flow

Let us now consider the flow of a Newtonian fluid between two motionless plane plates with afinite length along the x directionmdashyet still infinitely extended along the z directionmdash as illustratedin Fig VI2 The pressure is assumed to be different at both ends of the plates in the x directionamounting to the presence of a pressure gradient along x

Assuming for the flow velocity ~v(~r) the same form v(y)~ex independent of x as in the case ofthe plane Couette flow the equations of motion governing v(y) and pressure P (~r) are the same asin the previous Section VI12 namely Eqs (VI3)ndash(VI4) The boundary conditions are howeverdifferent(ae)G I Taylor 1886ndash1975

6yP 1 P 2

Figure VI2 ndash Flow between two motionless plates for P 1 gt P 2 ie ∆P gt 0

Thus P 1 6= P 2 results in a finite constant pressure gradient along x α = partP (~r)partx = minus∆PL 6= 0with ∆P equiv P 1 minus P 2 the pressure drop Equation (VI4) then leads to

v(y) = minus 1

∆PLy2 + γy + δ

with γ and δ two new constantsThe ldquono-sliprdquo boundary conditions for the velocity at the two plates read

v(y=0) = 0 v(y=h) = 0

which leads to δ = 0 and γ =1

∆PLh The flow velocity thus has the parabolic profile

v(y) =1

[y(hminus y)

]for 0 le y le h (VI5)

directed along the direction of the pressure gradient

Remark The flow velocity (VI5) becomes clearly problematic in the limit η rarr 0 Tracing theproblem back to its source the equations of motion (VI3) cannot hold with a finite gradient alongthe x direction and a vanishing viscosity One quickly checks that the only possibility in the caseof a perfect fluid is to drop one of the assumptions either incompressibility or laminarity

VI14 HagenndashPoiseuille flow

The previous two examples involved plates with an infinite length in at least one directionthus were idealized constructions In contrast an experimentally realizable fluid motion is that ofthe HagenndashPoiseuille flow (af) in which a Newtonian fluid flows under the influence of a pressuregradient in a cylindrical tube with finite length L and radius a (Fig VI3) Again the motion isassumed to be steady incompressible and laminar

P 1 P 2-z

Figure VI3 ndash Setup of the HagenndashPoiseuille flow

Using cylindrical coordinates the ansatz ~v(~r) = v(r)~ez with r =radicx2 + y2 satisfies the conti-

nuity equation ~nabla middot~v(~r) = 0 and gives for the incompressible NavierndashStokes equation

~nablaP (~r) = η4~v(~r) hArr

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= η

[part2v(r)

partx2+part2v(r))

party2

[d2v(r)

(af)G Hagen 1797ndash1884

The right member of the equation in the second line is independent of z implying that the pressuregradient along the z direction is constant

partP (~r)

partz= minus∆P

with ∆P equiv P 1 minus P 2 The z component of the NavierndashStokes equation (VI6) thus becomesd2v(r)

dr= minus∆P

ηL (VI7)

As always this linear differential equation is solved in two successive steps starting with theassociated homogeneous equation To find the general solution of the latter one may introduceχ(r) equiv dv(r)dr which satisfies the simpler equation

dχ(r)

dr+χ(r)

The generic solution is lnχ(r) = minus ln r+ const ie χ(r) = Ar with A a constant This then leadsto v(r) = A ln r +B with B an additional constant

A particular solution of the inhomogeneous equation (VI7) is v(r) = Cr2 with C = minus∆P4ηLThe general solution of Eq (VI7) is then given by

v(r) = A ln r +B minus ∆P4ηL

where the two integration constants still need to be determinedTo have a regular flow velocity at r = 0 the constant A should vanish In turn the boundary

condition at the tube wall v(r= a) = 0 determines the value of the constant B = (∆P4ηL)a2All in all the velocity profile thus reads

v(r) =∆P4ηL

(a2 minus r2

)for r le a (VI8)

This is again parabolic with~v in the same direction as the pressure drop

The mass flow rate across the tube cross section follows from a straightforward integration

0ρv(r) 2πr dr = 2πρ

∆P4ηL

(a2r minus r3

)dr = 2πρ

∆P4ηL

4=πρa4

∆PL (VI9)

This result is known as HagenndashPoiseuille law (or equation) and means that the mass flow rate isproportional to the pressure drop per unit length

Remarkslowast The HagenndashPoiseuille law only holds under the assumption that the flow velocity vanishes at thetube walls The experimental confirmation of the lawmdashwhich was actually deduced from experimentby Hagen (1839) and Poiseuille (1840)mdashis thus a proof of the validity of the no-slip assumption forthe boundary condition

lowast The mass flow rate across the tube cross section may be used to define that average flow velocityas Q = πa2ρ〈v〉 with

〈v〉 equiv 1

0v(r) 2πr dr =

2v(r=0)

The HagenndashPoiseuille law then expresses a proportionality between the pressure drop per unit lengthand 〈v〉 in a laminar flow

Viewing ∆PL as the ldquogeneralized forcerdquo driving the motion the corresponding ldquoresponserdquo 〈v〉 ofthe fluid is thus linear

The relation is quite different in the case of a turbulent flow with the same geometry for instancemeasurements by Reynolds [19] gave ∆PL prop 〈v〉1722

VI2 Dynamical similarityThe incompressible motion of a Newtonian fluid is governed by the continuity equation ~nablamiddot~v(t~r) = 0and the NavierndashStokes equation (III32) In order to determine the relative influence of the variousterms of the latter it is often convenient to consider dimensionless forms of the incompressibleNavierndashStokes equation which leads to the introduction of a variety of dimensionless numbers

For instance the effect of the fluid mass density ρ and shear viscosity η (or equivalently ν) whichare uniform throughout the fluid on a flow in the absence of volume forces is entirely encoded inthe Reynolds number (Sec VI21) Allowing for volume forces either due to gravity or to inertialforces their relative influence is controlled by similar dimensionless parameters (Sec VI22)

Let Lc resp vc be a characteristic length resp velocity for a given flow Since the NavierndashStokesequation itself does not involve any parameter with the dimension of a length or a velocity both arecontrolled by ldquogeometryrdquo by the boundary conditions for the specific problem under considerationThus Lc may be the size (diameter side length) of a tube in which the fluid flows or of an obstaclearound which the fluid moves In turn vc may be the uniform velocity far from such an obstacle

With the help of Lc and vc one may rescale the physical quantities in the problem so as toobtain dimensionless quantities which will hereafter be denoted with lowast

~rlowast equiv ~r

Lc ~vlowast equiv

vc tlowast equiv t

Lcvc P lowast equiv P minus P 0

(VI10)

where P 0 is some characteristic value of the (unscaled) pressure

VI21 Reynolds number

Consider first the incompressible NavierndashStokes equation in the absence of external volumeforces Rewriting it in terms of the dimensionless variables and fields (VI10) yields

part~vlowast(tlowast~rlowast)

parttlowast+[~vlowast(tlowast~rlowast) middot ~nablalowast

]~vlowast(tlowast~rlowast) = minus~nablalowastP lowast(tlowast~rlowast) +

ρvcLc4lowast~vlowast(tlowast~rlowast) (VI11)

mit ~nablalowast resp 4lowast the gradient resp Laplacian with respect to the reduced position variable ~rlowastBesides the reduced variables and fields this equation involves a single dimensionless parameterthe Reynolds number

Re equiv ρvcLcη

=vcLcν

(VI12)

This number measures the relative importance of inertia and viscous friction forces on a fluid elementor a body immersed in the moving fluid at large resp small Re viscous effects are negligible resppredominant

Remark As stated above Eq (VI10) both Lc and vc are controlled by the geometry and boundaryconditions The Reynolds numbermdashand every similar dimensionless we shall introduce hereaftermdashisthus a characteristic of a given flow not of the fluid

Law of similitude(lxii)

The solutions for the dynamical fields ~vlowast P lowast at fixed boundary conditions and geometrymdashspecified in terms of dimensionless ratios of geometrical lengthsmdashare functions of the independentvariables tlowast ~rlowast and of the Reynolds number

~vlowast(tlowast~rlowast) =~f1(tlowast~rlowastRe) P lowast(tlowast~rlowast) = f2(tlowast~rlowastRe) (VI13)

with~f1 resp f2 a vector resp scalar function Flow velocity and pressure are then given by

(lxii)Aumlhnlichkeitsgesetz

VI2 Dynamical similarity 95

~v(t~r) = vc~f1

) P (t~r) = P 0 + ρv2

These equations underlie the use of fluid dynamical simulations with experimental models at areduced scale yet possessing the same (rescaled) geometry Let Lc vc resp LM vM be the charac-teristic lengths of the real-size flow resp of the reduced-scale experimental flow for simplicity weassume that the same fluid is used in both cases If vMvc = LcLM the Reynolds number for theexperimental model is the same as for the real-size fluid motion both flows then admit the samesolutions~vlowast and P lowast and are said to be dynamically similar

Remark The functional relationships between the ldquodependent variablesrdquo~vlowast P lowast and the ldquoindependentvariablesrdquo tlowast ~rlowast and a dimensionless parameter (Re) represent a simple example of the more general(Vaschy(ag)ndash)Buckingham(ah) π-theorem [20] in dimensional analysis see eg Refs [21 22] Chapter 7or [23]

VI22 Other dimensionless numbers

If the fluid motion is likely to be influenced by gravity the corresponding volume force density~fV = minusρ~g must be taken into account in the right member of the incompressible NavierndashStokesequation (III32) Accordingly if the latter is written in dimensionless form as in the previousSection there will come an additional term on the right hand side of Eq (VI11) proportional to1Fr2 with

Fr equiv vcradicgLc

(VI14)

the Froude number (ai) This dimensionless parameter measures the relative size of inertial andgravitational effects in the flow the latter being important when Fr is small

In the presence of gravity the dimensionless dynamical fields ~vlowast P lowast become functions of thereduced variables tlowast ~rlowast controlled by both parameters Re and Fr

The NavierndashStokes equation (III31) holds in an inertial frame In a non-inertial reference framethere come additional terms which may be expressed as fictive force densities on the right hand sidewhich come in addition to the ldquophysicalrdquo volume force density ~fV In the case of a reference frame inuniform rotation (with respect to an inertial frame) with angular velocity ~Ω0 there are thus two extracontributions corresponding to centrifugal and Coriolis forces namely ~fcent = minusρ~nabla

[minus 1

(~Ω0times~r

)2]and ~fCor = minus2ρ~Ω0times~v respectively

The relative importance of the latter in a given flow can be estimated with dimensionless num-bers Thus the Ekman number (aj)

Ek equiv η

ρΩL2c

(VI15)

measures the relative size of (shear) viscous and Coriolis forces with the latter predominating overthe former when Ek 1

One may also wish to compare the influences of the convective and Coriolis terms in the NavierndashStokes equation This is done with the help of the Rossby number (ak)

Ro equiv vcΩLc

(VI16)

which is small when the effect of the Coriolis force is the dominant one

Remark Quite obviously the Reynolds (VI12) Ekman (VI15) and Rossby (VI16) numbers obeythe simple identity

Ro = Re middot Ek

(ag)A Vaschy 1857ndash1899 (ah)E Buckingham 1867ndash1940 (ai)W Froude 1810ndash1879 (aj)V Ekman 1874ndash1954(ak)C-G Rossby 1898ndash1957

VI3 Flows at small Reynolds numberThis Section deals with incompressible fluid motions at small Reynolds number Re 1 ie inthe situation in which shear viscous effects predominate over those of inertia in the NavierndashStokesequation Such fluid motions are also referred to as Stokes flows or creeping flows(lxiii)

VI31 Physical relevance Equations of motion

Flows of very different nature may exhibit a small Reynolds number (VI12) because the lattercombines physical quantities whose value can vary by many orders of magnitude in Nature(29) Afew examples of creeping flows are listed hereafter

bull The motion of fluids past microscopic bodies the small value of the Reynolds number thenreflects the smallness of the length scale Lc for instance

ndash In water (η asymp 10minus3 Pamiddots ie ν asymp 10minus6 m2 middot sminus1) a bacteria of size Lc asymp 5 microm ldquoswimsrdquowith velocity vc asymp 10 microm middot sminus1 so that Re asymp 5 middot 10minus5 for the motion of the water past thebacteria if the bacteria stops propelling itself the friction exerted by the water bringsit immediately to rest(30) Similarly creeping flows are employed to describe the motionof reptiles in sandmdashor more precisely the flow of sand a past an undulating reptile [25]

ndash The motion of a fluid past a suspension of small size (Brownian) particles This will bestudied at further length in Sec VI32

bull The slow-velocity motion of geological material in that case the small value of vc and thelarge shear viscosity compensate the possibly large value of the typical length scale Lc

For example the motion of the Earthrsquos mantle(31) with Lc asymp 100 km vc asymp 10minus5 m middot sminus1ρ asymp 5 middot 103 kg middotmminus3 and η asymp 1022 Pa middot s corresponds to a Reynolds number Re asymp 5 middot 10minus19

Note that the above examples all represent incompressible flows For the sake of simplicity we shallalso only consider steady motions

VI31 a

Stokes equation

Physically a small Reynolds number means that the influence of inertia is much smaller thanthat of shear viscosity That is the convective term

(~v middot ~nabla

)~v in the NavierndashStokes equation is

negligible with respect to the viscous contribution Assuming further stationaritymdashwhich allows usto drop the time variablemdashand incompressibility the NavierndashStokes equation (III31) simplifies tothe Stokes equation

~nablaP (~r) = η4~v(~r) + ~fV (~r) (VI17)

This constitutes a linearization of the incompressible NavierndashStokes equation

Using the relation~nablatimes

[~nablatimes ~c(~r)

]= ~nabla

[~nabla middot ~c(~r)

]minus4~c(~r) (VI18)

(29)This is mostly true of the characteristic length and velocity scales and of the shear viscosity in (non-relativistic)fluids the mass density is always of the same order of magnitude up to a factor 103

(30)A longer discussion of the motion of bacteriamdashfrom a physicist point of viewmdash together with the original formu-lation of the ldquoscallop theoremrdquo can be found in Ref [24]

(31)From the mass density the shear viscosity and the typical speed of sound cs asymp 5000 mmiddotsminus1 of transverse wavesmdashie shear waves that may propagate in a solid but not in a fluidmdash one constructs a characteristic time scaletmantle = ηρc2s asymp 3000 years For motions with a typical duration tc tmantle the Earthrsquos mantle behaves like adeformable solid for instance with respect to the propagation of sound waves following an earthquake On theother hand for motions on a ldquogeologicalrdquo time scale tc tmantle the mantle may be modeled as a fluid

(lxiii)schleichende Stroumlmungen

VI3 Flows at small Reynolds number 97

valid for any vector field ~c(~r) the incompressibility condition and the definition of vorticity theStokes equation can be rewritten as

~nablaP (~r) = minusη~nablatimes ~ω(~r) (VI19)

As a result the pressure satisfies the differential Laplace equation

4P (~r) = 0 (VI20)

In practice however this equation is not the most useful because the boundary conditions in aflow are mostly given in terms of the flow velocity in particular at walls or obstacles not of thepressure

Taking the curl of Eq (VI19) and invoking again relation (VI18) remembering that the vorticityvector is itself already a curl one finds

4~ω(~r) = ~0 (VI21)ie the vorticity also obeys the Laplace equation We shall see in Sec VI5 that the more generaldynamical equation obeyed by vorticity in Newtonian fluids does indeed yield Eq (VI21) in thecase of stationary small Reynolds number flows

VI31 b

Properties of the solutions of the Stokes equation

Thanks to the linearity of the Stokes equation (VI17) its solutions possess various properties(32)

bull Uniqueness of the solution at fixed boundary conditions

bull Additivity of the solutions if~v1 and~v2 are solutions of Eq (VI17) with respective boundaryconditions then the sum λ1~v1 +λ2~v2 with real numbers λ1 λ2 is also a solution for a problemwith adequate boundary conditionsPhysically the multiplying factors should not be too large to ensure that the Reynolds numberof the new problem remains small The multiplication of the velocity field~v(~r) by a constantλ represents a change in the mass flow rate while the streamlines (I15) remain unchanged

The dimensionless velocity field~vlowast associated with the two solutions~v(~r) and λ~v(~r) is the sameprovided the differing characteristic velocities vc resp λvc are used In turn these define differentvalues of the Reynolds number For these solutions~vlowast as given by Eq (VI13) is thus independentof the parameter Re and thereby only depends on the variable ~rlowast ~v = vc ~f

) This also holds

for the corresponding dimensionless pressure P lowastUsing dimensional arguments only the tangential stress is ηpartvipartxj sim ηvcLc so that the

friction force on an object of linear size(33) Lc is proportional to ηvcLc This result will now beillustrated on an explicit example [cf Eq (VI26)] for which the computation can be performedanalytically

VI32 Stokes flow past a sphere

Consider a sphere with radius R immersed in a fluid with mass density ρ and shear viscosity ηwhich far from the sphere flows with uniform velocity ~vinfin as sketched in Fig VI4 The goal is todetermine the force exerted by the moving fluid on the sphere which necessitates the calculationof the pressure and velocity in the flow Given the geometry of the problem a system of sphericalcoordinates (r θ ϕ) centered on the sphere center will be used

The Reynolds number Re = ρ|~vinfin|Rη is assumed to be small so that the motion in the vicinityof the sphere can be modeled as a creeping flow which is further taken to be incompressibleFor the flow velocity one looks for a stationary solution of the equations of motion of the form(32)Proofs can be found eg in Ref [2 Chapter 823](33)As noted in the introduction to Sec VI2 the characteristic length and velocity scales in a flow are precisely

determined by the boundary conditions

~vinfin ~er~eϕ

Figure VI4 ndash Stokes flow past a sphere

~v(~r) = ~vinfin +~u(~r) with the boundary condition ~u(~r) = ~0 for |~r| rarr infin In addition the usualimpermeability and no-slip conditions hold at the surface of the sphere resulting in the requirement~u(|~r|=R) = minus~vinfin

Using the linearity of the equations of motion for creeping flows ~u obeys the equations

4[~nablatimes~u(~r)

]= ~0 (VI22a)

and~nabla middot~u(~r) = 0 (VI22b)

which comes from the incompressibility conditionThe latter equation is automatically satisfied if ~u(~r) is the curl of some vector field ~V (~r) Using

dimensional considerations the latter should depend linearly on the only explicit velocity scale inthe problem namely~vinfin Accordingly one makes the ansatz(34)

~V (~r) = ~nablatimes[f(r)~vinfin

]= ~nablaf(r)times~vinfin

with f(r) a function of r = |~r| ie f only depends on the distance from the sphere apart from thedirection of ~vinfin which is already accounted for in the ansatz there is no further preferred spatialdirection so that f should be spherically symmetric

Relation (VI18) together with the identity ~nabla middot [f(r)~vinfin] = ~nablaf(r) middot~vinfin then yield

~u(~r) = ~nablatimes ~V (~r) = ~nabla[~nablaf(r) middot~vinfin

]minus4f(r)~vinfin (VI23)

The first term on the right hand side has a vanishing curl and thus does not contribute wheninserting ~u(~r) in equation (VI22a)

~nablatimes~u(~r) = minus~nablatimes[4f(r)~vinfin

]= minus~nabla

[4f(r)

]times~vinfin

so that4(~nabla[4f(r)

])times~vinfin = ~0

Since f(r) does not depend on the azimuthal and polar angles 4(~nabla[4f(r)

])only has a com-

ponent along the radial direction with (unit) basis vector ~er as thus it cannot be always parallelto ~vinfin Therefore 4

(~nabla[4f(r)

])must vanish identically for the above equation to hold One can

checkmdashfor instance using componentsmdashthe identity 4(~nabla[4f(r)

])= ~nabla

(4[4f(r)]

) so that the

equation obeyed by f(r) becomes4[4f(r)] = const

The integration constant must be zero since it is a fourth derivative of f(r) while the velocity~u(~r)which according to Eq (VI23) depends on the second derivatives must vanish as r rarr infin Onethus has

4[4f(r)] = 0

(34)The simpler guesses~u(~r) = f(r)~vinfin or~u(~r) = ~nablaf(r)times~vinfin are both unsatisfactory the velocity~u(~r) is then alwaysparallel resp orthogonal to~vinfin so that~v(~r) cannot vanish everywhere at the surface of the sphere

In spherical coordinates the Laplacian reads

4 =part2

partr2+

partrminus `(`+ 1)

with ` an integer that depends on the angular dependence of the function given the sphericalsymmetry of the problem for f one should take ` = 0 Making the ansatz 4f(r) = Crα theequation 4[4f(r)] = 0 is only satisfied for α = 0 or 1 Using Eq (VI23) and the condition~u(~r)rarr ~0 for r rarrinfin only α = 1 is possible

The general solution of the linear differential equation

4f(r) =d2f(r)

r(VI24a)

is then given byf(r) = A+

2r (VI24b)

where the first two terms in the right member represent the most general of the associated homo-geneous equation while the third term is a particular solution of the inhomogeneous equation

Equations (VI23) and (VI24) lead to the velocity field

~u(~r) = ~nabla[(minusB ~r

)middot~vinfin

]minus C

r~vinfin = minusB

~vinfin minus 3(~er middot~vinfin

~vinfin minus(~er middot~vinfin

rminus C

r~vinfin

= minusB~vinfin minus 3

(~er middot~vinfin

r3minus C

~vinfin +(~er middot~vinfin

The boundary condition ~u(|~r|=R) = minus~vinfin at the surface of the sphere translates into(1minus B

R3minus C

)~vinfin +

R3minus C

)(~er middot~vinfin

)~er = ~0

This must hold for any ~er which requires B = R34 and C = 6BR2 = 3R2 leading to

~v(~r) =~vinfin minus3R

[~vinfin +

(~er middot~vinfin

)~er]minus R3

[~vinfin minus 3

(~er middot~vinfin

)~er] (VI25)

Inserting this flow velocity in the Stokes equation (VI17) gives the pressure

P (~r) =3

~er middot~vinfinr2

+ const

With its help one can then compute the mechanical stress (III28) at a point on the surface of thesphere The total force exerted by the flow on the latter follows from integrating the mechanicalstress over the whole surface and equals

~F = 6πRη~vinfin (VI26)

This result is referred as Stokesrsquo law Inverting the point of view a sphere moving with velocity~vsphere in a fluid at rest undergoes a friction force minus6πRη~vsphere

Remarkslowast For the potential flow of a perfect fluid past a sphere with radius R the flow velocity is(35)

~v(~r) =~vinfin +R3

[~vinfin minus 3

(~er middot~vinfin

That is the velocity varies much faster in the vicinity of the sphere than for the Stokes flow (VI25)in the latter case momentum is transported not only convectively but also by viscosity whichredistributes it over a wider region

The approximation of a flow at small Reynolds number described by the Stokes equation actuallyonly holds in the vicinity of the sphere Far from it the flow is much less viscous(35)The proof can be found eg in LandaundashLifshitz [3 4] sect 10 problem 2

lowast In the limit η rarr 0 corresponding to a perfect fluid the force (VI26) exerted by the flow on thesphere vanishes this is again the drsquoAlembert paradox encountered in sect IV43 c

lowast The proportionality factor between the sphere velocity and the friction force it experiences iscalled the mobility(lxiv) micro According to Stokesrsquo law (VI26) for a sphere in the creeping-flow regimeone has micro = 1(6πRη)In his famous article on Brownian motion [26] A Einstein related this mobility with the diffusioncoefficient D of a suspension of small spheres in a fluid at rest

D = microkBT =kBT

6πRη

This formula (StokesndashEinstein equation) was checked experimentally by J Perrin which allowedhim to determine a value of the Avogadro constant and to prove the ldquodiscontinuous structure ofmatterrdquo [27]

VI4 Boundary layerThe Reynolds number defined in Sec VI21 which quantifies the relative importances of viscousand inertial effects in a given flow involves characteristic length and velocity scales Lc vc thatdepend on the geometry of the fluid motion When the flow involves an obstacle as was the case inthe example presented in Sec VI32 a natural choice when studying the details of the fluid motionin the vicinity of the obstacle is to adopt the typical size R of the latter as characteristic length Lcdefining the Reynolds number

Far from the obstacle however it is no longer obvious that R is really relevant For Lc a betterchoice might be the distance to the obstaclemdashor to any other wall or object present in the problemSuch a characteristic length gives a Reynolds number which can be orders of magnitude larger thanthe value computed with Lc That is even if the flow is viscous (small Re) close to the obstacle farfrom it the motion could still be to a large extent inviscid (large Re) and thus well approximatedby a perfect-fluid description

The above argumentation suggests that viscous effects may not be relevant throughout the wholefluid but only in the region(s) in the vicinity of walls or obstacles This is indeed the case andthe corresponding region surrounding walls or obstacles is referred to as boundary layer (lxv) In thelattermdashwhich often turns out to be rather thinmdash the velocity grows rapidly from its vanishing valueat the surface of objects (no-slip condition) to the finite value it takes far from them and which ismostly imposed by the boundary conditions ldquoat infinityrdquo

In this Section we shall first illustrate on an example flow the existence of the boundary layercomputing in particular its typical width (Sec VI41) The latter can also be estimated in a moregeneral approach to the description of the fluid motion inside the boundary layer as sketched inSec VI42

VI41 Flow in the vicinity of a wall set impulsively in motion

Consider an incompressible Newtonian fluid with uniform kinematic shear viscosity ν situatedin the upper half-space y gt 0 at rest for t lt 0 The volume forces acting on the fluid are supposedto be negligible

At t = 0 the plane y = 0 is suddenly set in uniform motion parallel to itself with constantvelocity ~u(t gt 0) = u~ex As a consequence the fluid in the vicinity of the plane is being draggedalong thanks to the viscous forces the motion is transfered to the next fluid layers The resultingflow is assumed to be laminar with a fluid velocity parallel to ~ex(lxiv)Beweglichkeit Mobilitaumlt (lxv)Grenzschicht

VI4 Boundary layer 101

The invariance of the problem geometry under translations in the x- or z-directions justifiesan ansatz ~v(t~r) = v(t y)exmdashwhich automatically fulfills the incompressibility conditionmdash andsimilarly for the pressure field That is there are no gradients along the x- and z-directions As aresult the incompressible NavierndashStokes equation (III32) projected onto the x-direction reads

partv(t y)

partt= ν

part2v(t y)

party2 (VI27a)

The boundary conditions are on the one hand the no-slip requirement at the moving plane namely

v(t y=0) = u for t gt 0 (VI27b)

on the other hand the fluid infinitely far from the moving plane remains unperturbed ie

limyrarrinfin

v(t y) = 0 for t gt 0 (VI27c)

Eventually there is the initial condition

v(t=0 y) = 0 forally gt 0 (VI27d)

The equations governing the motion (VI27) involve only two dimensionful quantities namelythe plane velocity u and the fluid kinematic viscosity ν With their help one can construct acharacteristic time νu2 and a characteristic length νu in a unique manner up to numerical factorsInvoking dimensional arguments one thus sees that the fluid velocity is necessarily of the form

v(t y) = uf1

with f1 a dimensionless function of dimensionless variables Since t and y are independent so aretheir reduced versions u2tν and uyν Instead of the latter one may adopt the equivalent setu2tν ξ equiv y(2

radicνt) ie write

v(t y) = uf2

2radicνt

with f2 again a dimensionless functionThe whole problem (VI27) is clearly linear in u since the involved differential equationsmdash

continuity equation and NavierndashStokes equation (VI27a)mdashare linear this allows us to exclude anydependence of f2 on the variable u2tν so that the solution is actually of the form

v(t y) = uf

2radicνt

)(VI28)

with f dimensionless ie dependent on a single reduced variableInserting the latter ansatz in Eq (VI27a) leads after some straightforward algebra to the ordi-

nary differential equationf primeprime(ξ) + 2ξ f prime(ξ) = 0 (VI29a)

with f prime f primeprime the first two derivatives of f Meanwhile the boundary conditions (VI27b)ndash(VI27c)become

f(0) = 1 limξrarrinfin

f(ξ) = 0 (VI29b)

The corresponding solution isf(ξ) = erfc(ξ) = 1minus erf(ξ) (VI30)

where erf denotes the (Gauss) error function defined as(36)

erf(ξ) equiv 2radicπ

int ξ

0eminusυ

2dυ (VI31)

(36)The reader interested in its properties can have a look at the NIST Handbook of mathematical functions [28]Chapter 7

while erfc is the complementary error function(36)

erfc(ξ) equiv 2radicπ

int infinξ

eminusυ2dυ (VI32)

All in all the solution of the problem (VI27) is thus

v(t y) = u

[1minus erf

2radicνt

)] (VI33)

For ξ = 2 erf(2) = 099532 ie erfc(2) asymp 0005 That is at given t the magnitude of thevelocity at

y = δl(t) equiv 4radicνt (VI34)

is reduced by a factor 200 with respect to its value at the moving plane This length δl(t) is atypical measure for the width of the boundary layer over which momentum is transported from theplane into the fluid ie the region in which the fluid viscosity plays a role

The width (VI34) of the boundary layer increases with the square root of time this is thetypical behavior expected for a diffusive processmdashwhich is understandable since Eq (VI27a) isnothing but the classical diffusion equation

Remark The above problem is often referred to as first Stokes problem or Rayleigh problem(al) Inthe second Stokes problem the plane is not set impulsively into motion it oscillates sinusoidallyin its own plane with a frequency ω In that case the amplitude of the induced fluid oscillationsdecrease ldquoonlyrdquo exponentially with the distance to the plane and the typical extent of the regionaffected by shear viscous effects is

radicνω

VI42 Modeling of the flow inside the boundary layer

As argued in the introduction to the present Section the existence of a ldquosmallrdquo boundary layer towhich the effects induced by viscosity in the vicinity of an obstaclemdashmore specifically the influenceof the no-slip condition at the boundariesmdashare confined can be argued to be a general featureTaking its existence as grantedmdashwhich is not necessary the case for every flowmdash we shall nowmodel the fluid motion inside such a boundary layer

For simplicity we consider a steady incompressible two-dimensional flow past an obstacle oftypical size Lc in the absence of relevant volume forces At each point of the surface of the obstaclethe curvature radius is assumed to be large with respect to the local width δl of the boundary layerThat is using local Cartesian (x y) coordinates with x resp y parallel resp orthogonal to thesurface the boundary layer has a large sizemdashof order Lcmdashalong the x-direction while it is muchthinnermdashof order δlmdashalong y For the sake of brevity the variables (x y) of the various dynamicalfields vx vy P will be omitted

For the fluid inside the boundary layer the equations of motion are on the one hand the incom-pressibility condition ~nabla middot~v = 0 ie

partvxpartx

+partvyparty

= 0 (VI35a)

on the other hand the incompressible NavierndashStokes equation (III32) projected on the x- andy-axes gives (

vxpart

partx+ vy

)vx = minus1

partPpartx

(part2

partx2+part2

party2

)vx (VI35b)(

vxpart

partx+ vy

)vy = minus1

partPparty

(part2

partx2+part2

party2

)vy (VI35c)

(al)J W Strutt Lord Rayleigh 1842ndash1919

Since the boundary layer is much extended along the tangential direction than along the normalone the range of x values is much larger than that of y values To obtain dimensionless variablestaking their values over a similar interval one defines

xlowast equiv x

Lc ylowast equiv y

δl(VI36)

where the typical extent in the normal direction ie the width of the boundary layer

δl Lc (VI37)

has to be determined be requiring that both xlowast ylowast are of order unity

Remarkslowast In realistic cases the width δl may actually depend on the position x along the flow boundaryyet this complication is ignored here

lowast If the local radius of curvature of the boundary is not much larger than the width δl of the bound-ary layer one should use curvilinear coordinates x1 (tangential to the boundary) and x2 instead ofCartesian ones yet within that alternative coordinate system the remainder of the derivation stillholds

Similarly the dynamical fields are rescaled to yield dimensionless fields

vlowastx equivvxvinfin

vlowasty equivvyu

P lowast equiv Pρv2infin (VI38)

where in order to account for the expectation that the normal velocity vy is (in average) muchsmaller than the tangential one vx which is of order vinfin at the outer edge of the boundary layer asecond velocity scale

u vinfin (VI39)

was introduced such that vlowastx vlowasty and P lowast are of order unity These fields are functions of thedimensionless variables (xlowast ylowast) although this shall not be written explicitly

Eventually the Reynolds number corresponding to the motion along x is

Re equiv Lcvinfinν

(VI40)

With the help of definitions (VI36)ndash(VI40) the equations of motion (VI35) can be recast in adimensionless form

+Lcδl

vinfin

partvlowastypartylowast

= 0 (VI41a)

vlowastxpartvlowastxpartxlowast

+Lcδl

vinfinvlowastypartvlowastxpartylowast

= minuspartP lowast

partxlowast+

(part2vlowastxpartylowast2

part2vlowastxpartxlowast2

) (VI41b)

vinfinvlowastxpartvlowastypartxlowast

+Lcδl

v2infin

vlowastypartvlowastypartylowast

= minusLcδ

partP lowast

partylowast+

vinfin

(part2vlowastypartylowast2

part2vlowastypartxlowast2

) (VI41c)

Consider first the continuity equation (VI41a) It will only yield a non-trivial constraint on theflow if both terms have the same order of magnitude which is possible if

vinfin= 1 (VI42)

yielding a first condition on the unknown characteristic quantities δl and uIn turn a second constraint comes from the dimensionless NavierndashStokes equation (VI41b)

along the tangential direction In the boundary layer by definition the effects from inertia encodedin the convective term and those of viscosity are of the same magnitude which necessitates that

the prefactor of the viscous term be of order unity This suggests the conditionL2c

Re= 1 (VI43)

Equations (VI42)ndash(VI43) are then easily solved yielding for the unknown quantities charac-terizing the flow along the direction normal to the boundary

δl =LcradicRe

u =vinfinradicRe (VI44)

As in the first or second Stokes problems see eg Eq (VI34) the width of the boundary layer isproportional to the square root of the kinematic viscosity ν

Substituting the conditions (VI42)ndash(VI43) in the system of equations (VI41) and keeping onlythe leading terms one eventually obtains

+partvlowastypartylowast

= 0 (VI45a)

+ vlowastypartvlowastxpartylowast

= minuspartP lowast

partxlowast+part2vlowastxpartylowast2

(VI45b)

partP lowast

partylowast= 0 (VI45c)

These equations constitute the simplified system first by written down by Prandtl(am) that describethe fluid motion in a laminar boundary layermdashwhere the laminarity assumption is hidden in the useof the typical length scale Lc imposed by geometry rather than of a smaller one driven by turbulentpatterns

VI5 Vortex dynamics in Newtonian fluidsThe equations derived in Sec IV32 regarding the behavior of vorticity in a perfect fluid are easilygeneralized to the case of a Newtonian fluid

VI51 Vorticity transport in Newtonian fluids

As was done with the Euler equation when going from Eq (III18) to the Eq (III20) one mayrewrite the convective term in the NavierndashStokes equation (III31) as

(~v middot ~nabla

)~v = 1

2~nabla(~v2)

+~v times ~ωAssuming then that the volume forces are conservative ie ~fV = minusρ~nablaΦ and taking the rotationalcurl one easily finds

part~ω(t~r)

]= minus

ρ(t~r)2+ ν4~ω(t~r) (VI46)

which generalizes Eq (IV20) to the case of Newtonian fluids Note that even without assumingthat the flow is incompressible the term involving the bulk viscosity has already dropped out fromthe problem

As in Sec IV32 the second term in the left member can be further transformed which leadsto the equivalent forms

D~ω(t~r)

Dt=[~ω(t~r) middot ~nabla

]~v(t~r)minus

]~ω(t~r)minus

ρ(t~r)2+ ν4~ω(t~r) (VI47a)

involving the material derivative D~ωDt or else

D~v ~ω(t~r)

Dt= minus

]~ω(t~r)minus

ρ(t~r)2+ ν4~ω(t~r) (VI47b)

which makes use of the comoving time-derivative (IV22a)(am)L Prandtl 1875ndash1953

VI5 Vortex dynamics in Newtonian fluids 105

The right hand side of this equation simplifies in various cases In the particular of a barotropicfluid the second term vanishes In an incompressible flow the first two terms are zero

As we shall illustrate on an example the viscous term proportional to the Laplacian of vorticityis of diffusive nature and tends to spread out the vorticity lines over a larger region

VI52 Diffusion of a rectilinear vortex

As example of application of the equation of motion introduced in the previous Section let usconsider the two-dimensional motion in the (x y)-plane of an incompressible Newtonian fluid withconservative forces in which there is at t = 0 a rectilinear vortex along the z-axis

~ω(t=0~r) =Γ0

2πrδ(z)~ez (VI48)

with r the distance from the z-axis Obviously the circulation around any curve circling this vortexonce is simply Γ0

At time t gt 0 this vortex will start diffusing with its evolution governed by Eq (VI46) Giventhe symmetry of the problem round the z-axis which suggests the use of cylindrical coordinates(r θ z) the vorticity vector will remain parallel to ~ez and its magnitude should only depend on r

~ω(t~r) = ωz(t r)~ez (VI49)

This results in a velocity field ~v(t~r) in the (x y)-plane in the orthoradial direction As a conse-quence the convective derivative in the left hand side of Eq (VI47a) vanishes since ~ω(t r) has nogradient along eθ Similarly the term

(~ω middot ~nabla

)~v also vanishes since the velocity is independent of z

Eventually the term involving ~nabla middot~v vanishes thanks to the assumed incompressibility All in allthe vorticity thus obeys the diffusion equation

partωz(t r)

partt= ν4ωz(t r) = ν

[part2ωz(t r)

partr2+

partωz(t r)

] (VI50)

with the initial condition (VI48)

The problem is clearly linear in Γ0 so that the solution ωz(t r) should be proportional to Γ0without any further dependence on Γ0 This leaves the kinematic viscosity ν as only dimensionfulparameter available in the problem using a dimensional reasoning similar to that made in the studyof the first Stokes problem (Sec VI41) there is a single relevant dimensionless variable namelyξ = r2(νt) combining the time and space variables The only ansatz respecting the dimensionalrequirements is then

ωz(t r) =Γ0

νtf(ξ) with ξ equiv r2

νt(VI51)

with f a dimensionless function Inserting this ansatz into Eq (VI50) leads to the ordinary differ-ential equation

f(ξ) + ξ f prime(ξ) + 4[f prime(ξ) + ξ f primeprime(ξ)] = 0 (VI52)

A first integration yieldsξ f(ξ) + 4ξ f prime(ξ) = const

In order to satisfy the initial condition the integration constant should be zero leaving with thelinear differential equation f(ξ) + 4f prime(ξ) = 0 which is readily integrated to yield

f(ξ) = C eminusξ4

that isωz(t r) =

νtC eminusr

2(4νt) (VI53)

with C an integration which still has to be fixed

To determine the latter let us consider the circulation of the velocity at time t around a circleCR of radius R centered on the axis z = 0

Γ(t R) =

~v(t~r) middot d~=

int 2π

0ωz(t r) r dr dθ = 2π

0ωz(t r) r dr dθ (VI54)

where the second identity follows from the Stokes theorem while the third is trivial Inserting thesolution (VI53) yields

Γ(t R) = 4πΓ0C[1minus eminusR

2(4νt)]

showing the C should equal 14π to yield the proper circulation at t = 0 All in all the vorticityfield in the problem reads

ωz(t r) =Γ0

4πνteminusr

2(4νt) (VI55)

That is the vorticity extends over a region of typical width δ(t) =radic

4νt which increases with timeone recognizes the characteristic diffusive behavior proportional to

radictmdashas well as the typical

radicν

dependence of the size of the region affected by viscous effects encountered in Sec VI4The vorticity (VI55) leads to the circulation around a circle of radius R

Γ(t R) = Γ0

[1minus eminusR

2(4νt)] (VI56)

which at given R decreases with time in contrast to the perfect-fluid case in which the circulationwould be conserved

Eventually one can also easily compute the velocity field associated with the expanding vortexnamely

~v(t~r) =Γ0

[1minus eminusr

2(4νt)]~eθr (VI57)

where |~eθ| = r

VI6 Absorption of sound wavesUntil now we only investigated incompressible motions of Newtonian fluids in which the bulkviscosity can from the start play no role The simplest example of compressible flow is that of soundwaves which were already studied in the perfect-fluid case As in Sec V11 we consider smalladiabatic perturbations of a fluid initially at rest and with uniform propertiesmdashwhich implies thatexternal volume forces like gravity are neglected Accordingly the dynamical fields characterizingthe fluid are

ρ(t~r) = ρ0 + δρ(t~r) P (t~r) = P 0 + δP (t~r) ~v(t~r) = ~0 + δ~v(t~r) (VI58a)

with|δρ(t~r)| ρ0 |δP (t~r)| P 0

∣∣δ~v(t~r)∣∣ cs (VI58b)

where cs denotes the quantity which in the perfect-fluid case was found to coincide with the phasevelocity of similar small perturbations ie the ldquospeed of soundrdquo defined by Eq (V5)

c2s equiv

(partPpartρ

(VI58c)

As in Sec V11 this relation will allow us to relate the pressure perturbation δP to the variationof mass density δρ

Remark Anticipating on later findings the perturbations must actually fulfill a further conditionrelated to the size of their spatial variations [cf Eq (VI68)] This is nothing but the assumptionof ldquosmall gradientsrdquo that underlies the description of their propagation with the NavierndashStokesequation ie with first-order dissipative fluid dynamics

For the sake of simplicity we consider a one-dimensional problem ie perturbations propagatingalong the x-direction and independent of y and zmdashas are the properties of the underlying backgroundfluid Under this assumption the continuity equation (III9) reads

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (VI59a)

while the NavierndashStokes equation (III31) becomes

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

]= minuspartδP (t x)

partx+

3η + ζ

)part2δv(t x)

partx2 (VI59b)

Substituting the fields (VI58a) in these equations and linearizing the resulting equations so as tokeep only the leading order in the small perturbations one finds

partδρ(t x)

partt+ ρ0

partδv(t x)

partx= 0 (VI60a)

ρ0partδv(t x)

partt= minuspartδP (t x)

partx+

3η + ζ

)part2δv(t x)

partx2 (VI60b)

In the second equation the derivative part(δP )partx can be replaced by c2s part(δρ)partx Let us in addition

introduce the (traditional) notation(37)

ν equiv 1

3η + ζ

) (VI61)

so that Eq (VI60b) can be rewritten as

ρ0partδv(t x)

partt+ c2

partδρ(t x)

partx= ρ0ν

part2δv(t x)

partx2 (VI62)

Subtracting c2s times the time derivative of Eq (VI60a) from the derivative of Eq (VI62) with

respect to x and dividing the result by ρ then yields

part2δv(t x)

partt2minus c2

part2δv(t x)

partx2= ν

part3δv(t x)

partt partx2 (VI63a)

One easily checks that the mass density variation obeys a similar equation

part2δρ(t x)

partt2minus c2

part2δρ(t x)

partx2= ν

part3δρ(t x)

partt partx2 (VI63b)

In the perfect-fluid case ν = 0 one recovers the traditional wave equation (V9a)

Equations (VI63) are homogeneous linear partial differential equations whose solutions can bewritten as superposition of plane waves Accordingly let us substitute the Fourier ansatz

δρ(t~r) = δρ(ω~k) eminusiωt+i~kmiddot~r (VI64)

in Eq (VI63b) This yields after some straightforward algebra the dispersion relation

ω2 = c2sk

2 minus iωk2ν (VI65)

Obviously ω and k cannot be simultaneously real numbers

Let us assume k isin R and ω = ωr + iωi where ωr ωi are real The dispersion relation becomes

ω2r minus ω2

i + 2iωrωi = c2sk

2 minus iωrk2ν + ωik

(37)Introducing the kinetic shear resp bulk viscosity coefficients ν resp νprime one has ν = 43ν + νprime hence the notation

which can only hold if both the real and imaginary parts are equal The identity between theimaginary parts reads (for ωr 6= 0)

ωi = minus1

2νk2 (VI66)

which is always negative since ν is non-negative This term yields in the Fourier ansatz (VI64)an exponentially decreasing factor eminusi(iωi)t = eminusνk

2t2 which represents the damping or absorptionof the sound wave The perturbations with larger wave number k ie corresponding to smallerlength scales are more dampened that those with smaller k This is quite natural since a larger kalso means a larger gradient thus an increased influence of the viscous term in the NavierndashStokesequation

In turn the identity between the real parts of the dispersion relation yields

ω2r = c2

sk2 minus 1

4ν2k4 (VI67)

This gives for the phase velocity cϕ equiv ωk of the traveling waves

c2ϕ = c2

s minus1

4ν2k2 (VI68)

That is the ldquospeed of soundrdquo actually depends on its wave number k and is smaller for smallwavelength ie high-k perturbationsmdashwhich are also those which are more damped

Relation (VI68) also shows that the whole linear description adopted below Eqs (VI59) requiresthat the perturbations have a relatively large wavelength namely k 2csν so that cϕ remain real-valued This is equivalent to stating that the dissipative term ν4δv sim k2νδv in the NavierndashStokesequation should be much smaller than the local acceleration parttδv sim ωδv sim cskδv

Remarks

lowast Instead of considering ldquotemporal dampingrdquo as was done above by assuming k isin R but ω isin Cone may investigate ldquospatial dampingrdquo ie assume ω isin R and put the whole complex dependencein the wave number k = kr + iki For (angular) frequencies ω much smaller than the inverse of thetypical time scale τν equiv νc2

s one finds

ω2 c2sk

4ω2τ2

)hArr cϕ equiv

8ω2τ2

)ie the phase velocity increases with the frequency and on the other hand

ki νω2

(VI69)

The latter relation is known as Stokesrsquo law of sound attenuation ki representing the inverse of thetypical distance over which the sound wave amplitude decreases due to the factor ei(iki)x = eminuskix

in the Fourier ansatz (VI64) Larger frequencies are thus absorbed on a smaller distance from thesource of the sound wave

Substituting k = kr + iki = kr(1 + iκ) in the dispersion relation (VI65) and writing the identityof the real and imaginary parts one obtains the system

2κ = ωτν(1minus κ2)

ω2 = c2sk2r(1 + 2ωτνκ minus κ2)

The first equation is a quadratic equation in κ that admits one positive and one negative solutionthe latter can be rejected while the former is κ ωτν2 +O

((ωτν)2

) Inserting it in the second

equation leads to the wanted results

An exact solution of the system of equations exists yes it is neither enlightening mathematicallynor relevant from the physical point of view in the general case as discussed in the next remark

One may naturally also analyze the general case in which both ω and k are complex numbers Inany case the phase velocity is given by cϕ equiv ωkr although it is more difficult to recognize thephysical content of the mathematical relations

lowast For air or water the reduced kinetic viscosity ν is of order 10minus6ndash10minus5 m2 middot sminus1 With speeds ofsound cs 300ndash1500 m middot sminus1 this yields typical time scales τν of order 10minus12ndash10minus10 s That is thechange in the speed of sound (VI68) or equivalently deviations from the assumption ωτν 1 under-lying the attenuation coefficient (VI69) become relevant for sound waves in the gigahertzterahertzregime() This explains why measuring the bulk viscosity is a non-trivial task

The wavelengths csτν corresponding to the above frequencies τminus1ν are of order 10minus9ndash10minus7 m

This is actually not far from the value of the mean free path in classical fluids so that the wholedescription as a continuous medium starts being questionable

Bibliography for Chapter VIbull National Committee for Fluid Mechanics film amp film notes on Rotating flows Low ReynoldsNumber Flow Fundamentals of Boundary Layers and Vorticity

bull Faber [1] Chapters 66 69 and 611

bull Guyon et al [2] Chapters 45 732 9 amp 101ndash104

bull LandaundashLifshitz [3 4] Chapter II sect 17ndash20 amp 24 Chapter IV sect 39 and Chapter VIII sect 79

bull Sommerfeld [5 6] Chapters II sect 10 III sect 16 and VII sect 35

CHAPTER VII

Turbulence in non-relativistic fluids

All examples of flows considered until now in these notes either of perfect fluids (Chapters IV and V)or of Newtonian ones (Chapter VI) share a common property namely they are all laminar Thisassumptionmdashwhich often translates into a relative simplicity of the flow velocity profilemdashis howevernot the generic case in real flows which most often are to some more or less large extent turbulentThe purpose of this Chapter is to provide an introduction to the problematic of turbulence in fluidmotions

A number of experiments in particular those conducted by O Reynolds have hinted at thepossibility that turbulence occurs when the Reynolds number (VI12) is large enough in the flowie when convective effects predominate over the shear viscous ones that drive the mean fluid motionover which the instabilities develop This distinction between mean flow and turbulent fluctuationscan be modeled directly by splitting the dynamical fields into two parts and one recovers with thehelp of dimensional arguments the role of the Reynolds number in separating two regimes one inwhich the mean viscous flow dominates and one in which turbulence takes over (Sec VII1)

Despite its appeal the decomposition into a mean flow and a turbulent motion has the drawbackthat it leads to a system of equations of motion which is not closed A possibility to remedy thisissue is to invoke the notion of a turbulent viscosity for which various models have been proposed(Sec VII2)

Even when the system of equations of motion is closed it still involves averagesmdashwith an apriori unknown underlying probability distribution That is the description of turbulent part of themotion necessitates the introduction of a few concepts characterizing the statistics of the velocityfield (Sec VII3)

For the sake of simplicity we shall mostly consider turbulence in the three-dimensional incom-pressible motion of Newtonian fluids with constant and homogeneous properties (mass densityviscosity ) in the absence of relevant external bulk forces and neglecting possible temperaturegradientsmdashand thereby convective heat transport

VII1 Generalities on turbulence in fluidsIn this Section a few experimental facts on turbulence in fluid flows is presented and the first stepstowards a modeling of the phenomenon are introduced

VII11 Phenomenology of turbulence

VII11 a

Historical example HagenndashPoiseuille flow

The idealized HagenndashPoiseuille flow of a Newtonian fluid in a cylindrical tube was already partlydiscussed in Sec VI14 It was found that in the stationary laminar regime in which the velocityfield ~v is purely parallel to the walls of the tube the mass flow rate Q across the cylinder crosssection is given by the HagenndashPoiseuille law

Q = minusπρa4

∆PL (VI9)

with a the tube radius ∆PL the pressure drop per unit length and ρ η the fluid properties

VII1 Generalities on turbulence in fluids 111

Due to the viscous friction forces part of the kinetic energy of the fluid motion is transformedinto heat To compensate for these ldquolossesrdquo and keep the flow in the stationary regime energy hasto be provided to the fluid namely in the form of the mechanical work of the pressure forces drivingthe flow Thus the rate of energy dissipation per unit mass is(38)

Ediss = minus1

∆PL〈v〉 =

8ν〈v〉2

a2(VII1)

with 〈v〉 the average flow velocity across the tube cross section

〈v〉 =Q

πa2ρ= minusa

Thus in the laminar regime the rate Ediss is proportional to the kinematic viscosity ν and to thesquare of the average velocity

According to the HagenndashPoiseuille law (VI9) at fixed pressure gradient the average velocity 〈v〉grows quadratically with the tube radius In practice the rise is actually slower reflecting a higherrate of energy loss in the flow as given by the laminar prediction (VII1) Thus the mean rate ofenergy dissipation is no longer proportional to 〈v〉2 but rather to a higher power of 〈v〉 Besidesthe flow velocity profile across the tube cross section is no longer parabolic but (in average) flatteraround the cylinder axis with a faster decrease at the tube walls

VII11 b

Transition to a turbulent regime

Consider a given geometrymdashsay for instance that of the HagenndashPoiseuille flow or the motion ofa fluid in a tube with fixed rectangular cross section In the low-velocity regime the flow in thatgeometry is laminar and the corresponding state(39) is stable against small perturbations whichare damped by viscosity (see Sec VI6)

However when the average flow velocity exceeds some critical value while all other character-istics of the flow in particular the fluid properties are fixed the motion cannot remain laminarSmall perturbations are no longer damped but can grow by extracting kinetic energy from theldquomainrdquo regular part of the fluid motion As a consequence instead of simple pathlines the fluidparticles now follow more twisted ones the flow becomes turbulent

In that case the velocity gradients involved in the fluid motion are in average much larger thanin a laminar flow The amount of viscous friction is thus increased and a larger fraction of thekinetic energy is dissipated as heat

The role of a critical parameter in the onset of turbulence was discovered by Reynolds in thecase of the HagenndashPoiseuille flow of water in which he injected some colored water on the axis ofthe tube repeating the experiment for increasing flow velocities [19] In the laminar regime foundat small velocities the streakline formed by the colored water forms a thin band along the tube axiswhich does not mix with the surrounding water Above some flow velocity the streakline remainsstraight along some distance in the tube then suddenly becomes instable and fills the whole crosssection of the tube

As Reynolds understood himself by performing his experiments with tubes of various diametersthe important parameter is not the velocity itself but rather the Reynolds number Re (VI12)which is proportional to the velocity Thus the transition to turbulence in flows with shear occursat a ldquocritical valuerdquo Rec which however depends on the geometry of the flow For instance Rec isof order 2000 for the HagenndashPoiseuille flow but becomes of order 1000 for the plane Poiseuille flowinvestigated in Sec VI13 while Rec 370 for the plane Couette flow (Sec VI12)(38)In this Chapter we shall only discuss incompressible flows at constant mass density ρ and thus always consider

energies per unit mass(39)This term really refers to a macroscopic ldquostaterdquo of the system in the statistical-physical sense In contrast to the

global equilibrium states usually considered in thermostatics it is here a non-equilibrium steady state in whichlocal equilibrium holds at every point

112 Turbulence in non-relativistic fluids

The notion of a critical Reynolds number separating the laminar and turbulent regimes is actu-ally a simplification In theoretical studies of the stability of the laminar regime against linearperturbations such a critical value Rec can be computed for some very simple geometries yield-ing eg Rec = 5772 for the plane Poiseuille flow Yet the stability sometimes also depends onthe size of the perturbation the larger it is the smaller the associated critical Rec is whichhints at the role of nonlinear instabilities

In the following we shall leave aside the problem of the temporal onset of turbulencemdashandthereby of the (in)stability of laminar flowsmdash and focus on flows which are already turbulent whenwe start looking at them

VII12 Reynolds decomposition of the fluid dynamical fields

Since experiment as well as reasoning hint at the existence of an underlying ldquosimplerdquo laminarflow over which turbulence develops a reasonable ansatz for the description of the turbulent motionof a fluid is to split the relevant dynamical fields into two components a first one which variesslowly both in time t and position ~r and a rapidly fluctuating component which will be denotedwith primed quantities In the case of the flow velocity field~v(t~r) this Reynolds decomposition(lxvi)

reads [29]

~v(t~r) = ~v(t~r) +~vprime(t~r) (VII2)

with ~v resp~vprime the ldquoslowrdquo resp ldquofastrdquo component For the pressure one similarly writes

P (t~r) = P (t~r) + P prime(t~r) (VII3)

The fluid motion with velocity ~v and pressure P is then referred to as ldquomean flowrdquo that withthe rapidly varying quantities as ldquofluctuating motionrdquo

As hinted at by the notation ~v(t~r) represents an average with some underlying probabilitydistribution

Theoretically the Reynolds average middot should be an ensemble average obtained from an infinitelylarge number of realizations namely experiments or computer simulations in practice howeverthere is only a finite number N of realizations ~v(n)(t~r) If the turbulent flow is statisticalstationary one may invoke the ergodic assumption and replace the ensemble average by a timeaverage

~v(~r) equiv limNrarrinfin

Nsumn=1

~v(n)(t~r) asymp 1

int t+T 2

tminusT 2~v(tprime~r) dtprime

with T much larger than the autocorrelation time of the turbulent velocity ~vprime(t~r) If the flowis not statistically stationary so that ~v(t~r) also depends on time then T must also be muchsmaller than the typical time scale of the variations of the mean flow

Using the same averaging procedure the fluctuating velocity must obey the condition

~vprime(t~r) = ~0 (VII4)

Despite this fact the turbulent velocity~vprime(t~r) still plays a role in the dynamics in particular thatof the mean flow because its two-point three-point and higher (auto)correlation functions are stillin general non-vanishing For instance one can writemdashassuming that the mass density ρ is constantand uniform

ρ vi(t~r) vj(t~r) = ρ vi(t~r) vj(t~r) + ρ vprimei(t~r) vprimej(t~r)

The first term of the right member corresponds to convective part of the momentum-flux density(lxvi)Reynolds-Zerlegung

of the mean flow while the second one

TTTijR(t~r) equiv ρ vprimei(t~r) vprimej(t~r) (VII5)

which is simply the ij-component of the rank 2 tensor

TTTR(t~r) equiv ρ~vprime(t~r)otimes~vprime(t~r) (VII6)

is due to the rapidly fluctuating motion TTTR is called turbulent stress or Reynolds stress(lxvii)

VII13 Dynamics of the mean flow

For the sake of simplicity the fluid motion will from now on be assumed to be incompressibleThanks to the linearity of the averaging process the kinematic condition ~nabla middot~v(t~r) = 0 leads tothe two relations

~nabla middot~v(t~r) = 0 and ~nabla middot~vprime(t~r) = 0 (VII7)That is both the mean flow and the turbulent motion are themselves incompressible

The total flow velocity~v obeys the usual incompressible NavierndashStokes equation [cf Eq (III32)]

(part~v(t~r)

]~v(t~r)

)= minus~nablaP (t~r) + η4~v(t~r) (VII8)

from which the equations governing the mean and turbulent flows can be derived For the sake ofbrevity the variables (t~r) of the various fields will be omitted in the following

VII13 a

Equations for the mean flow

Inserting the Reynolds decompositions (VII2)ndash(VII3) in the NavierndashStokes equation (VII8)and averaging with the Reynolds average middot leads to the so-called Reynolds equation

[part~v

partt+(~v middot~nabla

]= minus~nablaP + η4~v minus ρ

(~vprime middot~nabla

)~vprime (VII9a)

To avoid confusion this equation is also sometimes referred to as Reynolds-averaged NavierndashStokesequation In terms of components in a given system of coordinates this becomes after dividing bythe mass density ρ

partvi

)vi = minus1

dPdximinus

3sumj=1

dvprimeivprimej

dxj+ ν4vi (VII9b)

These two equations involve the material derivative ldquofollowing the mean flowrdquo

Dtequiv part

partt+~v middot~nabla (VII10)

Using the incompressibility of the fluctuating motion the rightmost term in Eq (VII9a) canbe rewritten as

minusρ(~vprime middot ~nabla

)~vprime = minusρ~nabla middot

(~vprimeotimes~vprime

)= minus~nabla middotTTTR

The Reynolds equation can thus be recast in the equivalent form [cf Eq (III24b)]part

(ρ~v)

+ ~nabla middotTTT = minus~nabla middotTTTR (VII11)

with TTT the momentum-flux density of the mean flow given by [cf Eqs (III26b) (III26e)]

TTT equiv P gminus1 + ρ~v otimes~v minus 2ηSSS (VII12a)

ie component-wise(lxvii)Reynolds-Spannung

TTTij equiv P gij + ρvi vj minus 2ηSSSij (VII12b)

with SSS the rate-of-shear tensor [Eq (II15b)] for the mean flow with components [cf Eq (II15d)]

SSSij equiv 1

dximinus 2

3gij ~nabla middot~v

) (VII12c)

where the third term within the brackets actually vanishes due to the incompressibility of the meanflow Eq (VII7)

The form (VII11) of the Reynolds equation emphasizes perfectly the role of the Reynolds stressie the turbulent component of the flow as ldquoexternalrdquo force driving the mean flow In particular theoff-diagonal terms of the Reynolds stress describe shear stresses which will lead to the appearanceof eddies in the flow

Starting from the Reynolds equation one can derive the equation governing the evolution of thekinetic energy 1

2ρ(~v)2 associated with the mean flow namely

(ρ~v2

)= minus~nabla middot

[P~v +

(TTTR minus 2ηSSS

)middot~v]

+(TTTR minus 2ηSSS

) SSS (VII13)

This equation is conventionally rather written in terms of the kinetic energy per unit mass k equiv 12

in which case it reads

Dt= minus~nabla middot

ρP~v +

(~vprimeotimes~vprimeminus 2νSSS

)middot~v]

+(~vprimeotimes~vprimeminus 2νSSS

) SSS (VII14a)

or component-wise

Dt= minus

3sumj=1

ρP vj +

3sumi=1

(vprimeivprimej minus 2νSSSij

3sumij=1

)SSSij (VII14b)

In either form the physical meaning of each term is rather transparent first comes the convectivetransport of energy in the mean flow given by the divergence of the energy flux density inclusivea term from the turbulent motion The second term represents the energy which is ldquolostrdquo to themean flow namely either because it is dissipated by the viscous friction forces (term in νSSS SSS) orbecause it is transferred to the turbulent part of the motion (term involving the Reynolds stress)

To prove Eq (VII13) one should first average the inner product with ~v of the Reynolds equa-tion (VII9) and then rewrite~vmiddot~nablaP and~vmiddot

(~vprime middot ~nabla

)~vprime under consideration of the incompressibility

condition (VII7)

Remark While equations (VII9) or (VII14) do describe the dynamics of the mean flow they relyon the Reynolds stress which is not yet specified by the equations

VII13 b

Description of the transition to the turbulent regime

Turbulence takes place when the effects of Reynolds stress TTTRmdashwhich represents a turbulenttransport of momentummdashpredominates over those of the viscous stress tensor 2ρνSSS associated withthe mean flow ie when the latter can no longer dampen the fluctuations corresponding to theformer

Let vc resp Lc denote a characteristic velocity resp length scale of the fluid motion Assumingthat averagesmdashhere a simple over the volume is meantmdashover the flow yield the typical orders ofmagnitude lang

3sumij=1

∣∣vprimeivprimejSSSij∣∣rang sim v3c

lang3sum

∣∣νSSSijSSSij∣∣2rang sim νv2c

(VII15)

then in the turbulent regime the first of these terms is significantly larger than the second whichcorresponds to having a large value of the Reynolds number Re equiv vcLcν [Eq (VI12)]

In that situation the equation (VII14) describing the evolution of the kinetic energy of themean flow becomes

ρP~v +

)middot~v]

+(~vprimeotimes~vprime

) SSS (VII16a)

or component-wise

Dt= minus

3sumj=1

ρP vj +

3sumi=1

(vprimeivprimej

3sumij=1

vprimeivprimejSSSij (VII16b)

That is the viscosity is no longer a relevant parameter for the dynamics of the mean flowAs already argued above the first term in Eq (VII16) represents the convective transport of

energy in the mean flow while the second ldquomixedrdquo term models the transfer of energy from themean flow into the turbulent motion and thus corresponds to the energy ldquodissipatedrdquo by the meanflow Invoking the first relation in Eq (VII15) the rate of energy dissipation in the mean flow is

Ediss =

lang3sum

vprimeivprimejSSSij

rangsim v3

Lc (VII17)

This grows like the third power of the typical velocity ie faster than v2c as argued at the end of

sectVII11 a for the turbulent regime of the HagenndashPoiseuille flow In addition this energy dissipationrate is actually independent of the properties (mass density viscosity ) of the flowing fluidturbulence is a characteristic of the motion not of the fluid itself

Eventually the middle term in Eq (VII17) must be negative so that the energy really flowsfrom the mean flow to the turbulent motion not in the other direction

Remark Looking naively at the definition of the Reynolds number the limit of an infinitely largeRe corresponds to the case of a vanishing shear viscosity that is to the model of a perfect fluidAs was just discussed this is clearly not the case with growing Reynolds number ie increasinginfluence of the turbulent motion the number of eddies in the flow also increases in which energyis dissipated into heat In contrast the kinetic energy is conserved in the flow of a perfect fluidThe solution to this apparent paradox is simply that with increasing Reynolds number the velocitygradients in the flow also increase In the incompressible NavierndashStokes equation the growth of 4~vcompensates the decrease of the viscosity ν so that the corresponding term does not disappear andthe NavierndashStokes equation does not simplify to the Euler equation

VII14 Necessity of a statistical approach

As noted above the evolution equation for the mean flow involves the Reynolds stress for whichno similar equation has yet be determined

A first natural solution is simply to write down the evolution equation for the turbulent velocity~vprime(t~r) see Eq (VII25) below Invoking then the identity

[ρ~vprime(t~r)otimes~vprime(t~r)

part~vprime(t~r)

parttotimes~vprime(t~r) + ρ~vprime(t~r)otimes part~vprime(t~r)

one can derive a dynamical equation for TTTR the so-called Reynolds-stress equation(lxviii) which incomponent form reads

DTTTijRDt

= minus2P primeSSSprimeij+d

(P primevprimeigjk+P primevprimejgik+ρvprimeivprimejvprimekminusν

dTTTijRdxk

)minus(TTTikR

dxk+TTTjkR

)minus2η

dvprimei

dxkdvprimej

(VII18)(lxviii)Reynolds-Spannungsgleichung

Irrespective of the physical interpretation of each of the terms in this equation an important issueis that the evolution of ρvprimeivprimej involves a contribution from the components ρvprimeivprimejvprimek of a tensor ofdegree 3 In turn the evolution of ρvprimeivprimejvprimek involves the tensor with components ρvprimeivprimejvprimekvprimel andso on at each step the appearance of a tensor of higher degree simply reflects the nonlinearity ofthe NavierndashStokes equation

All in all the incompressible NavierndashStokes equation (VII8) is thus equivalent to an infinitehierarchy of equations relating the successive n-point autocorrelation functions of the fluctuationsof the velocity field Any subset of this hierarchy is not closed and involves more unknown fieldsthan equations A closure prescription based on some physical assumption is therefore necessaryto obtain a description with a finite number of equations governing the (lower-order) autocorrelationfunctions Such an approach is presented in Sec VII2

An alternative possibility is to assume directly some ansatz for the statistical behavior of theturbulent velocity especially for its general two-point autocorrelation function of which the equal-time and position correlator vprimei(t~r)vprimej(t~r) is only a special case This avenue will be pursued inSec VII3

VII2 Model of the turbulent viscosityA first possibility to close the system of equations describing turbulence consists in using the phe-nomenological concept of turbulent viscosity which is introduced in Sec VII21 and for whichvarious models are quickly presented in Sec VII22ndashVII24

VII21 Turbulent viscosity

The basic idea underlying the model is to consider that at the level of the mean flow effect of theldquoturbulent frictionrdquo is to redistribute momentum from the high mean-velocity regions to the ones inslower motion in the form of a diffusive transport Accordingly the traceless part of the turbulentReynolds stress is dealt with like the corresponding part of the viscous stress tensor (III26e) andassumed to be proportional to the rate-of-shear tensor of the mean flow (Boussinesq hypothesis(an))

TTTR(t~r)minus Tr[TTTR(t~r)

]gminus1(t~r) equiv minus2ρνturb(t~r)SSS(t~r) (VII19a)

where the proportionality factor involves the (kinematic) turbulent viscosity or eddy viscosity(40)

νturb which a priori depends on time and position In terms of components in a coordinate systemand replacing the Reynolds stress and its trace by their expressions in terms of the fluctuatingvelocity this reads

ρ vprimei(t~r) vprimej(t~r)minus 1

3ρ [~vprime(t~r)]2 gij(t~r) equiv 2ρνturb(t~r)SSSij(t~r) (VII19b)

Using the ansatz (VII19) and invoking the incompressibility of the mean flow from which follows~nabla middotSSS = 1

24~v the Reynolds equation (VII9) can be rewritten as

part~v(t~r)

]~v(t~r) = minus~nabla

P (t~r)

[~vprime(t~r)]2

+ 2νeff(t~r)4~v(t~r) (VII20)

with the effective viscosityνeff(t~r) = ν + νturb(t~r) (VII21)

while the term in curly brackets may be seen as an effective pressure

(40)turbulente Viskositaumlt Wirbelviskositaumlt(an)J Boussinesq 1842ndash1929

VII2 Model of the turbulent viscosity 117

Even if the intrinsic fluid properties in particular its kinematic viscosity ν are assumed tobe constant and uniform this does not hold for the turbulent and effective viscosities νturb νeff because they model not the fluid but also its flowmdashwhich is time and position dependent

Either starting from Eq (VII20) multiplied by ~v or substituting the Reynolds stress with theansatz (VII19) in Eq (VII14) one can derive the equation governing the evolution of the kineticenergy of the mean flow In particular one finds that the dissipative term is

Ediss = 2νeffSSS SSS = 2νeff

3sumij=1

SSSijSSSij

Comparing with the rightmost term in Eq (VII14) gives for the effective viscosity

νeff =

minussumij

vprimeivprimejSSSij

2sumij

SSSijSSSij

2νsumij

SSSijSSSij

2sumij

SSSijSSSij= ν

where the inequality holds in the turbulent regime There thus follows νeff asymp νturb ν

It has been argued that in plasmas the turbulent viscosity νturb could in some regimes benegativemdashand of the same magnitude as νmdash leading to an ldquoanomalaousrdquo effective viscosity νeff

much smaller than ν [30 31]

Remark To emphasize the distinction with the turbulent viscosity ν is sometimes referred to asldquomolecularrdquo viscosity

While the ansatz (VII19) allows the rewriting of the Reynolds equation in the seemingly simplerform (VII20)mdashin which the two terms contributing to the effective pressure are to be seen asconstituting a single fieldmdash it still involves an unknown flow-dependent quantity namely theeffective viscosity νeff which still needs to be specified

VII22 Mixing-length model

A first phenomenological hypothesis for the turbulent viscosity is that implied in the mixing-length model (lxix) of Prandtl which postulates the existence of a mixing length(lxx) `m representingthe typical scale over which momentum is transported by turbulence

The ansatz was motivated by an analogy with the kinetic theory of gases in which the kinematicviscosity ν is proportional to the mean free path and to the typical velocity of particles

In practice `m is determined empirically by the geometry of the flowUnder this assumption the turbulent viscosity is given by

νturb(t~r) = `m(t~r)2

∣∣∣∣partvx(t~r)

∣∣∣∣ (VII22)

in the case of a two-dimensional flow like the plane Couette flow (Sec VI12) or for a more generalmotion

νturb(t~r) = `2m(t~r)∣∣SSS(t~r)

∣∣with |SSS| a typical value of the rate-of-shear tensor of the mean flow In any case the turbulentviscosity is determined by local quantities

The latter point is actually a weakness of the model For instance it implies that the turbulentviscosity (VII22) vanishes at an extremum of the mean flow velocitymdashfor instance on the tube axisin the HagenndashPoiseuille flowmdash which is not realistic In addition turbulence can be transportedfrom a region into another one which also not describe by the ansatz(lxix)Mischungswegansatz (lxx)Mischungsweglaumlnge

Eventually the mixing-length model actually merely displaces the arbitrariness from the choiceof the turbulent viscosity νturb to that of the mixing length `m ie it is just a change of unknownparameter

VII23 k-model

In order to describe the possible transport of turbulence within the mean flow the so-calledk-model was introduced

Denoting by kprime equiv 12~vprime2 the average kinetic energy of the turbulent fluctuations the turbulent

viscosity is postulated to beνturb(t~r) = `m(t~r)kprime(t~r)

12 (VII23)

An additional relation is needed to describe the transport of kprime to close the system of equationsFor simplicity one the actual relation [see Eq (VII26) below] is replaced by a similar-lookingequation in which the material derivative following the main flow of the average turbulent kineticenergy equals the sum of a transport termmdashminus the gradient of a flux density taken to beproportional to the gradient of kprimemdash a production termmdashnamely the energy extracted from themean flowmdash and a dissipation term that describes the rate of energy release as heat and whoseform

Ediss = Ckprime32`m

is motivated by dimensional arguments with C a constant Due to the introduction of this extraphenomenological transport equation for kprime which was not present in the mixing-length model thek-model is referred to as a one-equation model (lxxi)

The k-model allows by construction the transport of turbulence However the mixing length`m remains an empirical parameter and two further ones are introduced in the transport equationfor the average turbulent kinetic energy

VII24 (k-ε)-model

In the k-model the dissipation term Ediss which stands for the ultimate transformation of turbu-lent kinetic energy into heat under the influence of viscous friction and should thus be proportionalto the viscosity ν is determined by a dimensional argument

Another possibility is to consider the energy dissipation rate Ediss(t~r)mdashwhich is usually ratherdenoted as εmdashas a dynamical variable whose evolution is governed by a transport equation of itsown This approach yields a two-equation model (lxxii) the so-called (k-ε)-model

A dimensional argument then gives `m sim kprime32Ediss and thus

νturb(t~r) = Ckprime(t~r)

Ediss(t~r) (VII24)

with C an empirical constant

In this modelmdashor rather this class of modelsmdash the mixing length is totally fixed by the dy-namical variables thus is no longer arbitrary On the other hand the two transport equationsintroduced for the average turbulent kinetic energy and the dissipation rate involve a handful ofparameters which have to be determined empirically for each flow

In addition the (k-ε)-model like all descriptions involving a turbulent viscosity relies on theassumption that the typical scale of variations of the mean flow velocity is clearly separated fromthe turbulent mixing length This hypothesis is often not satisfied in that many flows involve(lxxi)Eingleichungsmodell (lxxii)Zweigleichungsmodell

VII3 Statistical description of turbulence 119

turbulent motion over many length scales in particular with a larger scale comparable with thatof the gradients of the mean flow In such flows the notion of turbulent viscosity is not reallymeaningful

VII3 Statistical description of turbulenceInstead of handling the turbulent part of the motion like a source of momentum or a sink of ki-netic energy for the mean flow another approach consists in considering its dynamics more closely(Sec VII31) As already argued in Sec VII14 this automatically involves higher-order autocor-relation functions of the fluctuating velocity which hints at the interest of looking at the generalautocorrelation functions rather than just their values at equal times and equal positions Thismore general approach allows on the one hand to determine length scales of relevance for turbu-lence (Sec VII32) and on the other hand to motivate a statistical theory of (isotropic) turbulence(Sec VII33)

VII31 Dynamics of the turbulent motion

Starting from the incompressible NavierndashStokes equation (VII8) for the ldquototalrdquo flow velocity ~vand subtracting the Reynolds equation (VII9) for the mean flow one finds the dynamical equationgoverning the evolution of the turbulent velocity~vprime namely [for brevity the (t~r)-dependence of thefields is omitted]

[part~vprime

)~vprime]

= minus~nablaP prime+ η4~vprimeminus ρ(~vprime middot ~nabla

)~v minus ~nabla middot

(ρ~vprimeotimes~vprimeminusTTTR

) (VII25a)

or after dividing by ρ and projecting along the xi-axis of a coordinate system

partvprimei

)vprimei = minus1

dP prime

dxi+ ν4vprimei minus

)vi minus d

(vprimeivprimej minus vprimeivprimej

) (VII25b)

One recognizes in the left hand side of those equations the material derivative of the fluctuatingvelocity following the mean flow D~vprimeDt

From the turbulent NavierndashStokes equation (VII25) one finds for the average kinetic energy ofthe fluctuating motion kprime equiv 1

2~vprime2

Dkprime

Dt= minus

3sumj=1

ρP primevprimej +

3sumi=1

2vprimeivprimeivprimej minus 2ν vprimeiSSS

primeij)]minus

3sumij=1

vprimeivprimej SSSij minus 2ν

3sumij=1

SSSprimeijSSSprimeij (VII26)

with SSSprimeij equiv 1

(dvprimei

dvprimej

dximinus 2

3gij ~nabla middot~vprime

)the components of the fluctuating rate-of-shear tensor

bull The first term describes a turbulent yet conservative transportmdashdue to pressure convectivetransport by the fluctuating flow itself or diffusive transport due the viscous frictionmdash mixingthe various length scales the kinetic energy is transported without loss from the large scalescomparable to that of the variations of the mean flow to the smaller ones This process isreferred to as energy cascade

bull The second term describes the ldquocreationrdquo of turbulent kinetic energy which is actually ex-tracted from the mean flow it is preciselymdashup to the signmdashthe loss term in the Eq (VII16)describing the transport of kinetic energy in the mean flow

bull Eventually the rightmost term in Eq (VII26) represents the average energy dissipated asheat by the viscous friction forces and will hereafter be denoted as Ediss

In a statistically homogeneous and stationary turbulent flow the amount of energy dissipatedby viscous friction equals that extracted by turbulence from the mean flow ie

minus3sum

vprimeivprimej SSSij = 2ν

3sumij=1

VII32 Characteristic length scales of turbulence

VII32 a

Two-point autocorrelation function of the turbulent velocity fluctuations

The fluctuations of the turbulent velocity~vprime are governed by an unknown probability distributionInstead of knowing the latter it is equivalent to rely on the (auto)correlation functions

κ(n)i1i2in

(t1~r1 t2~r2 tn~rn) equiv vprimei1(t1~r1) vprimei2(t2~r2) middot middot middot vprimein(tn~rn)

in which the components of fluctuations at different instants and positions are correlated with eachother Remember that the 1-point averages vanish Eq (VII4)

The knowledge of all n-point autocorrelation functions is equivalent to that of the probabilitydistribution Yet the simplestmdashboth from the experimental point of view as well as in numericalsimulationsmdashof these functions are the two-point autocorrelation functions [32]

κ(2)ij (t~r tprime~rprime) equiv vprimei(t~r) vprimej(t

prime~rprime) (VII28)

which will hereafter be considered only at equal times tprime = tIn the case of a statistically stationary turbulent flow(41) the 2-point autocorrelation functions

κ(2)ij (t~r tprime~rprime) only depend on the time difference tprime minus t which vanishes if both instants are equal

yielding a function of~r ~rprime only If the turbulence is in addition statistically homogeneous(41)mdashwhichnecessitates that one considers it far from any wall or obstacle although this does not yet constitutea sufficient conditionmdash then the 2-point autocorrelation function only depends on the separation~X equiv ~rprime minus~r of the two positions

κij( ~X) = vprimei(t~r) vprimej(t~r + ~X) (VII29)

If the turbulence is statistically locally isotropic(41) the tensor κij only depends on the distanceX equiv | ~X| between the two points Such a statistical local isotropy often represents a good assumptionfor the structure of the turbulent motion on small scalesmdashagain far from the boundaries of theflowmdashand will be assumed hereafter

Consider two points at ~r and ~r + ~X Let ~e denote a unit vector along ~X ~eperp a unit vector ina direction orthogonal to ~e and ~e primeperp perpendicular to both ~e and ~eperp The component vprime of theturbulent velocitymdashat ~r or ~r + ~Xmdashalong ~e is referred to as ldquolongitudinalrdquo those along ~eperp or ~e primeperp(vprimeperp vprimeperpprime) as ldquolateralrdquo

The autocorrelation function (VII29) can be expressed with the help of the two-point functionsκ(X) equiv vprime(t~r) vprime(t~r + ~X) κperp(X) equiv vprimeperp(t~r) vprimeperp(t~r + ~X) and κprimeperp(X) equiv vprimeperp(t~r) vprimeperpprime(t~r + ~X) as

κij(X) =XiXj

[κ(X)minus κperp(X)

]+ κperp(X) δij + κprimeperp(X)

3sumk=1

εijkXk

with Xi the Cartesian components of ~X where the last term vanishes for statistically space-parityinvariant turbulence(42) which is assumed to be the case from now on(43)

(41)This means that the probability distribution of the velocity fluctuations~vprime is stationary (time-independent) resphomogeneous (position-independent) resp locally isotropic (the same for all Cartesian components of~vprime)

(42)Invariance under the space-parity operation is sometimes considered to be part of the isotropy sometimes not (43)In presence of a magnetic fieldmdashie in the realm of magnetohydrodynamicsmdash this last term is indeed present

Multiplying the incompressibility condition ~nabla middot~vprime = 0 with vj and averaging yields3sumi=1

partκij(X)

partXi= 0

resulting in the identity

κperp(X) = κ(X) +X

dκ(X)

which means that κij can be expressed in terms of the autocorrelationfunction κ only

VII32 b

Microscopic and macroscopic length scales of turbulence

The assumed statistical isotropy gives κ(0) = [v(t~r)]2 = 13 [~vprime(t~r)]2 let f(X) be the function

such that κ(X) equiv 13 [~vprime(t~r)]2 f(X) and that

bull f(0) = 1

bull the fluctuations of the velocity at points separated by a large distance X are not correlatedwith another so that κ(X) must vanish lim

Xrarrinfinf(X) = 0

bull In addition f is assumed to be integrable over R+ and such that its integral from 0 to +infinis convergent

The function f then defines a typical macroscopic length scale namely that over which f resp κdecreases(44) the integral scale or external scale(lxxiii)

LI equivint infin

0f(X) dX (VII30)

Empirically this integral scale is found to be comparable to the scale of the variations of the meanflow velocity ie characteristic for the production of turbulence in the flow For example in a flowpast an obstacle LI is of the same order of magnitude as the size of the obstacle

Assumingmdashas has been done till nowmdashlocally isotropic and space-parity invariant turbulencethe function f(X) is even so that its Taylor expansion around X = 0 defines a microscopic lengthscale

f(X) Xrarr0

1minus 1

+O(X4) with `2T equiv minus1

f primeprime(0)gt 0 (VII31)

`T is the Taylor microscale(lxxiv)(45)

Let x denote the coordinate along ~X One finds

`2T =[vprime(t~r)

]2[dvprime(t~r)dx

]2 (VII32)

ie `T is the typical length scale of the gradients of the velocity fluctuations

Using the definition of f the Taylor expansion (VII31) can be rewritten as

vprime(t~r) vprime(t~r + ~X)

[vprime(t~r)]2

Xrarr0

vprime(t~r) part2 vprime(t~r)

[vprime(t~r)]2

where part denotes the derivative with respect to x Invoking the statistical homogeneity of theturbulence [vprime(t~r)]

2 is independent of position thus of x which after differentiation leads suc-cessively to vprime(t~r) partv

prime(t~r) = 0 and then [partv

prime(t~r)]

2 +vprimepart2 vprime(t~r) = 0 proving relation (VII32)

(44)The reader should think of the example κ(X) = κ(0) eminusXLI or at least κ(X) propsim eminusXLI for X large enoughcompared to a microscopic scale much smaller than LI

(45) named after the fluid dynamics practitioner G I Taylor not after B Taylor of the Taylor series(lxxiii)Integralskala aumluszligere Skala (lxxiv)Taylor-Mikroskala

Remark Even if the Taylor microscale emerges naturally from the formalism it does not representthe length scale of the smallest eddies in the flow despite what one could expect

To find another physically more relevant microscopic scale it is necessary to investigate thebehavior of the longitudinal increment

δvprime(X) equiv vprime(t~r + ~X)minus vprime(t~r) (VII33)

of the velocity fluctuations which compares the values of the longitudinal component of the latterat different points According to the definition of the derivative dvprimedx is the limit when X rarr 0of the ratio δvprime(X)X The microscopic Kolmogorov length scale `K is then defined by

[δvprime(`K)]2

`2Kequiv lim

Xrarr0

[δvprime(X)]2

[dvprime(t~r)

(VII34)

The role of this length scale will be discussed in the following Section yet it can already be mentionedthat it is the typical scale of the smallest turbulent eddies thus the pendant to the integral scaleLI

Remark Squaring the longitudinal velocity increment (VII33) and averaging under considerationof the statistical homogeneity one finds when invoking Eq (VII31)

[δvprime(X)]2

2[vprime(X)]2sim

Xrarr0

On the other hand experiments or numerical simulations show that the left hand side of thisrelation equals about 1 when X is larger than the integral scale LI That is the latter and theTaylor microscale can also be recovered from the longitudinal velocity increment

VII33 The Kolmogorov theory (K41) of isotropic turbulence

A first successful statistical theory of turbulence was proposed in 1941 by Kolmogorov(ao) for sta-tistically locally isotropic turbulent motion assuming further stationarity homogeneity and space-parity invariance [33 34] This K41-theory describes the fluctuations of the velocity incrementsδvprimei(X) and relies on two assumptionsmdashoriginally termed similarity hypotheses by Kolmogorov

1st Kolmogorov hypothesis

The probability distributions of the turbulent-velocity increments δvprimei(X) i=1 2 3are universal on separation scales X small compared to the integral scale LI andare entirely determined by the kinematic viscosity ν of the fluid and by the averageenergy dissipation rate per unit mass Ediss

(K41-1)

Here ldquouniversalityrdquo refers to an independence from the precise process which triggers theturbulence

Considering eg the longitudinal increment this hypothesis gives for the second moment ofthe probability distribution

[δvprime(X)]2 =

radicνEdiss Φ

)for X LI with `K =

(VII35)

and Φ(2) a universal function irrespective of the flow under study The factorradicνEdiss and

the form of `K follow from dimensional considerationsmdashthe n-point autocorrelation functioninvolves another function Φ(n) multiplying a factor

(νEdiss

)n4(ao)A N Kolmogorov = A N Kolmogorov 1903ndash1987

The hypothesis (K41-1) amounts to assuming that the physics of the fluctuating motion farfrom the scale at which turbulence is created is fully governed by the available energy extractedfrom the mean flowmdashwhich in the stationary regime equals the average energy dissipated byviscous friction in the turbulent motionmdashand by the amount of friction

2nd Kolmogorov hypothesis

The probability distributions of the turbulent-velocity increments δvprimei(X) i=1 2 3is independent of the kinematic viscosity ν of the fluid on separation scales X largecompared to the microscopic scale `K

(K41-2)

The idea here is that viscous friction only plays a role at the microscopic scale while the restof the turbulent energy cascade is conservative

The assumption holds for the longitudinal increment (VII35) if and only if Φ(2)(x) simx1

B(2)x23

with B(2) a universal constant ie if

[δvprime(X)]2 sim B(2)(EdissX

)23 for `K X LI (VII36)

The Kolmogorov 23-law (VII36) does not involve any length scale this reflects the length-scale ldquoself-similarityrdquo of the conservative energy-cascading process in the inertial range(lxxv)

`K X LI in which the only relevant parameter is the energy dissipation rate

The increase of the autocorrelation function [δvprime(X)]2 as X23 is observed both experimentallyand in numerical simulations(46)

A further prediction of the K41-theory regards the energy spectrum of the turbulent motion Let~v prime(t~k) denote the spatial Fourier transform of the fluctuating velocity Up to a factor involving theinverse of the (infinite) volume of the flow the kinetic energy per unit mass of the turbulent motioncomponent with wave vector equal to ~k up to d3~k is 1

2 [~v prime(t~k)]2 d3~k In the case of statisticallyisotropic turbulence 1

2 [~v prime(t~k)]2 d3~k = 2πk2[~v prime(t~k)]2 dk equiv SE(k) dk with SE(k) the kinetic-energyspectral density

From the 23-law (VII36)(47) one can then derive the minus53-law for the latter namely

SE(k) = CK Ediss

23kminus53 for Lminus1

I k `minus1K =

(VII37)

with CK a universal constant the Kolmogorov constant independent from the fluid or the flowgeometry yet dependingmdashlike the minus53-law itselfmdashon the space dimensionality Experimentally(46)

one finds CK asymp 145

As already mentioned the laws (VII36) and (VII37) provide a rather satisfactory descriptionof the results of experiments or numerical simulations The K41-theory also predicts that thehigher-order moments of the probability distribution of the velocity increments should be universalas wellmdashand the reader can easily determine their scaling behavior [δvprime(X)]n sim B(n)

(EdissX

)n3in the inertial range using dimensional argumentsmdash yet this prediction is no longer supported byexperiment the moments do depend on X as power laws yet not with the predicted exponents

A deficiency of Kolmogorovrsquos theory is that in his energy cascade only eddies of similar sizeinteract with each other to transfer the energy from large to small length scales which is encodedin the self-similarity assumption In that picture the distribution of the eddy sizes is statisticallystationary

(46)Examples from experimental results are presented in Ref [35 Chapter 5](47) and assuming that SE(k) behaves properly ie decreases quickly enough at large k

(lxxv)Traumlgheitsbereich

In contrast turbulent motion itself tends to deform eddies by stretching vortices into tubesof smaller cross section until they become so small that shear viscosity becomes efficient tocounteract this process (see Sec VI5) This behavior somewhat clashes with Kolmogorovrsquospicture

Bibliography for Chapter VIIbull Chandrasekhar [36]

bull Feynman [8 9] Chapter 41-4ndash41ndash6

bull Faber [1] Chapter 91 92ndash96

bull Frisch [35]

bull LandaundashLifshitz [3 4] Chapter III sect 33ndash34

CHAPTER VIII

Convective heat transfer

The previous two Chapters were devoted to flows dominated by viscosity (Chap VI) or by convectivemotion (Chap VII) In either case the energy-conservation equation (III35) and in particular theterm representing heat conduction was never taken into account with the exception of a briefmention in the study of static Newtonian fluids (Sec VI11)

The purpose of this Chapter is to shift the focus and to discuss motions of Newtonian fluids inwhich heat is transfered from one region of the fluid to another A first such type of transfer is heatconduction which was already encountered in the static case Under the generic term ldquoconvectionrdquoor ldquoconvective heat transferrdquo one encompasses flows in which heat is also transported by the movingfluid not only conductively

Heat transfer will be caused by differences in temperature in a fluid Going back to the equationsof motion one can make a few assumptions so as to eliminate or at least suppress other effectsand emphasize the role of temperature gradients in moving fluids (Sec VIII1) A specific instanceof fluid motion driven by a temperature difference yet also controlled by the fluid viscosity whichallows for a richer phenomenology is then presented in Sec VIII2

VIII1 Equations of convective heat transferThe fundamental equations of the dynamics of Newtonian fluids seen in Chap III include heatconduction in the form of a term involving the gradient of temperature yet the change in timeof temperature does not explicitly appear To obtain an equation involving the time derivative oftemperature some rewriting of the basic equations is thus needed which will be done together witha few simplifications (Sec VIII11) Conduction in a static fluid is then recovered as a limitingcase

In many instances the main effect of temperature differences is however rather to lead to varia-tions of the mass density which in turn trigger the fluid motion To have a more adapted descriptionof such phenomena a few extra simplifying assumptions are made leading to a new closed set ofcoupled equations (Sec VIII12)

VIII11 Basic equations of heat transfer

Consider a Newtonian fluid submitted to conservative volume forces ~fV = minusρ~nablaΦ Its motion isgoverned by the laws established in Chap III namely by the continuity equation the NavierndashStokesequation and the energy-conservation equation or equivalently the entropy-balance equation whichwe now recall

Expanding the divergence of the mass flux density the continuity equation (III9) becomes

Dρ(t~r)

Dt= minusρ(t~r)~nabla middot~v(t~r) (VIII1a)

In turn the NavierndashStokes equation (III30a) may be written in the form

ρ(t~r)D~v(t~r)

Dt= minus~nablaP (t~r)minus ρ(t~r)~nablaΦ(t~r) + 2~nablamiddot

[η(t~r)SSS(t~r)

]+ ~nabla

[ζ(t~r)~nablamiddot~v(t~r)

] (VIII1b)

126 Convective heat transfer

Eventually straightforward algebra using the continuity equation allows one to rewrite the entropybalance equation (III40b) as

ρ(t~r)D

[s(t~r)

ρ(t~r)

]= ~nabla middot

2η(t~r)

T (t~r)SSS(t~r) SSS(t~r) +

ζ(t~r)

T (t~r)

]2 (VIII1c)

Since we wish to isolate effects directly related with the transfer of heat or playing a role in itwe shall make a few assumptions so as to simplify the above set of equations

bull The transport coefficients η ζ κ depend on the local thermodynamic state of the fluid ieon its local mass density ρ and temperature T and thereby indirectly on time and positionNevertheless they will be taken as constant and uniform throughout the fluid and taken outof the various derivatives in Eqs (VIII1b)ndash(VIII1c) This is a reasonable assumption as longas only small variations of the fluid properties are considered which is consistent with thenext assumption

Somewhat abusively we shall in fact even allow ourselves to consider η resp κ as uniform inEq (VIII1b) resp (VIII1c) later replace them by related (diffusion) coefficients ν = ηρresp α = κρcP and then consider the latter as uniform constant quantitiesThe whole procedure is only ldquojustifiedrdquo in that one can checkmdashby comparing calculationsusing this assumption with numerical computations performed without the simplificationsmdashthat it does not lead to omitting a physical phenomenon

bull The fluid motions under consideration will be assumed to be ldquoslowrdquo ie to involve a small flowvelocity in the following sense

ndash The incompressibility condition ~nablamiddot~v(t~r) = 0 will hold on the right hand sides of each ofEqs (VIII1) Accordingly Eq (VIII1a) simplifies to Dρ(t~r)Dt = 0 while Eq (VIII1b)becomes the incompressible NavierndashStokes equation

part~v(t~r)

]~v(t~r) = minus 1

ρ(t~r)~nablaP (t~r)minus ~nablaΦ(t~r) + ν4~v(t~r) (VIII2)

in which the kinematic viscosity ν is taken to be constantndash The rate of shear is small so that its square can be neglected in Eq (VIII1c) Accord-

ingly that equation simplifies to

ρ(t~r)D

[s(t~r)

ρ(t~r)

]= κ4T (t~r) (VIII3)

The left member of that equation can be further rewritten Dividing the fundamental relationof thermodynamics dU = T dS minus P dV (at constant particle number) by the mass of the atoms ofthe fluid yields the relation

)= T d

)minus P d

In keeping with the assumed incompressibility of the motion the rightmost term vanishes whilethe change in specific energy can be related to the variation of temperature as d(eρ) = cP dT withcP the specific heat capacity at constant pressure In a fluid particle one may thus write

)= cP dT (VIII4)

which translates into a relation between material derivatives when the fluid particles are followedin their motion The left member of Eq (VIII3) may then be expressed in terms of the substantialderivative of the temperature Introducing the thermal diffusivity(lxxvi)

α equiv κ

ρcP (VIII5)

(lxxvi)Temperaturleitfaumlhigkeit

VIII1 Equations of convective heat transfer 127

which will be assumed to be constant and uniform in the fluid where ρcP is the volumetric heatcapacity at constant pressure one eventually obtains

DT (t~r)

Dt=partT (t~r)

]T (t~r) = α4T (t~r) (VIII6)

which is sometimes referred to as (convective) heat transfer equationIf the fluid is at rest or if its velocity is ldquosmallrdquo enough that the convective part ~v middot ~nablaT be

negligible Eq (VIII6) simplifies to the classical heat diffusion equation with diffusion constant αThe thermal diffusivity α thus measures the ability of a medium to transfer heat diffusively just

like the kinematic shear viscosity ν quantifies the diffusive transfer of momentum Accordingly bothhave the same dimension L2Tminus1 and their relative strength can be measured by the dimensionlessPrandtl number

Pr equiv ν

α=ηcP

κ(VIII7)

which in contrast to the Mach Reynolds Froude Ekman Rossby numbers encountered in theprevious Chapters is entirely determined by the fluid independent of any flow characteristics

VIII12 Boussinesq approximation

If there is a temperature gradient in a fluid it will lead to a heat flux density and thereby to atransfer of heat thus influencing the fluid motion However heat exchanges by conduction are oftenslowmdashexcept in metalsmdash so that another effect due to temperature differences is often the firstto play a significant role namely thermal expansion (or contraction) which will lead to buoyancy(Sec IV14) when a fluid particle acquires a mass density different from that of its surroundings

The simplest approach to account for this effect due to Boussinesq(48) consists in consideringthat even though the fluid mass density changes nevertheless the motion can be to a very goodapproximation viewed as incompressiblemdashwhich is what was assumed in Sec VIII11

~nablamiddot~v(t~r) 0 (VIII8)

where is used to allow for small relative variations in the mass density which is directly relatedto the expansion rate [Eq (VIII1a)]

Denoting by T0 a typical temperature in the fluid and ρ0 the corresponding mass density (strictlyspeaking at a given pressure) the effect of thermal expansion on the latter reads

ρ(Θ) = ρ0(1minus α(V )Θ) (VIII9)

withΘ equiv T minus T0 (VIII10)

the temperature difference measured with respect to the reference value and

α(V ) equiv minus1

(partρ

(VIII11)

the thermal expansion coefficient for volume where the derivative is taken at the thermodynamicpoint corresponding to the reference value ρ0 Strictly speaking the linear regime (VIII9) onlyholds when α(V )Θ 1 which will be assumed hereafter

(48)Hence its denomination Boussinesq approximation (for buoyancy)

Consistent with relation (VIII9) the pressure term in the incompressible NavierndashStokes equationcan be approximated as

minus 1

ρ(t~r)~nablaP (t~r) minus

~nablaP (t~r)

[1 + α(V )Θ(t~r)

Introducing an effective pressure P eff which accounts for the leading effect of the potential fromwhich the volume forces derive

P eff(t~r) equiv P (t~r) + ρ0Φ(t~r)

one finds

minus 1

ρ(t~r)~nablaP (t~r)minus ~nablaΦ(t~r) minus

~nablaP eff(t~r)

ρ0+ α(V )Θ(t~r)~nablaΦ(t~r)

where a term of subleading order α(V )Θ~nablaP eff has been dropped To this level of approximationthe incompressible NavierndashStokes equation (VIII2) becomes

part~v(t~r)

]~v(t~r) = minus

~nablaP eff(t~r)

ρ0+ α(V )Θ(t~r)~nablaΦ(t~r) + ν4~v(t~r) (VIII12)

This form of the NavierndashStokes equation emphasizes the role of a finite temperature difference Θin providing an extra force density which contributes to the buoyancy supplementing the effectivepressure term

Eventually definition (VIII10) together with the convective heat transfer equation (VIII6) leadat once to

partΘ(t~r)

]Θ(t~r) = α4Θ(t~r) (VIII13)

The (Oberbeck (ap)ndash)Boussinesq equations (VIII8) (VIII12) and (VIII13) represent a closedsystem of five coupled scalar equations for the dynamical fields~v Θmdashwhich in turn yields the wholevariation of the mass densitymdashand P eff

VIII2 RayleighndashBeacutenard convectionA relatively simple example of flow in which thermal effects play a major role is that of a fluidbetween two horizontal plates at constant but different temperatures the lower plate being at thehigher temperature in a uniform gravitational potential minus~nablaΦ(t~r) = ~g in the absence of horizontalpressure gradient

The distance between the two plates will be denoted by d and the temperature differencebetween them by ∆T where ∆T gt 0 when the lower plate is warmer When needed a system ofCartesian coordinates will be used with the (x y)-plane midway between the plates and a verticalz-axis with the acceleration of gravity pointing towards negative values of z

VIII21 Phenomenology of the RayleighndashBeacutenard convection

VIII21 a

Experimental findings

If both plates are at the same temperature or if the upper one is the warmer (∆T lt 0) the fluidbetween them can simply be at rest with a stationary linear temperature profile

As a matter of fact denoting by T0 resp P 0 the temperature resp pressure at a point at z = 0and ρ0 the corresponding mass density one easily checks that equations (VIII8) (VIII12) (VIII13)admit the static solution

~vst(t~r) = ~0 Θst(t~r) = minuszd

∆T P effst(t~r) = P 0 minus ρ0gz2

2dα(V )∆T (VIII14)

(ap)A Oberbeck 1849ndash1900

VIII2 RayleighndashBeacutenard convection 129

with the pressure given by P st(t~r) = P effst(t~r) minus ρ0gz Since |zd| lt 12 and α(V )∆T 1 one

sees that the main part of the pressure variation due to gravity is already absorbed in the definitionof the effective pressure

If ∆T = 0 one recognizes the usual linear pressure profile of a static fluid at constant tempera-ture in a uniform gravity field

One can check that the fluid state defined by the profile (VIII14) is stable against small per-turbations of any of the dynamical fields To account for that property that state (for a giventemperature difference ∆T ) will be referred to as ldquoequilibrium staterdquo

Increasing now the temperature of the lower plate with respect to that of the upper plate forsmall positive temperature differences ∆T nothing happens and the static solution (VIII14) stillholdsmdashand is still stable

When ∆T reaches a critical value ∆Tc the fluid starts developing a pattern of somewhat regularcylindrical domains rotating around their longitudinal horizontal axes two neighboring regionsrotating in opposite directions These domains in which warmer and thus less dense fluid rises onthe one side while colder denser fluid descends on the other side are called Beacutenard cells(aq)

Figure VIII1 ndash Schematic representation of Beacutenard cells between two horizontal plates

The transition from a situation in which the static fluid is a stable state to that in which motiondevelopsmdashie the static case is no longer stablemdash is referred to as (onset of the) RayleighndashBeacutenardinstability Since the motion of the fluid appears spontaneously without the need to impose anyexternal pressure gradient it is an instance of free convection or natural convectionmdashin oppositionto forced convection)

Remarks

lowast Such convection cells are omnipresent in Nature as eg in the Earth mantle in the Earthatmosphere or in the Sun convective zone

lowast When ∆T further increases the structure of the convection pattern becomes more complicatedeventually becoming chaotic

In a series of experiments with liquid helium or mercury A Libchaber(ar) and his collaboratorsobserved the following features [37 38 39] Shortly above ∆Tc the stable fluid state involvecylindrical convective cells with a constant profile Above a second threshold ldquooscillatory convec-tionrdquo develops that is undulatory waves start to propagate along the ldquosurfacerdquo of the convectivecells at first at a unique (angular) frequency ω1 thenmdashas ∆T further increasesmdashalso at higherharmonics n1ω1 n1 isin N As the temperature difference ∆T reaches a third threshold a secondundulation frequency ω2 appears incommensurate with ω1 later accompanied by the combina-tions n1ω1 +n2ω2 with n1 n2 isin N At higher ∆T the oscillator with frequency ω2 experiences ashift from its proper frequency to a neighboring submultiple of ω1mdasheg ω12 in the experimentswith Hemdash illustrating the phenomenon of frequency locking For even higher ∆T submultiples ofω1 appear (ldquofrequency demultiplicationrdquo) then a low-frequency continuum and eventually chaos

(aq)H Beacutenard 1874ndash1939 (ar)A Libchaber born 1934

Each appearance of a new frequency may be seen as a bifurcation The ratios of the experimentallymeasured lengths of consecutive intervals between successive bifurcations provide an estimate ofthe (first) Feigenbaum constant (as) in agreement with its theoretical valuemdashthereby providing thefirst empirical confirmation of Feigenbaumrsquos theory

VIII21 b

Qualitative discussion

Consider the fluid in its ldquoequilibriumrdquo state of rest in the presence of a positive temperaturedifference ∆T so that the lower layers of the fluid are warmer than the upper ones

If a fluid particle at altitude z acquires for some reason a temperature that differs from theequilibrium temperaturemdashmeasured with respect to some reference valuemdashΘ(z) then its massdensity given by Eq (VIII9) will differ from that of its environment As a result the Archimedesforce acting on it no longer exactly balances its weight so that it will experience a buoyancy forceFor instance if the fluid particle is warmer that its surroundings it will be less dense and experiencea force directed upwards Consequently the fluid particle should start to move in that direction inwhich case it encounters fluid which is even colder and denser resulting in an increased buoyancyand a continued motion According to that reasoning any vertical temperature gradient shouldresult in a convective motion

There are however two effects that counteract the action of buoyancy and explain why theRayleighndashBeacutenard instability necessitates a temperature difference larger than a given thresholdFirst the rising particle fluid will also experience a viscous friction force from the other fluid regionsit passes through which slows its motion Secondly if the rise of the particle is too slow heat hastime to diffusemdashby heat conductionmdashthrough its surface this tends to equilibrate the temperatureof the fluid particle with that of its surroundings thereby suppressing the buoyancy

Accordingly we can expect to find that the RayleighndashBeacutenard instability will be facilitated whenα(V )∆Tgmdashie the buoyancy per unit massmdashincreases as well as when the thermal diffusivity α andthe shear viscosity ν decrease

Translating the previous argumentation in formulas let us consider a spherical fluid particlewith radius R and assume that it has some vertically directed velocity v while its temperatureinitially equals that of its surroundings

With the fluid particle surface area proportional to R2 and the thermal diffusivity κ one canestimate the characteristic time for heat exchanges between the particle and the neighboring fluidnamely

τQ = CR2

αwith C a geometrical factor If the fluid particle moves with constant velocity v during that du-ration τQ while staying at almost constant temperature since heat exchanges remain limited thetemperature difference δΘ it acquires with respect to the neighboring fluid is

δΘ =partΘ

partzδz =

partΘ

partzvτQ = C

where ∆Td is the temperature gradient imposed by the two plates in the fluid This temperaturedifference gives rise to a mass density difference

δρ = minusρ0α(V )δΘ = minusCρ0vR2

α(V )∆T

between the particle and its surroundings As a result fluid particle experiences an upwards directedbuoyancy

minus 4π

3R3δρg =

3ρ0gv

α(V )∆T

d (VIII15)

(as)M Feigenbaum born 1944

On the other hand the fluid particle is slowed in its vertical motion by the downwards orientedStokes friction force acting on it namely in projection on the z-axis

FStokes = minus6πRηv (VIII16)

Note that assuming that the velocity v remains constant with a counteracting Stokes force that isautomatically the ldquogoodrdquo one relies on the picture that viscous effects adapt instantaneously iethat momentum diffusion is fast That is the above reasoning actually assumes that the Prandtlnumber (VIII7) is much larger than 1 yet its result is independent from that assumption

Comparing Eqs (VIII15) and (VIII16) buoyancy will overcome friction and thus the RayleighndashBeacutenard instability take place when

3ρ0gv

α(V )∆T

dgt 6πRρ0νv hArr

α(V )∆T gR4

ανdgt

Note that the velocity v which was invoked in the reasoning actually drops out from this conditionTaking for instance R = d2mdashwhich maximizes the left member of the inequalitymdash this becomes

Ra equivα(V )∆T g d

ναgt

C= Rac

Ra is the so-called Rayleigh number and Rac its critical value above which the static-fluid state isinstable against perturbation and convection takes place The ldquovaluerdquo 72C found with the abovesimple reasoning on force equilibrium is totally irrelevantmdashboth careful experiments and theoreticalcalculations agree with Rac = 1708 for a fluid between two very large platesmdash the important lessonis the existence of a threshold

VIII22 Toy model for the RayleighndashBeacutenard instability

A more refinedmdashalthough still crudemdashtoy model of the transition to convection consists inconsidering small perturbations ~v δΘ δP eff around a static state ~vst = ~0 Θst P effst and tolinearize the Boussinesq equations to first order in these perturbations As shown by Eq (VIII14)the effective pressure P effst actually already includes a small correction due to α(V )∆T being muchsmaller than 1 so that we may from the start neglect δP eff

To first order in the perturbations Eqs (VIII12) projected on the z-axis and (VIII13) giveafter subtraction of the contributions from the static solution

partvzpartt

= ν4vz + α(V )δΘg (VIII17a)

partδΘ

parttminus ∆T

dvz = α4δΘ (VIII17b)

Moving the second term of the latter equation to the right hand side increases the parallelism ofthis set of coupled equations In addition there is also the projection of Eq (VIII12) along thex-axis and the velocity field must obey the incompressibility condition (VIII8)

The proper approach would now be to specify the boundary conditions namely the vanish-ing of vz at both platesmdashimpermeability conditionmdash the vanishing of vx at both platesmdashno-slipconditionmdash and the identity of the fluid temperature at each plate with that of the correspondingplate that is all in all 6 conditions By manipulating the set of equations it can be turned intoa 6th-order linear partial differential equation for δΘ on which the boundary conditions can beimposed

Instead of following this long road(49) we refrain from trying to really solve the equationsbut rather make a simple ansatz namely vz(t~r) = v0 eγt cos(kx)mdashwhich automatically fulfills the(49)The reader may find details in Ref [40 Chap II]

incompressibility equation but clearly violates the impermeability conditionsmdash and a similar onefor δΘ with γ a constant Substituting these forms in Eqs (VIII17) yield the linear system

γv0 = minusk2νv0 + α(V )δΘ0g hArr(γ + νk2

)v0 minus gα(V )δΘ0 = 0

γδΘ0 = minusk2αδΘ0 +∆T

dv0 hArr ∆T

dv0 minus

(γ + αk2

)δΘ0 = 0

for the amplitudes v0 δΘ0 This admits a non-trivial solution only if(γ + νk2

)(γ + αk2

)minusα(V )∆T

dg = 0 (VIII18)

This is a straightforward quadratic equation for γ It always has two real solutions one of whichis negativemdashcorresponding to a dampened perturbationmdashsince their sum is minus(α + ν)k2 lt 0 theother solution may change sign since their product

ανk4 minusα(V )∆T

is positive for ∆T = 0 yielding a second negative solution yet changes sign as ∆T increases Thevanishing of this product thus signals the onset of instability Taking for instance k = πd to fixideas this occurs at a critical Rayleigh number

Rac =α(V )∆T g d

αν= π4

where the precise value (here π4) is irrelevantFrom Eq (VIII18) also follows that the growth rate of the instability is given in the neighborhood

of the threshold byγ =

Raminus RacRac

α+ νk2

ie it is infinitely slow at Rac This is reminiscent of a similar behavior in the vicinity of the criticalpoint associated with a thermodynamic phase transition

By performing a more rigorous calculation including non-linear effects one can show that thevelocity amplitude at a given point behaves like

v prop(

Raminus RacRac

)βwith β =

2(VIII19)

in the vicinity of the critical value and this prediction is borne out by experiments [41] Sucha power law behavior is again reminiscent of the thermodynamics of phase transitions morespecifically heremdashsince v vanishes below Rac and is finite abovemdashof the behavior of the orderparameter in the vicinity of a critical point Accordingly the notation β used for the exponentin relation (VIII19) is the traditional choice for the critical exponent associated with the orderparameter of phase transitions

Eventually a last analogy with phase transitions regards the breaking of a symmetry at the thresholdfor the RayleighndashBeacutenard instability Below Rac the system is invariant under translations parallelto the plates while above Rac that symmetry is spontaneously broken

Bibliography for Chapter VIIIbull A nice introduction to the topic is to be found in Ref [42] which is a popular science account

of part of Ref [43]

bull Faber [1] Chapter 85ndash87 amp 92

bull LandaundashLifshitz [3 4] Chapter V sect 49ndash53 amp 56ndash57

CHAPTER IX

Fundamental equations of relativisticfluid dynamics

When the energy density becomes largemdashas may happen for instance in compact astrophysicalobjects in the early Universe or in high-energy collisions of heavy nucleimdashthe ldquoatomsrdquo constitutinga fluid can acquire very high kinetic energies that become comparable to their (rest) mass energyA non-relativistic description of the medium is then no longer adapted and must be replaced by arelativistic model Some introductory elements of such a description are presented in this Chaptermdashin which the basic laws governing the dynamics of relativistic fluids are formulated and discussedmdashand the following onemdashwhich will deal with a few simple analytically tractable solutions of theequations

As in the non-relativistic case the basic equations governing the motion of a fluid in the rel-ativistic regime are nothing but formulations of the most fundamental laws of physics namelyconservation laws for ldquoparticle numberrdquomdashin fact for the conserved quantum numbers carried byparticlesmdash and for energy and momentum (Sec IX1)

Precisely because the equations simply express general conservation laws they are not veryspecific and contain at first too many degrees of freedom to be tractable To make progress it isnecessary to introduce models for the fluid under consideration leading for instance to distinguishingbetween perfect and dissipative fluids A convenient way to specify the constitutive equationscharacteristic of such models is to do so in terms of a fluid four-velocity which generalizes thenon-relativistic flow velocity yet in a non-unique way (Sec IX2)

Such a fluid four-velocity also automatically singles out a particular reference frame the localrest frame in which the conserved currents describing the physics of the fluid take a simpler formwhose physical interpretation is clearer The perfect fluids are thus those whose properties at eachpoint are spatially isotropic in the corresponding local rest frame from which there follows thatthe conserved currents can only depend on the flow four-velocity not on its derivatives (Sec IX3)Conversely when the conserved currents involve (spatial) gradients of the fluid four-velocity thesederivatives signal (real) fluids with dissipative effects (Sec IX4)

Two topics that lie beyond the main stream of this Chapter are given in appendices namely theexpression of the conserved currents of relativistic fluid dynamics in terms of underlying microscopicquantities (Sec IXA) and a discussion of relativistic kinematics (Sec IXB)

Throughout this Chapter and the next one the fluids occupy domains of the four-dimensionalMinkowski space-time M 4 of Special Relativity The position of a generic point of M 4 will bedesignated by a 4-vector x Given a reference frame R and a system of coordinates those of x willbe denoted by xmicro equiv (x0 x1 x2 x3)mdashwhere in the case of Minkowski coordinates(50) x0 = ct witht the time measured by an observer at rest in R

(50)We shall call Minkowski coordinates the analog on the space-time M 4 of the Cartesian coordinates on EuclideanspaceE 3 ie those corresponding to a set of four mutually orthogonal 4-vectors (e0 e1 e2 e3) such that themetric tensor has components gmicroν = emicro middot eν = diag(minus1+1+1+1) for micro ν = 0 1 2 3 They are also alternativelyreferred to as Lorentz coordinates

134 Fundamental equations of relativistic fluid dynamics

For the metric tensor g onM 4 we use the ldquomostly plusrdquo convention with signature (minus+++)ie in the case of Minkowski coordinates x0 = minusx0 while xi = xi for i = 1 2 3 Thus time-likeresp space-like 4-vectors have a negative resp positive semi-norm

IX1 Conservation lawsAs stated in the introduction the equations governing the dynamics of fluids in the relativistic justas in the non-relativistic case embody conservation principles More precisely they are differentialformulations of these laws Instead of proceeding as in Chap III in which the local formulationswere derived from integral ones we shall hereafter postulate the differential conservation laws andcheck or argue that they lead to the expected macroscopic behavior

Starting from the local level is more natural here since one of the tenets underlying relativistictheories as eg quantum field theory is precisely localitymdashthe absence of action at distancemdashbesides causality Thus both conservation equations (IX2) and (IX7) actually emerge asthose expressing the invariance of microscopic theories under specific transformations involvingassociated Noether currents

We first discuss the conservation of ldquoparticle numberrdquo (Sec IX11)mdashwhere that denomination hasto be taken with a grain of saltmdash then that of energy and momentum which in a relativistic contextare inseparable (Sec IX12)

IX11 Particle number conservation

The conservation law that was discussed first in the Chapter III introducing the equations ofnon-relativistic hydrodynamics was that of mass which in the case of a single-component fluidis fully equivalent to the conservation of particle number In a relativistic system the number ofparticles is strictly speaking not conserved even if the system is closed Indeed thanks to thehigh kinetic energies available particlendashantiparticle pairs can continuously either be created orannihilate

If the particles carry some conserved additive quantum numbermdashas eg electric charge or baryonnumbermdash then the difference between the respective amounts of particles and antiparticles isconserved in a creation resp annihilation process both amounts vary simultaneously by +1 respminus1 but the difference remains constant Accordingly throughout this Chapter and the followingldquoparticle numberrdquo is a shorthand for the difference between the numbers of particles and antiparticlesSimilarly ldquoparticle number densityrdquo or ldquoparticle flux densityrdquo also refer to differences between therespective quantities for particles and antiparticles

For the sake of simplicity we shall consider relativistic fluids comprising a single species ofparticles together with their antiparticles with mass m

IX11 a

Local formulation of particle number conservation

By definition the local particle (number) density n(t~r) in a fluid is such that the productn(t~r) d3~r represents the number of particles (minus that of antiparticles) in the infinitesimal spatialvolume d3~r about position ~r at time t Since the volume element d3~r depends on the referenceframe in which it is measuredmdashremember that in special relativity there is the length contractionphenomenonmdash this is also the case of the particle density n(t~r) so that the particle number ind3~r remain independent of the reference frame Hereafter n(t~r) will also be denoted by n(x)

The particle flux density ~N (t~r) is defined in a similar way as the number of particle that crossa unit surface per unit time interval where both ldquounit surfacerdquo and ldquounit time intervalrdquo are referenceframe-dependent concepts

Together n(x) and ~N (x) make up a particle number four-current (lxxvii) N(x) whose Minkowskicoordinates at every x are N0(x) = c n(x) N i(x) = ji

N(x) for i = 1 2 3 This is conveniently

(lxxvii)(Teilchen-)Viererstrom

IX1 Conservation laws 135

summarized in the formN(x) =

(c n(x)~N (x)

)(IX1)

or somewhat improperly

Nmicro(x) =

(c n(x)~N (x)

With the help of the particle number four-current the local formulation of the conservation ofparticle number in the motion of the system reads using coordinates

dmicroNmicro(x) = 0 (IX2a)

where dmicro equiv d dxmicro denote the components of the 4-gradient Denoting the latter which is aone-form by d one may write the even shorter ldquogeometricrdquo (ie coordinate-invariant) equation

d middot N(x) = 0 (IX2b)

with d middot the four-divergence

Remarks

lowast Whether N(x) defined by Eq (IX1) is a 4-vectormdashthat is whether it behaves as it should underLorentz transformationsmdashis at first far from clear That n(x) d3~r need be a numbermdashie a Lorentzscalar like d4x = dx0 d3~rmdashsuggests that n(x) should transform like the time-like component of a4-vector Yet it is admittedly not clear that the associated spatial part should be the particle fluxdensityWe shall see in Sec IX33 that assuming that there exists a 4-vector field obeying the conservationequation (IX2) leads in the non-relativistic limit to the above interpretation of its time-like andspace-like parts which may be viewed as a justification(51)

lowast More generally one associates to each conserved additive quantum number a 4-current J(x) withcomponents Jmicro(x) obeying a similar conservation equation d middot J(x) = 0 resp dmicroJ

micro(x) = 0

lowast If Minkowski coordinates xmicro are used the components of the 4-gradient d are simply the partialderivatives partmicro equiv part partxmicro so that Eq (IX2a) becomes partmicroNmicro(x) = 0

IX11 b

Global formulation

Consider in M 4 a space-like 3-dimensional hypersurface Σmdashie a hypersurface at every pointof which the normal 4-vector is time-likemdashwhich extends far enough so that the whole fluid passesthrough it in its motion that is Σ intercepts the worldlines of all fluid particles

Figure IX1

(51)A better argument is to introduce the particle number 4-current from a microscopic definition see App IXA1

The total (net) number N of particles in the fluid is the flux of the particle number 4-currentN(x) across Σ

Nmicro(x) d3σmicro =

N(x) middot d3σ (IX3)

where d3σmicro denotes the components of the 3-hypersurface element

d3σmicro equiv1

radicminusdetg εmicroνρλ dxνdxρdxλ (IX4)

with εmicroνρλ the four-dimensional Levi-Civita symbol with the convention ε0123 = +1(52)

Let Ω denote a 4-volume in M 4 and partΩ its 3-surface Applying the Gauss theorem the fluxof the particle number 4-current across partΩ is the integral of the 4-divergence of N(x) over Ω∮

partΩN(x) middot d3σ =

d middot N(x) d4x (IX5)

where the right member vanishes thanks to the local expression (IX2) of particle number conser-vation Splitting partΩ into two space-like parts through which particles enter resp leave Ω in theirmotionmdashthe technical criterion is the sign of N(x) middotd3σmdash one finds that there are as many particlesthat leave as those that enter which expresses particle number conservation globally

IX12 Energy-momentum conservation

In a relativistic theory energy and momentum constitute the temporal and spatial componentsof a four-vector the four-momentum To express the local conservationmdashin the absence of externalforcesmdashof the latter the densities and flux densities of energy and momentum at each space-timepoint x must be combined into a four-tensor of degree 2 the energy-momentum tensor(lxxviii)mdashalsocalled stress-energy tensormdashTTT(x) of type

This energy-momentum tensor(53) may be defined by the physical content of its 16 Minkowskicomponents Tmicroν(x) in a given reference frame R

bull T 00(x) is the energy densitybull cT 0j(x) is the j-th component of the energy flux density with j = 1 2 3

bull 1

cT i0(x) is the density of the i-th component of momentum with i = 1 2 3

bull T ij(x) for i j = 1 2 3 is the momentum flux-density tensor

All physical quantities are to be measured with respect to the reference frame R

Remarks

lowast The similarity of the notations TTT resp TTT for the energy-momentum four-tensor resp the three-dimensional momentum flux-density tensor is not accidental The former is the natural general-ization to the 4-dimensional relativistic framework of the latter just like four-momentum p withcomponents pmicro is the four-vector associated to the three-dimensional momentum ~p That is the3-tensor TTT is the spatial part of the 4-tensor TTT just like the momentum ~p is the spatial part offour-momentum p

lowast Starting from a microscopic description of the fluid one can show that the energy-momentumtensor is symmetric ie Tmicroν(x) = T νmicro(x) for all micro ν = 0 1 2 3

(52)This choice is not universal the alternative convention ε0123 = +1 results in ε0123 lt 0 due to the odd number ofminus signs in the signature of the metric tensor

(53)As in the case of the particle number 4-current the argument showing that TTT(x) is a Lorentz tensor is to defineit microscopically as a tensormdashsee App IXA2mdashand to later interpret the physical meaning of the components

(lxxviii)Energieimpulstensor

In the absence of external force acting on the fluid the local conservation of the energy-momentum tensor reads component-wise

dmicroTmicroν(x) = 0 forallν = 0 1 2 3 (IX7a)

which represents four equations the equation with ν = 0 is the conservation of energy while theequations dmicroT

microj(x) = 0 for j = 1 2 3 are the components of the momentum conservation equationIn geometric formulation Eq (IX7a) becomes

d middotTTT(x) = 0 (IX7b)

This is exactly the same form as Eq (IX2b) just like Eqs (IX2a) and (IX7a) are similar up tothe difference in the tensorial degree of the conserved quantity

As in sect IX11 b one associates to the energy-momentum tensor TTT(x) a 4-vector P by

P equivintΣ

TTT(x) middot d3σ hArr Pmicro =

Tmicroν(x) d3σν (IX8)

with Σ a space-like 3-hypersurface P represents the total 4-momentum crossing Σ and invokingthe Gauss theorem Eq (IX7) implies that it is a conserved quantity

IX2 Four-velocity of a fluid flow Local rest frameThe four-velocity of a flow is a field defined at each point x of a space-time domain D of time-like4-vectors u(x) with constant magnitude c ie

[u(x)]2 = umicro(x)umicro(x) = minusc2 forallx (IX9)

with umicro(x) the (contravariant) components of u(x)At each point x of the fluid one can define a proper reference frame the so-called local rest

frame(lxxix) hereafter abbreviated as LR(x) in which the space-like Minkowski components of thelocal flow 4-velocity vanish

umicro(x)∣∣LR(x)

= (c 0 0 0) (IX10)

Let~v(x) denote the instantaneous velocity of (an observer at rest in) the local rest frame LR(x)with respect to a fixed reference frame R In the latter the components of the flow four-velocityare

umicro(x)∣∣R

(γ(x)c

γ(x)~v(x)

) (IX11)

with γ(x) = 1radic

1minus~v(x)2c2 the corresponding Lorentz factor

The local rest frame represents the reference frame in which the local thermodynamic variablesof the systemmdashparticle number density n(x) and energy density ε(x)mdashare defined in their usualsense

n(x) equiv n(x)∣∣LR(x)

ε(x) equiv T 00(x)∣∣LR(x)

(IX12)

For the remaining local thermodynamic variables in the local rest frame it is assumed that theyare related to n(x) and ε(x) in the same way as when the fluid is at thermodynamic equilibriumThus the pressure P (x) is given by the mechanical equation of state

P (x)∣∣LR(x)

= P (ε(x) n(x)) (IX13)

the temperature T (x) is given by the thermal equation of state the entropy density s(x) is definedby the Gibbs fundamental relation and so on(lxxix)lokales Ruhesystem

Remarks

lowast A slightly more formal approach to define 4-velocity and local rest frame is to turn the reasoninground Namely one introduces the latter first as a reference frame LR(x) in which ldquophysics at pointx is easyrdquo that is in which the fluid is locally motionless Introducing then an instantaneous inertialreference frame that momentarily coincides with LR(x) one considers an observer O who is at restin that inertial frame The four-velocity of the fluid u(x) with respect to some fixed reference frameR is then the four-velocity of O in Rmdashdefined as the derivative of Orsquos space-time trajectory withrespect to his proper time

The remaining issue is that of the local absence of motion which defines LR(x) In particularthere must be no energy flow ie T 0j(x) = 0 One thus looks for a time-like eigenvector u(x) ofthe energy-momentum tensor TTT(x)

TTT(x) middot u(x) = minusεu(x) hArr Tmicroν(x)uν(x) = minusεumicro(x)

with minusε lt 0 the corresponding eigenvalue and u(x) normalized to c Writing that thanks to thesymmetry of TTT(x) u(x) is also a left-eigenvector ie umicro(x)Tmicroν(x) = minusεuν(x) one finds that theenergy flux density vanishes in the reference frame in which the Minkowski components of u(x)have the simple form (IX10) This constitutes an appropriate choice of local rest frame and onehas at the same time the corresponding four-velocity u(x)

lowast The relativistic energy density ε differs from its at first sight obvious non-relativistic counterpartthe internal energy density e The reason is that ε also contains the contribution from the massenergy of the particles and antiparticlesmdashmc2 per (anti)particlemdash which is conventionally not takeninto account in the non-relativistic internal energy density

lowast To distinguish between the reference frame dependent quantities like particle number densityn(x) or energy density T 00(x) and the corresponding quantities measured in the local rest framenamely n(x) or ε(x) the latter are referred to as comoving

The comoving quantities can actually be computed easily within any reference frame and coor-dinate system Writing thus

cN0(x)

∣∣LR(x)

=N0(x)u0(x)

[u0(x)]2

∣∣∣∣LR(x)

=N0(x)u0(x)

g00(x)[u0(x)]2

∣∣∣∣LR(x)

=Nmicro(x)umicro(x)

uν(x)uν(x)

∣∣∣∣LR(x)

where we used that u0(x) = g00(x)u0(x) in the local rest frame the rightmost term of the aboveidentity is a ratio of two Lorentz-invariant scalars thus itself a Lorentz scalar field independent ofthe reference frame in which it is computed

n(x) =Nmicro(x)umicro(x)

uν(x)uν(x)=

N(x) middot u(x)

[u(x)]2 (IX14)

Similarly one shows that

= c2 umicro(x)Tmicroν(x)uν(x)

[uρ(x)uρ(x)]2

∣∣∣∣LR(x)

c2umicro(x)Tmicroν(x)uν(x) =

c2u(x)middotTTT(x)middotu(x) (IX15)

where the normalization of the 4-velocity was used

In the following Sections we introduce fluid models defined by the relations between the con-served currentsmdashparticle number 4-current N(x) and energy-momentum tensor TTT(x)mdashand the fluid4-velocity u(x) and comoving thermodynamic quantities

IX3 Perfect relativistic fluid 139

IX3 Perfect relativistic fluidBy definition a fluid is perfect when there is no dissipative current in it see definition (III16a)As a consequence one can at each point x of the fluid find a reference frame in which the localproperties in the neighborhood of x are spatially isotropic [cf definition (III23)] This referenceframe represents the natural choice for the local rest frame at point x LR(x)

The forms of the particle-number 4-current and the energy-momentum tensor of a perfect fluidare first introduced in Sec IX31 It is then shown that the postulated absence of dissipativecurrent automatically leads to the conservation of entropy in the motion (Sec IX32) Eventuallythe low-velocity limit of the dynamical equations is investigated in Sec IX33

IX31 Particle four-current and energy-momentum tensor of a perfect fluid

To express the defining feature of the local rest frame LR(x) namely the spatial isotropy ofthe local fluid properties it is convenient to adopt a Cartesian coordinate system for the space-likedirections in LR(x) since the fluid characteristics are the same in all spatial directions this inparticular holds along the three mutually perpendicular axes defining Cartesian coordinates

Adopting momentarily such a systemmdashand accordingly Minkowski coordinates on space-timemdashthe local-rest-frame values of the particle number flux density ~(x) the j-th component cT 0j(x) ofthe energy flux density and the density cminus1T i0(x) of the i-th component of momentum should allvanish In addition the momentum flux-density 3-tensor TTT(x) should also be diagonal in LR(x)All in all one thus necessarily has

N0(x)∣∣LR(x)

= cn(x) ~(x)∣∣LR(x)

= ~0 (IX16a)

T 00(x)∣∣LR(x)

= ε(x)

T ij(x)∣∣LR(x)

= P (x)δij foralli j = 1 2 3 (IX16b)

T i0(x)∣∣LR(x)

= T 0j(x)∣∣LR(x)

= 0 foralli j = 1 2 3

where the definitions (IX12) were taken into account while P (x) denotes the pressure In matrixform the energy-momentum tensor (IX16b) becomes

Tmicroν(x)∣∣LR(x)

ε(x) 0 0 0

0 P (x) 0 00 0 P (x) 00 0 0 P (x)

(IX16c)

Remark The identification of the diagonal spatial components with a ldquopressurerdquo term is warrantedby the physical interpretation of the T ii(x) Referring to it as ldquotherdquo pressure anticipates the fact thatit behaves as the thermodynamic quantity that is related to energy density and particle number bythe mechanical equation of state of the fluid

In an arbitrary reference frame and allowing for the possible use of curvilinear coordinates thecomponents of the particle number 4-current and the energy-momentum tensor of a perfect fluidare

Nmicro(x) = n(x)umicro(x) (IX17a)

Tmicroν(x) = P (x)gmicroν(x) +[ε(x) + P (x)

]umicro(x)uν(x)

c2(IX17b)

respectively with umicro(x) the components of the fluid 4-velocity

Relation (IX17a) resp (IX17b) is an identity between the components of two 4-vectors resp two4-tensors which transform identically under Lorentz transformationsmdashie changes of referenceframemdashand coordinate basis changes Since the components of these 4-vectors resp 4-tensorsare equal in a given reference framemdashthe local rest framemdashand a given basismdashthat of Minkowskicoordinatesmdash they remain equal in any coordinate system in any reference frame

In geometric formulation the particle number 4-current and energy-momentum tensor respec-tively read

N(x) = n(x)u(x) (IX18a)

TTT(x) = P (x)gminus1(x) +[ε(x) + P (x)

]u(x)otimes u(x)

c2 (IX18b)

The latter is very reminiscent of the 3-dimensional non-relativistic momentum flux density (III22)similarly the reader may also compare the component-wise formulations (III21b) and (IX17a)

Remarks

lowast The energy-momentum tensor is obviously symmetricmdashwhich is a non-trivial physical statementFor instance the identity T i0 = T 0i means that (1c times) the energy flux density in directioni equals (c times) the density of the i-th component of momentummdashwhere one may rightly arguethat the factors of c are historical accidents in the choice of units This is possible in a relativistictheory only because the energy density also contains the mass energy

lowast In Eq (IX17b) or (IX18b) the sum ε(x) + P (x) is equivalently the enthalpy density w(x)

lowast Equation (IX17b) (IX18b) or (IX19a) below represents the most general symmetric(

)-tensor

that can be constructed using only the metric tensor and the 4-velocity

The component form (IX17b) of the energy-momentum tensor can trivially be recast as

Tmicroν(x) = ε(x)umicro(x)uν(x)

c2+ P (x)∆microν(x) (IX19a)

with∆microν(x) equiv gmicroν(x) +

umicro(x)uν(x)

c2(IX19b)

the components of a tensor ∆∆∆ whichmdashin its(

)-formmdashis actually a projector on the 3-dimensional

vector space orthogonal to the 4-velocity u(x) while umicro(x)uν(x)c2 projects on the time-like directionof the 4-velocity

One easily checks the identities ∆microν(x)∆ν

ρ(x) = ∆microρ(x) and ∆micro

ν(x)uν(x) = 0

From Eq (IX19a) follows at once that the comoving pressure P (x) can be found in any referenceframe as

P (x) =1

3∆microν(x)Tmicroν(x) (IX20)

which complements relations (IX14) and (IX15) for the particle number density and energy densityrespectively

Remark Contracting the energy-momentum tensor TTT with the metric tensor twice yields a scalarthe so-called trace of TTT

TTT(x) g(x) = Tmicroν(x)gmicroν(x) = Tmicromicro(x) = 3P (x)minus ε(x) (IX21)

IX32 Entropy in a perfect fluid

Let s(x) denote the (comoving) entropy density of the fluid as defined in the local rest frameLR(x) at point x

IX32 a

Entropy conservation

For a perfect fluid the fundamental equations of motion (IX2) and (IX7) lead automaticallyto the local conservation of entropy

dmicro[s(x)umicro(x)

]= 0 (IX22)

with s(x)umicro(x) the entropy four-current

Proof The relation U = TSminus PV +microNN with U resp micro

Nthe internal energy resp the chemical

potential gives for the local thermodynamic densities ε = TsminusP +microN

n Inserting this expressionof the energy density in Eq (IX17b) yields (dropping the x variable for the sake of brevity)

Tmicroν = Pgmicroν + (Ts+ microN

n)umicrouν

c2= Pgmicroν +

[T (sumicro) + micro

N(numicro)

]uνc2

Taking the 4-gradient dmicro of this identity gives

dmicroTmicroν = dνP +

[T (sumicro)+micro

N(numicro)

]dmicrouνc2

+[sdmicroT+n dmicromicroN

]umicrouνc2

+[T dmicro(sumicro)+micro

Ndmicro(numicro)

]uνc2

Invoking the energy-momentum conservation equation (IX7) the leftmost member of this iden-tity vanishes The second term between square brackets on the right hand side can be rewrittenwith the help of the GibbsndashDuhem relation as sdmicroT + n dmicromicroN = dmicroP Eventually the parti-cle number conservation formulation (IX7) can be used in the rightmost term Multiplyingeverything by uν yields

0 = uν dνP +[T (sumicro) + micro

N(numicro)

]uν dmicrouν

c2+ (dmicroP )

umicrouνuνc2

+[T dmicro(sumicro)

]uνuνc2

The constant normalization uνuν = minusc2 of the 4-velocity implies uν dmicrouν = 0 for micro = 0 3

so that the equation becomes

0 = uν dνP minus (dmicroP )umicro minus T dmicro(sumicro)

leading to dmicro(sumicro) = 0

IX32 b

Isentropic distribution

The local conservation of entropy (IX22) implies the conservation of the entropy per particles(x)n(x) along the motion where n(x) denotes the comoving particle number density

Proof the total time derivative of the entropy per particle reads

)+~v middot ~nabla

γu middot d

where the second identity makes use of Eq (IX11) with γ the Lorentz factor The rightmostterm is then

u middot d(s

nu middot dsminus s

n2u middot dn =

(u middot dsminus s

nu middot dn

The continuity equation d middot (nu) = 0 gives u middot dn = minusn d middot u implying

γu middot d

γn(u middot ds+ s d middot u

γnd middot (su) = 0

where the last identity expresses the conservation of entropy

IX33 Non-relativistic limit

We shall now consider the low-velocity limit |~v| c of the relativistic equations of motion (IX2)and (IX7) in the case when the conserved currents are those of perfect fluids namely as given byrelations (IX17a) and (IX17b) Anticipating on the result we shall recover the equations governingthe dynamics of non-relativistic perfect fluids presented in Chapter III as could be expected for thesake of consistency

In the small-velocity limit the typical velocity of the atoms forming the fluid is also much smallerthan the speed of light which has two consequences On the one hand the available energies are toolow to allow the creation of particlendashantiparticle pairsmdashwhile their annihilation remains possiblemdashso that the fluid consists of either particles or antiparticles Accordingly the ldquonetrdquo particle numberdensity n(x) difference of the amounts of particles and antiparticles in a unit volume actuallycoincides with the ldquotruerdquo particle number density

On the other hand the relativistic energy density ε can then be expressed as the sum of thecontribution from the (rest) masses of the particles and of a kinetic energy term By definitionthe latter is the local internal energy density e of the fluid while the former is simply the numberdensity of particles multiplied by their mass energy

ε(x) = n(x)mc2 + e(x) = ρ(x)c2 + e(x) (IX23)

with ρ(x) the mass density of the fluid constituents It is important to note that the internal energydensity e is of order ~v2c2 with respect to the mass-energy term The same holds for the pressureP which is of the same order of magnitude as e(54)

Eventually Taylor expanding the Lorentz factor associated with the flow velocity yields

γ(x) sim|~v|c

~v(x)2

(~v(x)4

) (IX24)

Accordingly to leading order in~v2c2 the components (IX11) of the flow 4-velocity read

umicro(x) sim|~v|c

) (IX25)

Throughout the Section we shall omit for the sake of brevity the variables x resp (t ~r) of thevarious fields In addition we adopt for simplicity a system of Minkowski coordinates

IX33 a

Particle number conservation

The 4-velocity components (IX25) give for those of the particle number 4-current (IX17a)

Nmicro sim|~v|c

(n cn~v

Accordingly the particle number conservation equation (IX2) becomes

0 = partmicroNmicro asymp 1

part(n c)partt

3sumi=1

part(n vi)

partxi=partnpartt

+ ~nabla middot (n~v) (IX26)

That is one recovers the non-relativistic continuity equation (III10)

IX33 b

Momentum and energy conservation

The (components of the) energy-momentum tensor of a perfect fluid are given by Eq (IX17b)Performing a Taylor expansion including the leading and next-to-leading terms in |~v|c yields underconsideration of relation (IX23)(54)This is exemplified for instance by the non-relativistic classical ideal gas in which the internal energy density is

e = ncV kBT with cV a number of order 1mdashthis results eg from the equipartition theoremmdashwhile its pressure isP = nkBT

T 00 = minusP + γ2(ρc2 + e+ P ) sim|~v|c

ρc2 + e+ ρ~v2 +O(~v2

) (IX27a)

T 0j = T j0 = γ2(ρc2 + e+ P )vj

csim|~v|c

ρcvj +(e+ P + ρ~v2

(|~v|3

) (IX27b)

T ij = P gij + γ2(ρc2 + e+ P )vivj

c2sim|~v|c

P gij + ρ vivj +O(~v2

)= TTTij +O

) (IX27c)

In the last line we have introduced the components TTTij defined in Eq (III21b) of the three-dimensional momentum flux-density tensor for a perfect non-relativistic fluid As emphasized belowEq (IX23) the internal energy density and pressure in the rightmost terms of the first or secondequations are of the same order of magnitude as the term ρ~v2 with which they appear ie they arealways part of the highest-order term

Momentum conservationConsidering first the components (IX27b) (IX27c) the low-velocity limit of the relativistic

momentum-conservation equation partmicroTmicroj = 0 for j = 1 2 3 reads

part(ρcvj)

partt+

3sumi=1

partTTTij

partxi+O

)=part(ρvj)

partt+

3sumi=1

partTTTij

partxi+O

) (IX28)

This is precisely the conservation-equation formulation (III24a) of the Euler equation in absenceof external volume forces

Energy conservationGiven the physical interpretation of the components T 00 T i0 with i = 1 2 3 the component

ν = 0 of the energy-momentum conservation equation (IX7) partmicroTmicro0 = 0 should represent theconservation of energy

As was mentioned several times the relativistic energy density and flux density actually alsocontain a term from the rest mass of the fluid constituents Thus the leading order contribution topartmicroT

micro0 = 0 coming from the first terms in the right members of Eqs (IX27a) and (IX27b) is

0 =part(ρc)

partt+

3sumi=1

part(ρcvi)

partxi+O

that is up to a factor c exactly the continuity equation (III9) which was already shown to be thelow-velocity limit of the conservation of the particle-number 4-current

To isolate the internal energy contribution it is thus necessary to subtract that of mass energyIn the fluid local rest frame relation (IX23) shows that one must subtract ρc2 from ε The formersimply equals ρcu0|LR while the latter is the component micro = 0 of Tmicro0|LR whose space-like compo-nents vanish in the local rest frame To fully subtract the mass energy contribution in any framefrom both the energy density and flux density one should thus consider the 4-vector Tmicro0 minus ρcumicro

Accordingly instead of simply using partmicroTmicro0 = 0 one should start from the equivalentmdashthanksto Eq (IX2) and the relation ρ = mnmdashequation partmicro(Tmicro0 minus ρcumicro) = 0 With the approximations

ρcu0 = γρc2 = ρc2 +1

2ρ~v2 +O

ρcuj = γρcvj = ρcvj +

2ρ~v2

(|~v|5

)one finds

0 = partmicro(Tmicro0 minus ρcumicro

2ρ~v2 + e

3sumj=1

partxj

2ρ~v2 + e+ P

that ispart

2ρ~v2 + e

)+ ~nabla middot

2ρ~v2 + e+ P

]asymp 0 (IX29)

This is the non-relativistic local formulation of energy conservation (III33) for a perfect fluid inabsence of external volume forces Since that equation had been postulated in Section III41 theabove derivation may be seen as its belated proof

IX33 c

Using the approximate 4-velocity components (IX25) the entropy conservation equation (IX22)becomes in the low-velocity limit

0 = partmicro(sumicro) asymp 1

part(sc)

partt+

3sumi=1

part(svi)

partxi=parts

partt+ ~nabla middot (s~v) (IX30)

ie gives the non-relativistic equation (III34)

IX4 Dissipative relativistic fluidsIn a dissipative relativistic fluid the transport of particle number and 4-momentum is no longeronly convectivemdashie caused by the fluid motionmdash but may also diffusive due eg to spatial gra-dients of the flow velocity field the temperature or the chemical potential(s) associated with theconserved particle number(s) The description of these new types of transport necessitate the in-troduction of additional contributions to the particle-number 4-current and the energy-momentumtensor (Sec IX41) that break the local spatial isotropy of the fluid As a matter of fact the localrest frame of the fluid is no longer uniquely but there are in general different choices that lead toldquosimplerdquo expressions for the dynamical quantities (Sec IX42)

For the sake of brevity we adopt in this Section a ldquonaturalrdquo system of units in which the speedof light c and the Boltzmann constant kB equal 1

IX41 Dissipative currents

To account for the additional types of transport present in dissipative fluids extra terms areadded to the particle-number 4-current and energy-momentum tensor Denoting with a subscript(0) the quantities for a perfect fluid their equivalent in the dissipative case thus read

Nmicro(x) = Nmicro(0)(x) + nmicro(x) Tmicroν(x) = Tmicroν(0)(x) + τmicroν(x) (IX31a)

or equivalently in geometric formulation

N(x) = N(0)(x) + n(x) TTT(x) = TTT(0)(x) + τττ(x) (IX31b)

with n(x) resp τττ(x) a 4-vector resp 4-tensor of degree 2 with components nmicro(x) resp τmicroν(x) thatrepresents a dissipative particle-number resp energy-momentum flux density

In analogy by the perfect-fluid case it is natural to introduce a 4-velocity u(x) in terms ofwhich the quantities n(0)(x) TTT(0)(x) have a simple ldquoisotropicrdquo expression Accordingly let u(x) bean arbitrary time-like 4-vector field with constant magnitude minusc2 = minus1 with components umicro(x)micro isin 0 1 2 3 The reference frame in which the spatial components of this ldquo4-velocityrdquo vanisheswill constitute the local rest frame LR(x) associated with u(x)

The projector ∆∆∆ on the 3-dimensional vector space orthogonal to the 4-velocity u(x) is definedas in Eq (IX19b) ie has components

∆microν(x) equiv gmicroν(x) + umicro(x)uν(x) (IX32)

with gmicroν(x) the components of the inverse metric tensor gminus1(x) For the comprehension it is im-portant to realize that ∆∆∆ plays the role of the identity in the 3-space orthogonal to u(x)

IX4 Dissipative relativistic fluids 145

In analogy with Eqs (IX17a) (IX18) and (IX19a) one thus writes

Nmicro(x) = n(x)umicro(x) + nmicro(x) (IX33a)

or equivalently

N(x) = n(x)u(x) + n(x) (IX33b)

andTmicroν(x) = ε(x)umicro(x)uν(x) + P (x)∆microν(x) + τmicroν(x) (IX34a)

ie in geometric formTTT(x) = ε(x)u(x)otimesu(x) + P (x)∆∆∆(x) + τττ(x) (IX34b)

The precise physical content and mathematical form of the additional terms can now be furtherspecified

Tensor algebra

In order for n(x) to represent the (net) comoving particle density the dissipative 4-vector n(x)may have no timelike component in the the local rest frame LR(x) defined by the 4-velocity seedefinition (IX12) Accordingly the condition

umicro(x)nmicro(x)∣∣LR(x)

must hold in the local rest frame Since the left hand side of this identity is a Lorentz scalar itholds in any reference frame or coordinate system

umicro(x)nmicro(x) = u(x) middot n(x) = 0 (IX35a)

Equations (IX33a) (IX33) thus represent the decomposition of a 4-vector in a component parallelto the flow 4-velocity and a component orthogonal to it In keeping one can write

nmicro(x) = ∆microν(x)Nν(x) (IX35b)

Physically n(x) represents a diffusive particle-number 4-current in the local rest frame which de-scribes the non-convective transport of particle number

Similarly the dissipative energy-momentum current $(x) can have no 00-component in the localrest frame to ensure that T 00(x) in that frame still define the comoving energy density ε(x) Thismeans that the components τmicroν(x) may not be proportional to the product umicro(x)uν(x) The mostgeneral symmetric tensor of degree 2 which obeys that condition is of the form

τmicroν(x) = qmicro(x)uν(x) + qν(x)umicro(x) + πmicroν(x) (IX36a)

with qmicro(x) resp πmicroν(x) the components of a 4-vector q(x) resp πππ(x) such that

umicro(x)qmicro(x) = u(x) middot q(x) = 0 (IX36b)

andumicro(x)πmicroν(x)uν(x) = u(x) middotπππ(x) middot u(x) = 0 (IX36c)

Condition (IX36b) expresses that q(x) is a 4-vector orthogonal to the 4-velocity u(x) which physi-cally represents the heat current or energy flux density in the local rest frame

In turn the symmetric tensor πππ(x) can be decomposed into the sum of a traceless tensor $$$(x)with components $microν(x) and a tensor proportional to the projector (IX19b) orthogonal to the4-velocity

πmicroν(x) = $microν(x) + Π(x)∆microν(x) (IX36d)

The tensor $$$(x) is the shear stress tensor in the local rest frame of the fluid that describes thetransport of momentum due to shear deformations Eventually Π(x) represents a dissipative pressureterm since it behaves as the thermodynamic pressure P (x) as shown by Eq (IX37) below

All in all the components of the energy-momentum tensor in a dissipative relativistic fluid maythus be written as

Tmicroν(x) = ε(x)umicro(x)uν(x) +[P (x) + Π(x)

]∆microν(x) + qmicro(x)uν(x) + qν(x)umicro(x) +$microν(x) (IX37a)

which in geometric formulation reads

TTT(x) = ε(x)u(x)otimesu(x) +[P (x) + Π(x)

]∆∆∆(x) + q(x)otimesu(x) + u(x)otimesq(x) +$$$(x) (IX37b)

One can easily check the identities

qmicro(x) = ∆microν(x)Tνρ(x)uρ(x) (IX38a)

$microν(x) =1

[∆micro

ρ(x)∆νσ(x) + ∆ν

ρ(x)∆microσ(x)minus 2

3∆microν(x)∆ρσ(x)

]T ρσ(x) (IX38b)

P (x) + Π(x) = minus1

3∆microν(x)Tmicroν(x) (IX38c)

which together with Eq (IX15)

ε(x) = umicro(x)Tmicroν(x)uν(x) = u(x) middotTTT(x) middot u(x) (IX38d)

allow one to recover the various fields in which the energy-momentum tensor has been decomposed

Remarks

lowast The energy-momentum tensor comprises 10 unknown independent fields namely the componentsTmicroν with ν ge micro In the decomposition (IX37) written in the local rest frame ε(x) P (x)+Π(x) thespace-like components qi(x) and $ij(x) represent 1+1+3+5=10 equivalent independent fieldsmdashoutof the 6 components $ij(x) with j ge i one of the diagonal ones is fixed by the condition on thetrace This in particular shows that the decomposition of the left hand side of Eq (IX38c) intotwo terms is as yet prematuremdashthe splitting actually requires of an equation of state to properlyidentify P (x)

Similarly the 4 unknown components Nmicro of the particle-number 4-current are expressed in termsof n(x) and the three spatial components ni(x) ie an equivalent number of independent fields

lowast Let amicroν denote the (contravariant) components of an arbitrary(

)-tensor One encounters in the

literature the various notationsa(microν) equiv 1

(amicroν + aνmicro

which represents the symmetric part of the tensor

a[microν] equiv 1

(amicroν minus aνmicro

)for the antisymmetric partmdashso that amicroν = a(microν) + a[microν]mdash and

a〈microν〉 equiv(

∆ (microρ ∆ν)

σ minus1

3∆microν∆ρσ

)aρσ

which is the symmetrized traceless projection on the 3-space orthogonal to the 4-velocity Usingthese notations the dissipative stress tensor (IX36a) reads

τmicroν(x) = q(micro(x)uν)(x) +$microν(x)minusΠ(x)∆microν(x)

while Eq (IX38b) becomes $microν(x) = T 〈microν〉(x)

IX42 Local rest frames

At a given point in a dissipative relativistic fluid the net particle number(s) and the energy canflow in different directions This can happen in particular because particlendashantiparticle pairs whichdo not contribute to the net particle-number density still transport energy Another not exclusivepossibility is that different conserved quantum numbers flow in different directions In any caseone can in general not find a preferred reference frame in which the local properties of the fluid areisotropic

As a consequence there is also no unique ldquonaturalrdquo choice for the 4-velocity u(x) of the fluidmotion On the contrary several definitions of the flow 4-velocity are possible which imply varyingrelations for the dissipative currents although the physics that is being described remains the same

bull A first natural possibility proposed by Eckart(at) [44] is to take the 4-velocity proportionalto the particle-number 4-current(55) namely

umicroEckart(x) equiv Nmicro(x)radicNν(x)Nν(x)

(IX39)

Accordingly the dissipative particle-number flux n(x) vanishes automatically so that theexpression of particle-number conservation is simpler with that choiceThe local rest frame associated with the flow 4-velocity (IX39) is then referred to as Eckartframe

A drawback of that definition of the fluid 4-velocity is that the net particle number can possiblyvanish in some regions of a given flow so that uEckart(x) is not defined unambiguously in suchdomains

bull An alternative natural definition is that of Landau(au) (and Lifshitz(av)) according to whomthe fluid 4-velocity is taken to be proportional to the energy flux density The corresponding4-velocity is defined by the implicit equation

umicroLandau(x) =Tmicroν(x)uνLandau(x)radic

uλLandau(x)T ρλ (x)Tρσ(x)uσLandau(x)

(IX40a)

or equivalently

uρLandau(x)Tρσ(x)uσLandau(x) (IX40b)

With this choice which in turn determines the Landau frame the heat current q(x) vanishesso that the dissipative tensor τττ(x) satisfies the condition

umicroLandau(x)τmicroν(x) = 0 (IX40c)

and reduces to its ldquoviscousrdquo part πππ(x)

For a fluid without conserved quantum number the Landau definition of the 4-velocity is theonly natural one However in the presence of a conserved quantum number heat conductionnow enters the dissipative part of the associated current n(x) which conflicts with the intuitiongained in the non-relativistic case This implies that the Landau choice does not lead to asimple behavior in the limit of low velocities

(55) or to one of the quantum-number 4-currents in case there are several conserved quantum numbers

(at)C Eckart 1902ndash1973 (au)L D Landau = L D Landau 1908ndash1968 (av)E M Lifxic = E M Lifshitz1915ndash1985

Eventually one may of course choose to work with a general 4-velocity u(x) and thus to keepboth the diffusive particle-number current and the heat flux density in the dynamical fields (IX33)ndash(IX37)

IX43 General equations of motion

By substituting the decompositions (IX33) (IX37) into the generic conservation laws (IX2)(IX7) one can obtain model-independent equations of motion that do not depend on any assump-tion on the various dissipative currents

For that purpose let us introduce the notation

nablamicro(x) equiv ∆microν(x)dν (IX41a)

where dν ν isin 0 1 2 3 denotes the components of the 4-gradient dmdashinvolving covariant deriva-tives in case a non-Minkowski system of coordinates is being used In geometric formulation thisdefinition reads

nablanablanabla(x) equiv∆∆∆(x) middot d (IX41b)

As is most obvious in the local rest frame at point x in which the timelike componentnabla0(x) vanishesnablanablanabla(x) is the projection of the gradient on the space-like 3-space orthogonal to the 4-velocity Letus further adopt the Landau definition for the flow 4-velocity(56) which is simply denoted by u(x)without subscript

The net particle-number conservation equation (IX2) first yields

dmicroNmicro(x) = umicro(x)dmicron(x) + n(x)dmicrou

micro(x) + dmicronmicro(x) = 0 (IX42a)

In turn the conservation of the energy momentum tensor (IX7) projected perpendicular to respalong the 4-velocity gives

∆ρν(x)dmicroT

microν(x) =[ε(x) + P (x)

]umicro(x)dmicrou

ρ(x) +nablaρ(x)P (x) + ∆ρν(x)dmicroπ

microν(x) = 0 (IX42b)

respuν(x)dmicroT

microν(x) = minusumicro(x)dmicroε(x)minus[ε(x) + P (x)

]dmicrou

micro(x) + uν(x)dmicroπmicroν(x) = 0

In the latter equation one can substitute the rightmost term by

uν(x)dmicroπmicroν(x) = dmicro

[uν(x)πmicroν(x)

]minus[dmicrouν(x)

]πmicroν(x) = minus

[dmicrouν(x)

]πmicroν(x)

where the second equality follows from condition (IX40c) with τmicroν = πmicroν (since q = 0) Using theidentity dmicro = umicro(u middot d) +nablamicro and again the condition uνπmicroν = 0 this becomes

umicro(x)dmicroε(x) +[ε(x) + P (x)

]dmicrou

micro(x) + πmicroν(x)nablamicro(x)uν(x) = 0 (IX42c)

Equations (IX42a)ndash(IX42c) represent the relations governing the dynamics of a dissipative fluid inthe Landau frame

Remark If one adopts Eckartrsquos choice of velocity the resulting equations of motion differ from thosegiven heremdashfor instance the third term d middot n(x) in Eq (IX42a) drops out since n(x) = 0mdash yetthey are physically totally equivalent

Entropy law in a dissipative relativistic fluid

Combining the dynamical equation (IX42c) with the thermodynamic relations ε+P = Ts+microNnand dε = T ds+ microNdn one finds

T (x)dmicro[s(x)umicro(x)

]= minusπmicroν(x)nablamicro(x)uν(x) + microN(x)dmicron

micro(x)

(56)This choice of form for u(x) is often announced as ldquolet us work in the Landau framerdquo where frame is to beunderstood in its sense of framework

or equivalently using the identity nmicrodmicro = nmicronablamicro that follows from nmicroumicro = 0

dmicro

[s(x)umicro(x)minus microN(x)

T (x)nmicro(x)

]= minusπmicroν(x)

nablamicro(x)uν(x)

T (x)minus nmicro(x)nablamicro

[microN(x)

] (IX43a)

Using the symmetry of πmicroν one can replace nablamicrouν by its symmetric part 12(nablamicrouν + nablaνumicro) in

the first term on the right hand side With the decompositions πmicroν = $microν + Π∆microν [Eq (IX36d)]and

(nablamicrouν +nablaνumicro

[nablamicrouν +nablaνumicro minus

3∆microν

(nablanablanabla middot u

3∆microν

)equiv SSSmicroν +

3∆microν

where the SSSmicroν are the components of a traceless tensor(57)mdashcomparing with Eq (II15d) this is therate-of-shear tensormdash while nablanablanabla middot u is the (spatial) 3-divergence of the 4-velocity field one finds

dmicro

T (x)nmicro(x)

]= minus$

microν(x)

T (x)SSSmicroν(x)minus Π(x)

T (x)nablanablanabla(x) middot u(x)minus nmicro(x)nablamicro

[microN(x)

] (IX43b)

The left member of this equation is the 4-divergence of the entropy 4-current S(x) with componentsSmicro(x) comprising on the one hand the convective transport of entropymdashwhich is the only contribu-tion present in the perfect-fluid case see Eq (IX22)mdash and on the other hand a contribution fromthe dissipative particle-number current

Remark When working in the Eckart frame the dissipative particle-number current no longercontributes to the entropy 4-current Smdashwhich is obvious since n vanishes in that framemdash but theheat 4-current q does In an arbitrary framemdashie using a different choice of fluid 4-velocity andthereby of local rest framemdash both n and q contribute to S and to the right hand side of Eq (IX43b)

Let Ω be the 4-volume that represents the space-time trajectory of the fluid between an initialand a final times Integrating Eq (IX43b) over Ω while using the same reasoning as in sect IX11 bone sees that the left member will yield the change in the total entropy of the fluid during these twotimes This entropy variation must be positive to ensure that the second law of thermodynamicsholds Accordingly one requests that the integrand be positive dmicroS

micro(x) ge 0 This requirementcan be used to build models for the dissipative currents

IX44 First order dissipative relativistic fluid dynamics

The decompositions (IX33) (IX37) are purely algebraic and do not imply anything regardingthe physics of the fluid Any such assumption involve two distinct elements an equation of staterelating the energy density ε to the (thermodynamic) pressure P and the particle-number density n and a constitutive equation(lxxx) that models the dissipative effects ie the diffusive particle-number4-current N(x) the heat flux density q(x) and the dissipative stress tensor τ(x)

Several approaches are possible to construct such constitutive equations A first one would be tocompute the particle-number 4-current and energy-momentum tensor starting from an underlyingmicroscopic theory in particular from a kinetic description of the fluid constituents Alternativelyone can work at the ldquomacroscopicrdquo level using the various constraints applying to such

A first constraint is that the tensorial structure of the various currents should be the correctone using as building blocks the 4-velocity u the 4-gradients of the temperature T the chemicalpotential micro and of u as well as the projector ∆∆∆ one writes the possible forms of n q Π and $$$A further condition is that the second law of thermodynamics should hold ie that when insertingthe dissipative currents in Eq (IX43b) one obtains a 4-divergence of the entropy 4-current that isalways positive(57)In the notation introduced in the remark at the end of Sec IX41 SSSmicroν = nabla〈microuν〉(lxxx)konstitutive Gleichung

Working like in Sec IX43 in the Landau frame(58) in which the heat flux density q(x) vanishesthe simplest possibility that satisfies all constraints is to require

Π(x) = minusζ(x)nablamicro(x)umicro(x) (IX44a)

for the dissipative pressure

$microν(x) = minusη(x)

[nablamicro(x)uν(x) +nablaν(x)umicro(x)minus 2

3∆microν(x)

[nablaρ(x)uρ(x)

]]= minus2η(x)SSSmicroν(x) (IX44b)

for the shear stress tensor and

nmicro(x) = κ(x)

[n(x)T (x)

ε(x)+P (x)

]2nablamicro(x)

[microN(x)

](IX44c)

for the dissipative particle-number 4-current with η ζ κ three positive numbersmdashwhich depend onthe space-time position implicitly inasmuch as they vary with temperature and chemical potentialThe first two ones are obviously the shear and bulk viscosity coefficients respectively as hintedat by the similarity with the form (III26f) of the shear stress tensor of a Newtonian fluid in thenon-relativistic case Accordingly the equation of motion (IX42b) in which the dissipative stresstensor is substituted by πmicroν = $microν + Π∆microν with the forms (IX44a) (IX44b) yields the relativisticversion of the NavierndashStokes equation

What is less obvious is that κ in Eq (IX44c) does correspond to the heat conductivitymdashwhichexplains why the coefficient in front of the gradient is written in a rather contrived way

Inserting the dissipative currents (IX44) in the entropy law (IX43b) the latter becomes

d middot S(x) =$$$(x) $$$(x)

2η(x)T (x)+

Π(x)2

ζ(x)T (x)+

[ε(x)+P (x)

n(x)T (x)

]2 n(x)2

κ(x)T (x) (IX45)

Since n(x) is space-like the right hand side of this equation is positive as it should

The constitutive equations (IX44) only involve first order terms in the derivatives of velocitytemperature or chemical potential In keeping the theory constructed with such Ansaumltze is referredto as first order dissipative fluid dynamicsmdashwhich is the relativistic generalization of the set of lawsvalid for Newtonian fluids

This simple relation to the non-relativistic case together with the fact that only 3 transportcoefficients are neededmdashwhen working in the Landau or Eckart frames in the more general caseone needs 4 coefficientsmdashmakes first-order dissipative relativistic fluid dynamics attractive Thetheory suffers however from a severe issue which does not affect its non-relativistic counterpartIndeed it has been shown that many solutions of the relativistic NavierndashStokes(ndashFourier) equationsare unstable against small perturbations [46] Such disturbances will grow exponentially with timeon a microscopic typical time scale As a result the velocity of given modes can quickly exceedthe speed of light which is of course unacceptable in a relativistic theory In addition gradientsalso grow quickly leading to the breakdown of the small-gradient assumption that underlies theconstruction of first-order dissipative fluid dynamics This exponential growth of perturbation isespecially a problem for numerical implementations of the theory in which rounding errors whichquickly propagate

Violations of causality actually occur for short-wavelength modes which from a physical pointof view should not be described by fluid dynamics since they involve length scales on which thesystem is not ldquocontinuousrdquo As such the issue is more mathematical than physical These modes

(58)The corresponding formulae for Π $microν and qmicro valid in the Eckart frame in which n vanishes can be found egin Ref [45 Sec 24]

do however play a role in numerical computations so that there is indeed a problem when oneis not working with an analytical solution

As a consequence including dissipation in relativistic fluid dynamics necessitates going beyonda first-order expansion in gradients ie beyond the relativistic NavierndashStokesndashFourier theory

IX45 Second order dissipative relativistic fluid dynamics

Coming back to an arbitrary 4-velocity u(x) the components of the entropy 4-current S(x) in afirst-order dissipative theory read

Smicro(x) =P (x)gmicroν(x)minus Tmicroν(x)

T (x)uν(x)minus microN(x)

T (x)Nmicro(x) (IX46a)

or equivalently

Smicro(x) = s(x)umicro(x)minus microN(x)

T (x)nmicro(x) +

T (x)qmicro(x) (IX46b)

which simplify to the expression between square brackets on the left hand side of Eq (IX43b) withLandaursquos choice of 4-velocity

This entropy 4-current is linear in the dissipative 4-currents n(x) and q(x) In addition it isindependent of the velocity 3-gradientsmdashencoded in the expansion rate nablanablanabla(x)middotu(x) and the rate-of-shear tensor SSS(x)mdash which play a decisive role in dissipation That is the form (IX46) can begeneralized A more general form for the entropy 4-current is thus

S(x) = s(x)u(x)minus microN(x)

T (x)n(x) +

T (x)q(x) +

T (x)Q(x) (IX47a)

or equivalently component-wise

T (x)nmicro(x) +

T (x)qmicro(x) +

T (x)Qmicro(x) (IX47b)

with Q(x) a 4-vector with componentsQmicro(x) that depends on the flow 4-velocity and its gradientsmdashwhere nablanablanabla(x) middotu(x) and SSS(x) are traditionally replaced by Π(x) and $$$(x)mdashand on the dissipativecurrents

Qmicro(x) = Qmicro(u(x) n(x) q(x)Π(x)$$$(x)

) (IX47c)

In second order dissipative relativistic fluid dynamics the most general form for the additional4-vector Q(x) contributing to the entropy density is [47 48 49]

Q(x) =β0(x)Π(x)2 + β1(x)qN(x)2 + β2(x)$$$(x) $$$(x)

2T (x)u(x)minus α0(x)

T (x)Π(x)qN(x)minus α1(x)

T (x)$$$(x)middotqN(x)

(IX48a)where

qN(x) equiv q(x)minus ε(x) + P (x)

n(x)n(x)

component-wise this reads

Qmicro(x) =β0(x)Π(x)2 +β1(x)qN(x)2 +β2(x)$νρ(x)$νρ(x)

2T (x)umicro(x)minus α0(x)

T (x)Π(x)qmicro

N(x)minusα1(x)

T (x)$micro

ρ(x)qρN

(IX48b)The 4-vector Q(x) is now quadratic (ldquoof second orderrdquo) in the dissipative currentsmdashin the widersensemdashq(x) n(x) Π(x) and $$$(x) and involves 5 additional coefficients depending on temperatureand particle-number density α0 α1 β0 β1 and β2

Substituting this form of Q(x) in the entropy 4-current (IX47) the simplest way to ensurethat its 4-divergence be positive is to postulate linear relationships between the dissipative currents

and the gradients of velocity chemical potential (or rather of minusmicroNT ) and temperature (or rather1T ) as was done in Eqs (IX44) This recipe yields differential equations for Π(x) $$$(x) qN(x)representing 9 coupled scalar equations of motion These describe the relaxationmdashwith appropriatecharacteristic time scales τΠ τ$$$ τqN respectively proportional to β0 β2 β1 while the involved ldquotimederivativerdquo is that in the local rest frame u middotdmdash of the dissipative currents towards their first-orderexpressions (IX44)

Adding up the new equations to the usual ones (IX2) and (IX7) the resulting set of equationsknown as (Muumlller(aw)ndash)Israel(ax)ndashStewart(ay) theory is no longer plagued by the issues that affectsthe relativistic NavierndashStokesndashFourier equations

Bibliography for Chapter IXbull Andersson amp Comer [50]

bull LandaundashLifshitz [3 4] Chapter XV sect 133134 (perfect fluid) and sect 136 (dissipative fluid)

bull Romatschke [51]

bull Weinberg [52] Chapter 2 sect 10 (perfect fluid) and sect 11 (dissipative fluid)

(aw)I Muumlller born 1936 (ax)W Israel born 1931 (ay)J M Stewart born 1943

Appendices to Chapter IX

IXA Microscopic formulation of the hydrodynamical fieldsIn Sec IX1 we have taken common non-relativistic quantitiesmdashparticle number density and fluxdensity energy density momentum flux density and so onmdashand claimed that they may be used todefine a 4-vector resp a Lorentz tensor namely the particle number 4-current N(x) resp the energy-momentum tensor TTT(x) However we did not explicitly show that the latter are indeed a 4-vectorresp a tensor For that purpose the best is to turn the reasoning round and to introduce quantitieswhich are manifestly by construction a Lorentz 4-vector or tensor In turn one investigates thephysical interpretation of their components and shows that it coincides with known non-relativisticquantities

Throughout this Appendix we consider a system Σ of N ldquoparticlesrdquomdashie carriers of some con-served additive quantum numbermdashlabeled by k isin 1 N with world-lines xk(τ) and associated4-velocities uk(τ) equiv dxk(τ)dτ where the scalar parameter τ along the world-line of a given particleis conveniently taken as its proper time

IXA1 Particle number 4-current

The particle-number 4-current associated with the collection of particles Σ is defined as

N(x) equivNsumk=1

intuk(τ)δ(4)

(xminusxk(τ)

)d(cτ) (IXA1a)

or component-wise

Nmicro(x) equivNsumk=1

intumicrok(τ)δ(4)

(xνminusxνk(τ)

)d(cτ) for micro = 0 1 2 3 (IXA1b)

where the k-th integral in either sum is along the world-line of particle k The right hand sides ofthese equations clearly define a 4-vector resp its components For the latter some simple algebrayields the identities

cN0(t~r) =

Nsumk=1

δ(3)(~r minus ~xk(t)

) (IXA2a)

N i(t~r) =

Nsumk=1

vik(t)δ(3)(~r minus ~xk(t)

)(IXA2b)

with ~xk(t) the spatial trajectory corresponding to the world-line xk(τ)

Using u0k(τ) = cdtk(τ)dτ and changing the parameter along the world-lines from τ to t one

N0(t~r) = c

Nsumk=1

intδ(ctminusctk(τ)

)δ(3)(~xminus~xk(τ)

)dtk(τ)

dτd(cτ) = c

Nsumk=1

intδ(tminustk(t)

)δ(3)(~xminus~xk(t)

ie N0(t~r) = c

Nsumk=1

δ(3)(~xminus~xk(t)

) The proof for Eq (IXA2b) is identical

Inspecting the right hand sides of relations (IXA2) they obviously represent the particle num-ber density and flux density for the system Σ respectively

IXA2 Energy-momentum tensor

Denoting by pk the 4-momentum carried by particle k the energy-momentum tensor associatedwith the collection of particles Σ is defined as

TTT(x) equivNsumk=1

intpk(τ)otimes uk(τ)δ(4)

(xminusxk(τ)

)d(cτ) (IXA3a)

where the k-th integral in the sum is along the world-line of particle k as above component-wisethis gives

Tmicroν(x) equivNsumk=1

intpmicrok(τ)uνk(τ)δ(4)

(xλminusxλ(τ)

)d(cτ) for micro ν = 0 1 2 3 (IXA3b)

The members of these equations clearly define a Lorentz tensor of type(

)resp its components

Repeating the same derivation as that leading to Eq (IXA2a) one shows that

Tmicro0(t~r) =

Nsumk=1

pmicrok(t)cδ(3)(~r minus ~xk(t)

) (IXA4a)

Recognizing in p0kc the energy of particle k T 00 represents the energy density of the system Σmdash

under the assumption that the potential energy associated with the interaction between particles ismuch smaller than their mass and kinetic energiesmdash while T i0 for i = 1 2 3 represents c times thedensity of the i-th component of momentum In turn

T 0j(t~r) =Nsumk=1

p0k(t)v

jk(t)δ

(3)(~r minus ~xk(t)

)(IXA4b)

with j isin 1 2 3 is the 1c times the j-th of the energy flux density of the collection of particlesEventually for i j = 1 2 3

T ij(t~r) =Nsumk=1

pik(t)vjk(t)δ

(3)(~r minus ~xk(t)

)(IXA4c)

is clearly the j-th component of the flux density of momentum along the i-th direction

Remark Invoking the relation p = mu between the 4-momentum mass and 4-velocity of a (massive)particle shows at once that the energy-momentum tensor (IXA3) is symmetric

IXB Relativistic kinematicsLater

IXC Equations of state for relativistic fluids

CHAPTER X

Flows of relativistic fluids

X1 Relativistic fluids at rest

X2 One-dimensional relativistic flows

X21 Landau flow

[53 54]

X22 Bjorken flow(az)

perfect fluid [55]first-order dissipative fluid

(az)J D Bjorken born 1934

156 Flows of relativistic fluids

Appendices

APPENDIX A

Basic elements of thermodynamics

To be written

U = TS minus PV + microN (A1)

dU = T dS minus P dV + microdN (A2)

e+ P = Ts+ micron (A3)

de = T ds+ microdn (A4)

dP = sdT + n dmicro (A5)

Die letztere Gleichung folgt aus

de = d

VdU minus U

VdS minus P

VdN minus TS

V 2dV +

dV minus microN

V 2dV = T d

)+ microd

wobei die Relation dU = T dS minus P dV + microdN benutzt wurde

APPENDIX B

Tensors on a vector space

In this Appendix we gather mathematical definitions and results pertaining to tensors The purposeis mostly to introduce the ldquomodernrdquo geometrical view on tensors which defines them by their actionon vectors or one-forms ie in a coordinate-independent way (Sec B1) in contrast to the ldquooldrdquodefinition based on their behavior under basis transformations (Sec B2)

The reader is assumed to already possess enough knowledge on linear algebra to know what arevectors linear (in)dependence (multi)linearity matrices Similarly the notions of group fieldapplicationfunctionmapping are used without further mention

In the remainder of these lecture notes we actually consider tensors on real vector spaces iefor which the underlying base field K of scalars is the set R of real numbers here we remain moregeneral Einsteinrsquos summation convention is used throughout

B1 Vectors one-forms and tensors

B11 Vectors

are by definition the elements ~c of a vector space V ie of a set with 1) a binary operation(ldquoadditionrdquo) with which it is an Abelian group and 2) a multiplication with ldquoscalarsrdquomdashelements of abase field Kmdashwhich is associative has an identity element and is distributive with respect to bothadditions on V and on K

Introducing a basis B = ~ei ie a family of linearly independent vectors that span the wholespace V one associates to each vector ~c its uniquely defined components ci elements of the basefield K such that

~c = ci~ei (B1)

If the number of vectors of a basis is finitemdashin which case this holds for all basesmdash and equal tosome integer Dmdashwhich is the same for all basesmdash the space V is said to be finite-dimensional andD is its dimension (over K) D = dim V We shall assume that this is the case in the remainder ofthis Section

B12 One-forms

on a vector space V are the linear applications hereafter denoted as h˜ from V into thebase field of scalars K

The set of 1-forms on V equipped with the ldquonaturalrdquo addition and scalar multiplication is itselfa vector space over the field K denoted by V lowast and said to be dual to V

If V is finite-dimensional so is V lowast with dim V lowast = dim V Given a basis B = ~ei in V onecan then construct its dual basis Blowast = ε˜j in V lowast such that

ε˜j(~ei) = δji (B2)

where δji denotes the usual Kronecker delta symbol

B1 Vectors one-forms and tensors 161

The components of a 1-form h˜ on a given basis will be denoted as hjh˜ = hj ε˜j (B3)

Remarkslowast The choice of notations in particular the position of indices is not innocent Thus if ε˜jdenotes the dual base to ~ei the reader can trivially check that

ci = ε˜i(~c) and hj = h˜(~ej) (B4)

lowast In the ldquooldrdquo language the vectors of V resp the 1-forms of V lowast were designated as ldquocontravariantvectorsrdquo resp ldquocovariant vectorsrdquo or ldquocovectorsrdquo and their coordinates as ldquocontravariantrdquo resp rdquoco-variantrdquo coordinatesThe latter two applying to the components remain useful short denominations especially whenapplied to tensors (see below) Yet in truth they are not different components of a same mathemat-ical quantity but components of different objects between which a ldquonaturalrdquo correspondence wasintroduced in particular by using a metric tensor as in sect B14

B13 Tensors

Definition and first results

Let V be a vector space with base field K and m n denote two nonnegative integersThe multilinear applications of m one-formsmdashelements of V lowastmdashand n vectorsmdashelements of V mdashintoK are referred to as the tensors of type

)on V where linearity should hold with respect to every

argument The integer m+ n is the order (or often but improperly rank) of the tensorAlready known objects arise as special cases of this definition when either m or n is zero

bull the(

)-tensors are simply the scalars of the base field K

bull the(

)-tensors coincide with vectors(59)

bull the(

)-tensors are the one-forms More generally the

)-tensors are also known as (multi-

linear) n-forms

bull Eventually(

)-tensors are sometimes called ldquobivectorsrdquo or ldquodyadicsrdquo

Tensors will generically be denoted as TTT irrespective of their rank unless the latter is 0 or 1

A tensor may be symmetric or antisymmetric under the exchange of two of its arguments eitherboth vectors or both 1-forms Generalizing it may be totally symmetricmdashas eg the metric tensorwe shall encounter belowmdash or antisymmetric An instance of the latter case is the determinantwhich is the only (up to a multiplicative factor) totally antisymmetric D-form on a vector space ofdimension D

Remark Consider a(mn

)-tensorTTT (V lowast)mtimes(V lowast)n rarr K and letmprime le m nprime le n be two nonnegative

integers For every mprime-uplet of one-forms h˜i and nprime-uplet of vectors ~cjmdashand correspondingmultiplets of argument positions although here we take for simplicity the first onesmdashthe object

TTT(h˜1 h˜mprime middot middot ~c1 ~cnprime middot middot

where the dots denote ldquoemptyrdquo arguments can be applied to mminusmprime one-forms and nminus nprime vectorsto yield a scalar That is the tensor TTT induces a multilinear application(60) from (V lowast)m

prime times (V lowast)nprime

into the set of(mminusmprimenminusnprime

)-tensors

For example the(

)-tensors are in natural correspondence with the linear applications from V into

V ie in turn with the square matrices of order dim V (59)More accurately they are the elements of the double dual of V which is always homomorphic to V (60)Rather the number of such applications is the number of independentmdashunder consideration of possible

symmetriesmdashcombinations of mprime resp nprime one-form resp vector arguments

162 Tensors on a vector space

Operations on tensors

The tensors of a given type with the addition and scalar multiplication inherited from V forma vector space on K Besides these natural addition and multiplication one defines two furtheroperations on tensors the outer product or tensor productmdashwhich increases the rankmdashand thecontraction which decreases the rank

Consider two tensors TTT and TprimeTprimeTprime of respective types(mn

(mprime

nprime

) Their outer product TTTotimesTprimeTprimeTprime is

a tensor of type(m+mprime

n+nprime

)satisfying for every (m+mprime)-uplet (h˜1 h˜m h˜m+mprime) of 1-forms and

every (n+ nprime)-uplet (~c1 ~cn ~cn+nprime) of vectors the identity

TTTotimesTprimeTprimeTprime(h˜1 h˜m+mprime ~c1 ~cn+nprime)

TTT(h˜1 h˜m~c1 ~cn

)TprimeTprimeTprime(h˜m+1 h˜m+mprime ~cn+1 ~cn+nprime

For instance the outer product of two 1-forms h˜ h˜prime is a 2-form h˜ otimes h˜prime such that for every pairof vectors (~c~c prime) h˜ otimes h˜prime(~c~c prime) = h˜(~c) h˜prime(~c prime) In turn the outer product of two vectors ~c ~c prime is a(

)-tensor ~cotimes ~cprime such that for every pair of 1-forms (h˜ h˜prime) ~cotimes ~cprime(h˜ h˜ prime) = h˜(~c) h˜prime(~c prime)Tensors of type

)that can be written as outer products of m vectors and n one-forms are

sometimes called simple tensors

Let TTT be a(mn

)-tensor where both m and n are non-zero To define the contraction over its j-th

one-form and k-th vector arguments the easiestmdashapart from introducing the tensor componentsmdashisto write TTT as a sum of simple tensors By applying in each of the summand the k-th one-form tothe j-th vector which gives a number one obtains a sum of simple tensors of type

(mminus1nminus1

) which is

the result of the contraction operationExamples of contractions will be given after the metric tensor has been introduced

Tensor coordinates

Let ~ei resp ε˜j denote bases on a vector space V of dimension D resp on its dual V lowastmdashinprinciple they need not be dual to each other although using dual bases is what is implicitly alwaysdone in practicemdashand m n be two nonnegative integersThe Dm+n simple tensors ~ei1 otimes middot middot middot otimes~eim otimes ε˜j1 otimes middot middot middot otimes ε˜jn where each ik or jk runs from 1 to Dform a basis of the tensors of type

) The components of a tensor TTT on this basis will be denoted

as TTTi1imj1jnTTT = TTTi1imj1jn ~ei1 otimes middot middot middot otimes~eim otimes ε˜j1 otimes middot middot middot otimes ε˜jn (B5a)

whereTTTi1imj1jn = TTT(ε˜i1 ε˜im ~ej1 ~ejn) (B5b)

The possible symmetry or antisymmetry of a tensor with respect to the exchange of two of itsarguments translates into the corresponding symmetry or antisymmetry of the components whenexchanging the respective indices

In turn the contraction of TTT over its j-th one-form and k-th vector arguments yields the tensorwith components TTTijminus1`ij+1

jkminus1`jk+1 with summation over the repeated index `

B14 Metric tensor

Nondegenerate(61) symmetric bilinear forms play an important role as they allow one to intro-duce a further structure on the vector space V namely an inner product(62)

Accordingly let ε˜j denote a basis on the dual space V lowast A 2-form g = gij ε˜i otimes ε˜j is a metrictensor on V if it is symmetricmdashie g(~a~b) = g(~b~a) for all vectors ~a~b or equivalently gij = gji

(61)This will be introduced 4 lines further down as a condition on the matrix with elements gij which is equivalentto stating that for every non-vanishing vector ~a there exists ~b such that g(~a~b) 6= 0

(62)More precisely an inner product if g is (positive or negative) definite a semi-inner product otherwise

for all i jmdashand if the square matrix with elements gij is regular The number g(~a~b) is then alsodenoted ~a middot~b which in particularly gives

gij = g(~ei~ej

)= ~ei middot~ej (B6)

where ~ei is the basis dual to ε˜jSince the DtimesD-matrix with elements gij is regular it is invertible Let gij denote the elements

of its inverse matrix gijgjk = δki gijgjk = δik The D

2 scalars gij define a(

)-tensor gij~ei otimes~ej the

inverse metric tensor denoted as gminus1

Using results on symmetric matrices the square matrix with elements gij is diagonalizablemdashieone can find an appropriate basis ~ei such that g

(~ei~ej

)= 0 for i 6= j Since g is nondegenerate

the eigenvalues are non-zero at the cost of multiplying the basis vectors ~ei by a numerical factorone may demand that every g

(~ei~ei

)be either +1 or minus1 which yields the canonical form

gij = diag(minus1 minus1 1 1) (B7)

for the matrix representation of the components of the metric tensorIn that specific basis the component gij of gminus1 coincides with gij yet this does not hold in an

arbitrary basis

Role of g in tensor algebra

In agreement with the remark at the end of sect B13 a for any given vector ~c = ci~ei the objectg(~c ) maps vectors into the base field K ie it is a one-form c˜= cj ε˜j such that

cj = c˜(~ej) = g(~c~ej) = g(ci~ei~ej) = cigij (B8a)

That is a metric tensor g provides a mapping from vectors onto one-forms Reciprocally its inversemetric tensor gminus1 maps one-forms onto tensors leading to the relation

ci = gijcj (B8b)

Generalizing a metric tensor and its inverse thus allow one ldquoto lower or to raise indicesrdquo whichare operations mapping a tensor of type

)on a tensor of type

(m∓1nplusmn1

) respectively

Remarks

lowast Lowering resp raising an index actually amounts to an outer product with g resp gminus1 followedby the contraction of two indices For instance

~c = ci~eiouter product7minusrarr ~cotimes g = cigjk~ei otimes ε˜j otimes ε˜k contraction7minusrarr c˜= cigik ε˜k = ck ε˜k

where the first and second arguments of ~cotimes g have been contracted

lowast Generalizing the ldquodot productrdquo notation for the inner product defined by the metric tensor thecontraction is often also denoted with a dot product For example for a 2-form TTT and a vector ~c

TTT middot ~c =(TTTij ε˜i otimes ε˜j) middot (ck~ek) = TTTij c

jε˜iwhere we implicitly used Eq (B2) Note that for the dot-notation to be unambiguous it is betterif TTT is symmetric so that which of its indices is being contracted plays no roleSimilarly if TTT denotes a dyadic tensor and TprimeTprimeTprime a 2-form

TTT middotTprimeTprimeTprime =(TTTij~ei otimes~ej

)middot(TprimeTprimeTprimekl ε˜k otimes ε˜l) = TTTijTprimeTprimeTprime

jl~ei otimes ε˜lwhich is different from TprimeTprimeTprime middotTTT if the tensors are not symmetric The reader may even find in theliterature the notation

TTT TprimeTprimeTprime equiv TTTijTprimeTprimeTprimeji

involving two successive contractions

B2 Change of basisLet B = ~ei and Bprime = ~ejprime denote two bases of the vector space V and Blowast = ε˜i Bprimelowast = ε˜jprimethe corresponding dual bases on V lowast The basis vector of Bprime can be expressed in terms of those ofB with the help of a non-singular matrix Λ with elements Λijprime such that

~ejprime = Λijprime~ei (B9)

Remark Λ is not a tensor for the two indices of its elements refer to two different basesmdashwhichis emphasized by the use of one primed and one unprimed indexmdashwhile both components of a(

)-tensor are with respect to the ldquosamerdquo basis(63)

Let Λkprimei denote the elements of the inverse matrix Λminus1 that is

ΛkprimeiΛijprime = δk

primejprime and ΛikprimeΛ

kprimej = δij

One then easily checks that the numbers Λkprimei govern the change of basis from Blowast to Bprimelowast namely

ε˜jprime = Λjprimei ε˜i (B10)

Accordingly each ldquovectorrdquo component transforms with Λminus1

cjprime

= Λjprimei ci TTTj

prime1j

primem = Λj

prime1i1 middot middot middotΛj

primemimTTTi1im (B11)

In turn every ldquo1-formrdquo component transforms with Λ

hjprime = Λijprimehi TTTjprime1jprimen = Λi1jprime1 middot middot middotΛinjprimenTTTi1in (B12)

One can thus obtain the coordinates of an arbitrary tensor in any basis by knowing just thetransformation of basis vectors and one-forms

Bibliography for Appendix Bbull Your favorite linear algebra textbook

bull A concise reminder can eg be found in Nakahara [56] Chapter 22

bull A more extensivemdashand elementarymdashtreatment biased towards geometrical applications oflinear algebra is provided in Postnikov [57](64) see eg Lectures 1 (beginning) 4ndash6 amp 18

(63)Or rather with respect to a basis and its dual(64)The reader should be aware that some of the mathematical terms usedmdashas translated from the Russianmdashare

non-standard eg (linear bilinear) ldquofunctionalrdquo for form or ldquoconjugaterdquo (space basis) for dual

APPENDIX C

Tensor calculus

Continuum mechanics and in particular fluid dynamics is a theory of (classical) fields The lattermay be scalars vectors or more generally tensorsmdashmainly of degree at most 2mdash whose dynamicalbehavior is governed by partial differential equations which obviously involve various derivatives oftensorial quantities

When describing vector or tensor fields by their respective components on appropriate (local)bases the basis vectors or tensors may actually vary from point to point Accordingly care mustbe taken when differentiating with respect to the space coordinates instead of the usual partialderivatives the quantities that behave in the expected manner are rather covariant derivatives(Sec C1) which are the main topic of this Appendix

To provide the reader with some elementary background on the proper mathematical frameworkto discuss vector and tensor fields and their differentiation some basic ideas of differential geometryare gathered in Sec C2

C1 Covariant differentiation of tensor fieldsThe purpose of this Section is to introduce the covariant derivative which is the appropriate math-ematical quantity measuring the spatial rate of change of a field on a space irrespective of thechoice of coordinates on that space The notion is first introduced for vector fields (Sec C11)and illustrated on the example of vector fields on a plane (Sec C12) The covariant derivative oftensors of arbitrary type in particular of one-forms is then given in Sec C13 Eventually theusual differential operators of vector analysis are discussed in Sec C14

Throughout this Section we mostly list recipes without providing proofs or the given resultsnor specifying for example in which space the vector or tensor fields ldquoliverdquo These more formal issueswill be shortly introduced in Sec C2

C11 Covariant differentiation of vector fields

Consider a set M of points generically denoted by P possessing the necessary properties sothat the following features are realized

(a) In a neighborhood of every point P isinM one can find a system of local coordinates xi(P )

(b) It is possible to define functions on M with sufficient smoothness properties as eg differen-tiable functions

(c) At each point P isin M one can attach vectorsmdashand more generally tensors Let ~ei(P )denote a basis of the vectors at P

From the physicistrsquos point of view the above requirements mean that we want to be able to definescalar vector or tensor fields at each point [property (c)] that depend smoothly on the position[property (b)] where the latter can be labeled by local coordinates [property (a)] Mathematicallyit will be seen in Sec C2 that the proper framework is to look at a differentiable manifold and itstangent bundle

166 Tensor calculus

Before we go any further let us emphasize that the results we state hereafter are independentof the dimension n of the vectors from 1 to which the indices i j k l run In addition we useEinsteinrsquos summation convention throughout

Assuming the above requirements are fulfilled which we now do without further comment wein addition assume that the local basis ~ei(P ) at every point is that which is ldquonaturally inducedrdquoby the coordinates xi(P )(65) and that for every possible i the mapping P 7rarr ~ei(P ) defines acontinuous and even differentiable vector field on M(66) The derivative of ~ei at P with respect toany of the (local) coordinate direction xk is then itself a vector ldquoat P rdquo which may thus be expandedon the basis ~el(P ) denoting by Γlik(P ) its coordinates

part~ei(P )

partxk= Γlik(P )~el(P ) (C1)

The numbers Γlik which are also alternatively denoted asli k

are called Christoffel symbols (of

the second type) or connection coefficients

Remark The reader should remember that the local coordinates also depend on P ie a betternotation for the left hand side of Eq (C1)mdashand for every similar derivative in the followingmdashcouldbe part~ei(P )partxk(P )

Let now ~c(P ) be a differentiable vector field defined on M whose local coordinates at each pointwill be denoted by ci(P ) [cf Eq (B1)]

~c(P ) = ci(P )~ei(P ) (C2)

The spatial rate of change in ~c between a point P and a neighboring point P prime situated in thexk-direction with respect to P is given by

part~c(P )

partxk=

dci(P )

dxk~ei(P ) (C3a)

where the component along ~ei(P ) is the so-called covariant derivative

dci(P )

dxk=partci(P )

partxk+ Γilk(P )cl(P ) (C3b)

Remark The covariant derivative dcidxk is often denoted by cik with a semicolon in front of theindex (or indices) related to the direction(s) along which one differentiates In contrast the partialderivative partcipartxk is then written as cik with a comma That is Eq (C3b) is recast as

cik(P ) = cik(P ) + Γilk(P )cl(P ) (C3c)

The proof of Eqs (C3) is rather straightforward Differentiating relation (C2) with the productrule first gives

part~c(P )

partxk=partci(P )

partxk~ei(P ) + ci

part~ei(P )

partxk=partci(P )

partxk~ei(P ) + ci(P ) Γlik(P )~el(P )

where we have used the derivative (C1) In the rightmost term the dummy indices i and l maybe relabeled as l and i respectively yielding ciΓlik~el = cl Γilk~ei ie

part~c(P )

partxk=partci(P )

partxk~ei(P ) + cl(P ) Γilk(P )~ei(P ) =

dci(P )

dxk~ei(P )

One can show that the covariant derivatives dci(P )dxk are the components of a(

)-tensor field

the (1-form-)gradient of the vector field ~c which may be denoted by nabla˜~c On the other hand neitherthe partial derivative on the right hand side of Eq (C3b) nor the Christoffel symbols are tensors(65)This requirement will be made more precise in Sec C2(66)This implicitly relies on the fact that the vectors attached to every point P isinM all have the same dimension

C1 Covariant differentiation of tensor fields 167

The Christoffel symbols can be expressed in terms of the (local) metric tensor g(P ) whosecomponents are in agreement with relation (B6) given by(67)

gij(P ) = ~ei(P ) middot~ej(P ) (C4)

and of its partial derivatives Thus

Γilk(P ) =1

2gip(P )

[partgpl(P )

partxk+partgpk(P )

partxlminus partgkl(P )

partxp

with gip(P ) the components of the inverse metric tensor gminus1(P )This relation shows that Γilk(P ) is symmetric under the exchange of the lower indices l and l

ie Γikl(P ) = Γilk(P )

C12 Examples differentiation in Cartesian and in polar coordinates

To illustrate the results introduced in the previous Section we calculate the derivatives of vectorfields defined at each point of the real plane R2 which plays the role of the set M

Cartesian coordinates

As a first trivial example let us associate to each point P isin R2 local coordinates x1(P ) = xx2(P ) = y that coincide with the usual global Cartesian coordinates on the plane Let ~e1(P ) = ~ex~e2(P ) = ~ey denote the corresponding local basis vectorsmdashwhich actually happen to be the same atevery point P ie which represent constant vector fields

Either by writing down the vanishing derivatives part~ei(P )partxk ie using Eq (C1) or by invokingrelation (C5)mdashwhere the metric tensor is trivial g11 = g22 = 1 g12 = g21 = 0 everywheremdash onefinds that every Christoffel symbol vanishes This means [Eq (C3b)] that covariant and partialderivative coincide which is why one need not worry about ldquocovariant differentiationrdquo whenworking in Cartesian coordinates

Polar coordinates

It is thus more instructive to associate to each point P isin R2 with the exception of the originpolar coordinates x1prime = r equiv xr x2prime = θ equiv xθ The corresponding local basis vectors are

~er(r θ) = cos θ~ex + sin θ~ey

~eθ(r θ) = minusr sin θ~ex + r cos θ~ey(C6)

To recover the usual inner product on R2 the metric tensor g(P ) should have components

grr(r θ) = 1 gθθ(r θ) = r2 grθ(r θ) = gθr(r θ) = 0 (C7a)

That is the components of gminus1(P ) are

grr(r θ) = 1 gθθ(r θ) =1

r2 grθ(r θ) = gθr(r θ) = 0 (C7b)

Computing the derivatives

part~er(r θ)

partxr= ~0

part~er(r θ)

partxθ=

r~eθ(r θ)

part~eθ(r θ)

partxr=

r~eθ(r θ)

part~eθ(r θ)

partxθ= minusr~er(r θ)

and using Eq (C1) or relying on relation (C5) one finds the Christoffel symbols

Γrrr = Γθrr = 0 Γθrθ = Γθθr =1

r Γrθθ = minusr Γrrθ = Γrθr = 0 Γθθθ = 0 (C8)

where for the sake of brevity the (r θ)-dependence of the Christoffel symbols was dropped(67)Remember that the metric tensor g actually defines the inner product

168 Tensor calculus

Remarkslowast The metric tensor in polar coordinates (C7a) has signature (0 2)mdashie 0 negative and 2 positiveeigenvaluesmdash just like it has in Cartesian coordinates the signature of the metric (tensor) isindependent of the choice of coordinates if it defines the same inner product

lowast It is also interesting to note that the Christoffel symbols for polar coordinates (C8) are not allzero while this is the case for the Christoffel symbols in Cartesian coordinates This shows thatthe Christoffel symbols are not the components of a tensormdasha tensor which is identically zero in abasis remains zero in any basis

Consider now a constant vector field ~c(P ) = ~c(r θ) = ~ex Obviously it is unchanged when goingfrom any point (r θ) to any neighboring point ie a meaningful derivative along either the r or θdirection should identically vanishLet us write

~c(r θ) = ~ex = cos θ~er(r θ)minussin θ

r~eθ(r θ) = cr(r θ)~er(r θ) + cθ(r θ)~eθ(r θ)

The partial derivatives partcrpartxθ partcθpartxr and partcθpartxθ are clearly non-vanishing On the otherhand all covariant derivatives are identically zero omitting the variables one finds

dxr=partcr

partxr= 0

dxr=partcθ

partxr+ Γθθrc

θ =sin θ

(minus sin θ)

ie d~cdxr = ~0 anddcr

dxθ=partcr

partxθ+ Γrθθc

θ = minus sin θ minus r (minus sin θ)

dxθ=partcθ

partxθ+ Γθrθc

r = minuscos θ

rcos θ = 0

ie d~cdxθ = ~0 Thus the covariant derivatives give the expected result while the partial derivativeswith respect to the coordinates do not

C13 Covariant differentiation of general tensor fields

Scalar fields

scalar field f(P )

df(P )

dxk=partf(P )

partxk (C9)

One-forms

one-form field h˜(P ) = hj(P ) ε˜j(P )

dhj(P )

dxk=parthj(P )

partxkminus Γljk(P )hl(P ) (C10)

Tensors of arbitrary type(

)-tensor field TTT(P )

dTTTi1middotmiddotmiddotimj1jn(P )

dxk=partTTTi1middotmiddotmiddotimj1jn

partxk+ Γi1kl(P )TTTli2middotmiddotmiddotimj1jn

(P ) + middot middot middot+ Γimkl (P )TTTi1middotmiddotmiddotimminus1lj1jn

minusΓlj1k(P )TTTi1middotmiddotmiddotimlj2jn(P )minus middot middot middot minus Γljnk(P )TTTi1middotmiddotmiddotimj1jnminus1l

C14 Gradient divergence Laplacian

to be completed

C2 Beginning of elements of an introduction to differential geome-try

attempt (C12)

Bibliography for Appendix Cbull Nakahara [56] Chapter 51ndash53 amp 71ndash72

bull Postnikov [57](64) see eg Lectures 1 (beginning) 4ndash6 amp 18

APPENDIX D

Elements on holomorphic functions ofa complex variable

D1 Holomorphic functions

D11 Definitions

A function Z = f (z) is defined to be complex-differentiable at a point z0 in its domain ofdefinition if the limit

f prime(z0) equiv limzrarrz0

f (z)minus f (z0)

z minus z0(D1)

exists independently of the direction along which z approaches z0If f is complex-differentiable at every point of an open set U resp of a neighborhood of a point z0it is said to be holomorphic on U resp at z0

D12 Some properties

CauchyndashRiemann equations

Let P (x y) resp Q(x y) denote the real resp imaginary part of a function f (z = x+iy) of acomplex variable

f (x+iy) = P (x y) + iQ(x y) (D2)

Theorem f is holomorphic if and only if the CauchyndashRiemann equations

partP (x y)

partx=partQ(x y)

partyand

partP (x y)

party= minuspartQ(x y)

partx(D3)

relating the first partial derivatives of its real and imaginary parts are satisfiedEquivalently the relations (D3) can be recast as

= 0 (D4)

where z = xminus iy

Corollary A function f (z=x+iy) is holomorphic on a domain if and only if its real and imaginaryparts are conjugate harmonic functions ie they obey the CauchyndashRiemann equations (D3) andthe Laplace equations

4P (x y) = 0 4Q(x y) = 0 (D5)

on the domain

Integration of holomorphic functionsint

Cf (z) dz =

af(γ(t)

)γprime(t) dt (D6)

Cauchyrsquos integral theorem

f (z) dz = 0 (D7)

Cauchyrsquos integral formula

f(z0) =1

z minus z0dz (D8)

D2 Multivalued functions

D3 Series expansions

D31 Taylor series

f (z) =

infinsumn=0

f (n)(z0)

n(z minus z0)n (D9)

f (n)(z0) =n

(z minus z0)n+1dz (D10)

which generalizes the Cauchy integral formula (D8) to the successive derivatives of f zeroes

D32 Isolated singularities and Laurent series

Definitions

isolated singularityremovable singularitypole of order messential singularity

Laurent series

f (z) =

infinsumn=minusinfin

an(z minus z0)n (D11)

where C denotesaminus1 residue

D33 Singular points

172 Elements on holomorphic functions of a complex variable

D4 Conformal mapsFunction Z = f (z) defines mapping from plane of complex variables z = x+ iy to plane of complexZ Such a function is said to be a conformal map if it preserves angles locally

If a function Z = f (z) is holomorphic at z0 and such that f prime(z0) 6= 0 it is invertible in aneighborhood of z0 and f and its inverse F define a conformal mapping between the planes z andZ

Proof dZ = |f prime(z0)| ei arg f prime(z0) dzSingular point f prime(z0) = 0 if zero of n angles are multiplied by n+ 1 in transformation z rarr Z

φ(z) complex potential on z-plane Then Φ(Z) equiv φ(F (Z)) potential on Z-plane with velocityw(F (Z))F prime(Z)

Bibliography for Appendix Dbull Cartan [58] Chapters II III amp VI

bull Whittaker amp Watson [59] Chapters 46 51ndash52 amp 56ndash57

Bibliography

[1] T E Faber Fluid dynamics for physicists (University Press Cambridge 1995)

[2] E Guyon J-P Hulin L Petit C D Mitescu Physical hydrodynamics 2nd ed (UniversityPress Oxford 2015)

[3] L Landau E Lifshitz Course of theoretical physics Vol VI Fluid mechanics 2nd ed (Perg-amon Oxford 1987)

[4] L Landau E Lifschitz Lehrbuch der theoretischen Physik Band VI Hydrodynamik 5 ed(Harri Deutsch Frankfurt am Main 1991)

[5] A Sommerfeld Lectures on Theoretical Physics Vol II Mechanics of deformable bodies (As-sociated Press New York NY 1950)

[6] A Sommerfeld Vorlesungen uumlber theoretische Physik Band II Mechanik der deformierbarenMedien 6 ed (Harri Deutsch Frankfurt am Main 1992)

[7] M Reiner The Deborah Number Phys Today 17(1) (1964) 62

[8] R P Feynman R B Leighton M Sands The Feynman Lectures on Physics Volume II Mainly Electromagnetism and Matter definitive ed (Addison-Wesley Reading MA 2005)

[9] R P Feynman R B Leighton M Sands Feynman-Vorlesungen uumlber Physik Band 2 Elek-tromagnetismus und Struktur der Materie 5 ed (Oldenbourg Wissenschaftsverlag Muumlnchen2007)

[10] L Sedov A course in continuum mechanics Vol I Basic equations and analytical techniques(WoltersndashNoordhoff Groningen 1971)

[11] R E Graves B M Argrow Bulk viscosity Past to present J Thermophys Heat Tr 13(1999) 337ndash342

[12] R K Agarwal K-Y Yun R Balakrishnan Beyond NavierndashStokes Burnett equations forflows in the continuumndashtransition regime Phys Fluids 13 (2001) 3061ndash3085

[13] T Flieszligbach Lehrbuch zur theoretischen Physik I Mechanik 4 ed (Spektrum AkademischerVerlag Heidelberg amp Berlin 2003)

[14] B F Schutz Geometrical methods of mathematical physics (University Press Cambridge1980)

[15] V I Arnold Lectures on Partial Differential Equations (Springer Berlin Heidelberg NewYork 2004)

[16] G F Carrier C E Pearson Partial differential equations Theory and techniques 2nd ed(Academic Press New York 1988)

[17] E C Zachmanoglou D W Thoe Introduction to Partial Differential Equations with Appli-cations (Dover New York 1986)

174 Bibliography

[18] M Peyrard T Dauxois Physics of solitons (University Press Cambridge 2006)

[19] O Reynolds An experimental investigation of the circumstances which determine whether themotion of water shall be direct or sinuous and of the law of resistance in parallel channelsPhil Trans R Soc Lond 174 (1883) 935ndash982

[20] E Buckingham On physically similar systems Illustrations of the use of dimensional equa-tions Phys Rev 4 (1914) 345ndash376

[21] F Durst Fluid mechanics An introduction to the theory of fluid flows (Springer Berlin ampHeidelberg 2008)

[22] F Durst Grundlagen der Stroumlmungsmechanik (Springer Berlin amp Heidelberg 2006)

[23] T Misic M Najdanovic-Lukic L Nesic Dimensional analysis in physics and the Buckinghamtheorem Eur J Phys 31 (2010) 893ndash906

[24] E M Purcell Life at low Reynolds number Am J Phys 45 (1977) 3ndash11

[25] N Cohen J H Boyle Swimming at low Reynolds number a beginners guide to undulatorylocomotion Contemp Phys 51 (2010) 103ndash123 [arXiv09082769]

[26] A Einstein Uumlber die von der molekularkinetischen Theorie der Waumlrme geforderte Bewegungvon in ruhenden Fluumlssigkeiten suspendierten Teilchen Annalen Phys 17 (1905) 549ndash560

[27] J Perrin Discontinuous structure of matter Nobel Lecture (1926)

[28] F W J Olver et al NIST Handbook of mathematical functions (University Press Cambridge2010) [available online at httpdlmfnistgov]

[29] O Reynolds On the dynamical theory of incompressible viscous fluids and the determinationof the criterion Phil Trans R Soc Lond 186 (1894) 123ndash164

[30] T Abe K Niu Anomalous viscosity in turbulent plasma due to electromagnetic instabilityI J Phys Soc Jpn 49 (1980) 717ndash724

[32] G I Taylor Statistical theory of turbulence Proc R Soc Lond A 151 (1935) 421ndash464

[33] A N Kolmogorov The local structure of turbulence in incompressible viscous fluid for verylarge Reynolds numbers Dokl Akad Nauk SSSR 30 (1941) 299ndash303 [English translation inProc R Soc Lond A 434 (1991) 9ndash13]

[34] A N Kolmogorov Dissipation of energy in the locally isotropic turbulence Dokl Akad NaukSSSR 32 (1941) 19ndash21 [English translation in Proc R Soc Lond A 434 (1991) 15ndash17]

[35] U Frisch Turbulence The legacy of A N Kolmogorov (University Press Cambridge 1995)

[36] E A Spiegel The Theory of Turbulence Subrahmanyan Chandrasekharrsquos 1954 Lectures (LectNotes Phys 810 Springer Dordrecht 2011)

[37] J Maurer A Libchaber Rayleigh-Beacutenard experiment in liquid helium frequency locking andthe onset of turbulence J Phys (Paris) Lett 40 (1979) L-419ndashL-423

[38] A Libchaber J Maurer Une expeacuterience de RayleighndashBeacutenard de geacuteomeacutetrie reacuteduite mul-tiplication accrochage et deacutemultiplication de freacutequences J Phys (Paris) Colloq 41 (1980)C3-51ndashC3-56

[39] A Libchaber C Laroche S Fauve Period doubling cascade in mercury a quantitative mea-surement J Phys (Paris) Lett 43 (1982) L-211ndashL-216

[40] S Chandrasekhar Hydrodynamic and hydromagnetic stability (University Press Oxford1961)

[41] J Wesfreid Y Pomeau M Dubois C Normand P Bergeacute Critical effects in RayleighndashBeacutenardconvection J Phys (Paris) 39 (1978) 725ndash731

[42] M G Velarde C Normand Convection Sci Am 243No1 (1980) 93ndash108

[43] C Normand Y Pomeau M G Velarde Convective instability A physicistrsquos approach RevMod Phys 49 (1977) 581ndash624

[44] C Eckart The thermodynamics of irreversible processes 3 Relativistic theory of the simplefluid Phys Rev 58 (1940) 919ndash924

[45] D H Rischke Fluid dynamics for relativistic nuclear collisions Lect Notes Phys 516 (1999)21ndash70 [arXivnucl-th9809044]

[46] W A Hiscock L Lindblom Generic instabilities in first-order dissipative relativistic fluidtheories Phys Rev D 31 (1985) 725ndash733

[47] I Muumlller Zum Paradoxon der Waumlrmeleitungstheorie Z Phys 198 (1967) 329ndash344

[48] W Israel Nonstationary irreversible thermodynamics A causal relativistic theory Ann Phys(NY) 100 (1976) 310ndash331

[49] W Israel J M Stewart Transient relativistic thermodynamics and kinetic theory Ann Phys(NY) 118 (1979) 341ndash372

[50] N Andersson G L Comer Relativistic fluid dynamics Physics for many different scalesLiving Rev Rel 10 (2005) 1ndash87 [arXivgr-qc0605010]

[51] P Romatschke New developments in relativistic viscous hydrodynamics Int J Mod Phys E19 (2010) 1ndash53 [arXiv09023663 [hep-ph]]

[52] S Weinberg Gravitation and Cosmology (John Wiley amp Sons New York 1972)

[53] L Landau On the multiparticle production in high-energy collisions Izv Akad Nauk Ser Fiz17 (1953) 51ndash64 [English translation in Collected papers of LDLandau (Gordon and BreachNew York NY 1965) pp 569ndash585]

[54] S Belenkij L D Landau Hydrodynamic theory of multiple production of particles NuovoCim Suppl 3 S1 (1956) 15ndash31

[55] J D Bjorken Highly relativistic nucleus-nucleus collisions The central rapidity region PhysRev D 27 (1983) 140ndash151

[56] M Nakahara Geometry Topology and Physics 2nd ed (Institute of Physics Bristol 2003)

[57] M Postnikov Lectures in Geometry Semester II Linear Algebra and Differential Geometry(Mir Publishers Moscow 1982)

[58] H Cartan Theacuteorie eacuteleacutementaire des fonctions analytiques drsquoune ou plusieurs variables com-plexes 6th ed (Herrmann Paris 1985)

[59] E T Whittaker G N Watson A course of modern analysis 4th ed (University PressCambridge 1927)

176 Bibliography

Contents

Introduction

I Basic notions on continuous media

Continuous medium a model for many-body systems

Basic ideas and concepts

General mathematical framework

Local thermodynamic equilibrium

Lagrangian description

Lagrangian coordinates

Continuity assumptions

Velocity and acceleration of a material point

Eulerian description

Eulerian coordinates Velocity field

Equivalence between the Eulerian and Lagrangian viewpoints

Streamlines

Material derivative

Mechanical stress

Forces in a continuous medium

Fluids

Bibliography for Chapter I

II Kinematics of a continuous medium

Generic motion of a continuous medium

Local distribution of velocities in a continuous medium

Rotation rate tensor and vorticity vector

Strain rate tensor

Classification of fluid flows

Geometrical criteria

Kinematic criteria

Physical criteria


Deformations in a continuous medium

III Fundamental equations of non-relativistic fluid dynamics

Reynolds transport theorem

Closed system open system

Material derivative of an extensive quantity

Mass and particle number conservation continuity equation

Integral formulation

Local formulation

Momentum balance Euler and NavierndashStokes equations

Material derivative of momentum

Perfect fluid Euler equation

Newtonian fluid NavierndashStokes equation

Higher-order dissipative fluid dynamics

Energy conservation entropy balance

Energy and entropy conservation in perfect fluids

Energy conservation in Newtonian fluids

Entropy balance in Newtonian fluids

IV Non-relativistic flows of perfect fluids

Hydrostatics of a perfect fluid

Incompressible fluid

Fluid at thermal equilibrium

Isentropic fluid

Archimedes principle

Steady inviscid flows

Bernoulli equation

Applications of the Bernoulli equation

Vortex dynamics in perfect fluids

Circulation of the flow velocity Kelvins theorem

Vorticity transport equation in perfect fluids

Potential flows

Equations of motion in potential flows

Mathematical results on potential flows

Two-dimensional potential flows

V Waves in non-relativistic perfect fluids

Sound waves

Sound waves in a uniform fluid at rest

Sound waves on moving fluids

Riemann problem Rarefaction waves

Shock waves

Formation of a shock wave in a one-dimensional flow

Jump equations at a surface of discontinuity

Gravity waves

Linear sea surface waves

Solitary waves

VI Non-relativistic dissipative flows

Statics and steady laminar flows of a Newtonian fluid

Static Newtonian fluid

Plane Couette flow

Plane Poiseuille flow

HagenndashPoiseuille flow

Dynamical similarity

Reynolds number

Other dimensionless numbers

Flows at small Reynolds number

Physical relevance Equations of motion

Stokes flow past a sphere

Boundary layer

Flow in the vicinity of a wall set impulsively in motion

Modeling of the flow inside the boundary layer

Vortex dynamics in Newtonian fluids

Vorticity transport in Newtonian fluids

Diffusion of a rectilinear vortex

Absorption of sound waves

VII Turbulence in non-relativistic fluids

Generalities on turbulence in fluids

Phenomenology of turbulence

Reynolds decomposition of the fluid dynamical fields

Dynamics of the mean flow

Necessity of a statistical approach

Model of the turbulent viscosity

Turbulent viscosity

Mixing-length model

k-model

(k-epsilon)-model

Statistical description of turbulence

Dynamics of the turbulent motion

Characteristic length scales of turbulence

The Kolmogorov theory (K41) of isotropic turbulence

VIII Convective heat transfer

Equations of convective heat transfer

Basic equations of heat transfer

Boussinesq approximation

RayleighndashBeacutenard convection

Phenomenology of the RayleighndashBeacutenard convection

Toy model for the RayleighndashBeacutenard instability

IX Fundamental equations of relativistic fluid dynamics

Conservation laws


Energy-momentum conservation

Four-velocity of a fluid flow Local rest frame

Perfect relativistic fluid

Particle four-current and energy-momentum tensor of a perfect fluid

Entropy in a perfect fluid

Non-relativistic limit

Dissipative relativistic fluids

Dissipative currents

Local rest frames

General equations of motion

First order dissipative relativistic fluid dynamics

Second order dissipative relativistic fluid dynamics

Bibliography for Chapter IX


Microscopic formulation of the hydrodynamical fields

Particle number 4-current

Energy-momentum tensor

Relativistic kinematics

Equations of state for relativistic fluids

X Flows of relativistic fluids

Relativistic fluids at rest

One-dimensional relativistic flows

Landau flow

Bjorken flow

Appendices

A Basic elements of thermodynamics

B Tensors on a vector space

Vectors one-forms and tensors

Vectors

One-forms

Tensors

Metric tensor

Change of basis

C Tensor calculus

Covariant differentiation of tensor fields

Covariant differentiation of vector fields

Examples differentiation in Cartesian and in polar coordinates

Covariant differentiation of general tensor fields

Gradient divergence Laplacian

Beginning of elements of an introduction to differential geometry

D Elements on holomorphic functions of a complex variable

Holomorphic functions

Definitions

Some properties

Multivalued functions

Series expansions

Taylor series

Isolated singularities and Laurent series

Singular points

Conformal maps

Bibliography

Nicolas BorghiniUniversitaumlt Bielefeld Fakultaumlt fuumlr PhysikHomepage httpwwwphysikuni-bielefeldde~borghiniEmail borghini at physikuni-bielefeldde

Foreword

Contents

I33 Streamlines 10

I42 Fluids 14

VII23 k-model 118

B11 Vectors 160

B12 One-forms 160

B13 Tensors 161

D11 Definitions 170

Introduction

CHAPTER I

observation scale

measuredlocalq

uantity

at every time t

L sim=

|G(t~r)|

]minus1

Kn equiv`mfp

L 1 (I4)

~r = ~r(t ~R) (I5a)

G = G(t ~R) (I6)

~R = ~R(t~r) (I7)

partt (I8)

partt (I9)

~vt(t~r) = ~v(t ~R(t~r)

) (I13a)

) (I13b)

part~r(t ~R)

partt=~vt

(t~r(t ~R)

)~r(t0 ~R) = ~R

(I14a)

) (I14c)

I33 Streamlines

d~x(λ)

dλ= α(λ)~v

(t ~x(λ)

)(I15a)

(xi)Stromlinien

d~xtimes~v(t ~x(λ)

)= ~0 (I15b)

dx1(λ)

v1(t ~x(λ)

) =dx2(λ)

v2(t ~x(λ)

) =dx3(λ)

v3(t ~x(λ)

) (I15c)

v1(t~r)=

v2(t~r)=

v3(t~r)

d~v part~v(t~r)

parttdt+

part~v(t~r)

partx1dx1 +

part~v(t~r)

partx2dx2 +

part~v(t~r)

partx3dx3

partx1+ dx2 part

partx2+ dx3 part

partx3

(xii)Stromroumlhre

d~v part~v(t~r)

parttdt+

(d~r middot ~nabla

)~v(t~r) (I16)

~a(t) =part~v(t~r)

]~v(t~r) (I17)

Dtequiv part

~a(t) =D~v(t~r)

Dt (I19)

Remarks

d2S ~en

d2 ~Fs

Figure I2

~Ts equivd2 ~Fsd2S

T is =

3sumj=1

σσσij ejn (I21b)

tensor

I42 Fluids

Remarks

CHAPTER II

δ~(t)

δ~(t+ δt)

~v(t~r + δ~(t)

~v(t~r)

resp~v(t~r + δ~(t)

) That is one finds

δ~(t+ δt) = δ~(t) +[~v(t~r + δ~(t)

)minus~v(t~r)]δt+O

(δt2) (II1)

δvi 3sumj=1

partvi(t~r)

partxjδxj (II2a)

Introducing the(

DDD(t~r) equiv 1

]T) (II3b)

DDDij(t~r) =1

[partvi(t~r)

partxj+partvj(t~r)

partxi

] RRRij(t~r) =

[partvi(t~r)

partxi

] (II3c)

DDDij(t~r) =

[partvi(t~r)

partxj+partvj(t~r)

partxi

DDDij(t~r) =

2δikδlj

[partvk(t~r)

partxl+partvl(t~r)

partxk

2gik(t~r)glj(t~r)

[partvk(t~r)

partxl+partvl(t~r)

partxk

DDDij(t~r) =

2gik(t~r)glj(t~r)

[dvk(t~r)

dvl(t~r)

](II4a)

RRRij(t~r) =1

2gik(t~r)glj(t~r)

[dvk(t~r)

dxlminus dvl(t~r)

](II4b)

~v(t~r + δ~r

)=~v(t~r)

)(II5)

]δt+O

(δt2)

)middot

3sumjk=1

(xxiv)Wirbeltensor

equivalently reads

~Ω(t~r) =1

)middot

ω1(t~r)=

ω2(t~r)=

ω3(t~r) (II11b)

II13 a

Diagonal components

δ`i(t+ δt) = δ`i(t) +[vi(t~r + δ~(t)

)minus vi

(t~r)]δt+O

(δt2)

partvi(t~r)

partxjδ`j(t) δt

i(t~r) δ`i(t) δt

direction i

δL3(t)

=[DDD1

3(t~r)]δt

DDD11(t~r) +DDD2

2(t~r) +DDD33(t~r) =

partv1(t~r)

partx1+partv2(t~r)

partx2+partv3(t~r)

II13 b

v1 δt

(v1+δv1)δt

v2 δt

(v2+δv2)δtδα1

δv1 =partv1(t~r)

partx2δ`2 δv2 =

partv2(t~r)

partx1d`1

δα1 δv2 δt

δ`1and δα2 minus

δv1 δt

δα1 partv2(t~r)

partv1(t~r)

partx2δt

δt partv2(t~r)

]111+SSS(t~r) (II15a)

]T minus 2

) (II15b)

DDDij(t~r) =1

with [cf Eq (II4a)]

SSSij(t~r) equiv1

[gki(t~r)g

lj(t~r)

(dvk(t~r)

dvl(t~r)

)minus 2

]gij(t~r)

] (II15d)

Summary

bull a translation

Ma equiv v

cs(II16)

CHAPTER III

boundary of Σ

at time t+ δt

streamlines

(t~r) =dG(t~r)

dM(t~r) (III1)

G(t) =

(t~r) dM =

δG =(G1 + G2+

= δG1 + δG2

δG1 equiv(G1

)t+δtminus(G1

)t δG2 equiv

)t+δtminus(G2minus

δG1 =dG1(t)

dtδt =

[ intV

(t~r) ρ(t~r) d3~r

]δt =

-|~v| δt

mρd3V with

δG2 =

(t~r) ρ(t~r) d3~r]

Remarks

Dt= 0 (III6)

m(t~r) = 1 reads

[ intVρ(t~r) d3~r

[ρ(t~r)~v(t~r)

[ intV

n(t~r) d3~r

[n(t~r)~v(t~r)

partρ(t~r)

[ρ(t~r)~v(t~r)

]d3~r = 0

resp intV

partn(t~r)

[n(t~r)~v(t~r)

]d3~r = 0

partρ(t~r)

[ρ(t~r)~v(t~r)

]= 0 (III9)

resppartn(t~r)

[n(t~r)~v(t~r)

]= 0 (III10)

Remarks

]= 0 ie

D~P (t)

Dt= ~F (t) (III11)

D~P (t)

minus~v(t~r)

partρ(t~r)

partt+ ρ(t~r)

]~v(t~r)

d3~r (III13)

D~P (t)

intVρ(t~r)

part~v(t~r)

]~v(t~r)

d3~r =

intVρ(t~r)

D~v(t~r)

Dtd3~r (III14)

J i(t) =

]middot d2~S =

]= minusvi(t~r)

partρ(t~r)

partt+ ρ(t~r)

]vi(t~r)

III32 a

~F (t) =

ie the(

III32 b

Euler equation

part~v(t~r)

]~v(t~r)

d3~r =

ρ(t~r)

part~v(t~r)

]~v(t~r)

Remarks

III32 c

Boundary conditions

III32 d

dM(t~r)=

~fV (t~r)

ρ(t~r)

D~v(t~r)

Dt= minus 1

)= ~nabla

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

III32 e

Remarks

(III23)

[ρ(t~r) vi(t~r)

3sumj=1

dTTTij(t~r)

part(ρvi)

partt+

3sumj=1

dTTTij

dxj=partρ

parttvi + ρ

partvi

partt+

3sumj=1

gijdPdxj

3sumj=1

vid(ρvj)

3sumj=1

ρvjdvi

= vi[partρ

[partvi

[ρ(t~r)~v(t~r)

III33 a

dxj dxk

minus η(t~r)

[dvi(t~r)

dvj(t~r)

dximinus 2

](III26a)

[DDD(t~r)minus 1

]gminus1(t~r)

]minusζ(t~r)

]gminus1(t~r)

[DDDij(t~r)minus 1

]gij(t~r)

]minus ζ(t~r)

]gij(t~r) (III26d)

]gij(t~r) (III26f)

Remarks

III33 b

σij(t~r) =

minusP (t~r)+

[ζ(t~r)minus 2

3η(t~r)

gij(t~r)+η(t~r)

[dvi(t~r)

dvj(t~r)

] (III27c)

3sumij=1

[minusP (t~r) +

(ζ(t~r)minus 2

3η(t~r)

]gij(t~r)

+ η(t~r)

(dvi(t~r)

dvj(t~r)

= minusint

~nablaP (t~r) d3V +

= minusint

~nablaP (t~r) d3V +

~fvisc(t~r) =3sum

η(t~r)

[dvi(t~r)

dvj(t~r)

]~ej(t~r)

3η(t~r)

(III29b)

III33 c

ρ(t~r)

part~v(t~r)

]~v(t~r)

or component-wise

ρ(t~r)

partvi(t~r)

]vi(t~r)

=minusdP (t~r)

[ζ(t~r)minus 2

3η(t~r)

3sumj=1

η(t~r)

[dvi(t~r)

dvj(t~r)

]+[~fV (t~r)

]i(III30b)

ρ(t~r)

part~v(t~r)

]~v(t~r)

]+ ~fV (t~r)

part~v(t~r)

]~v(t~r) = minus1

III33 d

Boundary conditions

]+ ~nabla middot

]~v(t~r)

(III33)

Remarks

parts(t~r)

[s(t~r)~v(t~r)

]= 0 (III34)

sumj π

ij vj to the

2ρ(t~r)~v(t~r)2 + e(t~r)

]+ ~nabla middot

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r)

]~v(t~r)

minus η(t~r)

)~v(t~r) + ~nabla

(~v(t~r)2

(III35)

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r)

2ρ~v2

)+ ~nabla middot

2ρ~v2

ρ~v middot part~v

partt+

partρ

partt~v2 +

3sumi=1

[partvi

)vi] (III37a)

]= T~nablamiddot

)= T~nablamiddot

partt= T

partt+ micro

partnpartt

= Tparts

(n~v) (III37c)

3sumi=1

[partvi

+ Tparts

3sumij=1

partxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot~v

3sumi=1

partxi

[ζ(~nabla middot~v

) (III37d)

[partvi

+ vipartPpartxi

=3sumj=1

vipart

partxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot~v

)]+ vi

Tparts

= η3sum

partvipartxj

(partvi

partxj+partvj

partximinus 2

)2+ ~nabla middot

(κ~nablaT

) (III38)

3sumij=1

(partvi

partxj+partvj

partximinus2

3gij~nablamiddot~v

)(partvipartxj

+partvjpartximinus2

3gij~nablamiddot~v

3sumij=1

(partvi

partxj+partvj

partximinus2

3gij~nablamiddot~v

)partvjpartxi

(III39a)

)= T~nabla middot

(κ~nablaTT

(~nablaT)2 (III39b)

(κ~nablaTT

3sumij=1

(partvi

partxj+partvj

partximinus 2

)(partvipartxj

3gij~nabla middot~v

(~nabla middot~v

)2+ κ

(~nablaT)2

T 2 (III40a)

parts(t~r)

~nablaT (t~r)

T (t~r)

]2+ κ(t~r)

[~nablaT (t~r)

]2T (t~r)

(III40b)

One thus findsdS

CHAPTER IV

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= minusρg

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= minus mg

kBTP (~r)

P (~r) = P (z) = P 0 exp

(minusmgzkBT

)= T d

ρdP (IV4)

~nabla[w(~r)

ρ(~r)+ Φ(~r)

]= ~0 (IV5)

that isw(z)

enthalpy density

w = e+ P =5

2nkBT + nkBT =

That isw

dT (z)

dz= minus mg7

[w(~r)

ρ(~r)

[s(~r)

ρ(~r)

] (IV6)

fluid 1

fluid 2 S

fluid 1

fluid 2 solid body

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

[s(t~r)

ρ(t~r)

]minus ~nabla

[w(t~r)

ρ(t~r)

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

2+w(t~r)

ρ(t~r)+ Φ(t~r)

[s(t~r)

ρ(t~r)

] (IV9)

~v(~r)2

2+w(~r)

~nabla middot[(

ρ+ Φ

P (~r)

IV22 a

Figure IV2

2+ gzA =

2+ gzB

v2B = v2

A + 2gh

vB =radic

IV22 b

Venturi effect

s-v1 -v2

Figure IV3

P 1minusP 2

s2minus 1

IV22 c

Pitot tube

Obull bullIbull

OprimebullA

-manometer

-bullAprime

PO + ρ~v2

2= PAprime + ρ

~v2Aprime

PI minus PB = ρ~v2

Remarks

IV22 d

Magnus effect

~v0~ωC

Γ~γ(t) =

intS~γ

[~nablatimes~v(t~r)

]middot d2~S =

DΓ~γ(t)

Dt= 0 (IV14b)

DΓ~γDt

part~γ(t λ)

[part2~γ

part~γ

partλmiddot(part~v

partt+sumi

part~v

partxipartγi

DΓ~γDt

part~v

partλmiddot ~v +

part~γ

partλmiddot[part~v

DΓ~γDt

DΓ~γ(t)

Dt= minus

∮~γ

~nablaP (t~r)

~nablatimes(~nablaP

ρ2 (IV16)

part~ω(t~r)

]= minus

ρ(t~r)2 (IV20)

(~nabla middot ~ω

)~v minus

(~v middot ~nabla

)~ω minus

(~nabla middot~v

part~ω(t~r)

]~ω(t~r)minus

]~v(t~r) = minus

]~ω(t~r)minus

ρ(t~r)2

D~v ~ω(t~r)

]~ω(t~r)minus

]~v(t~r) (IV22a)

Dtequiv D~ω(t~r)

]~v(t~r) (IV22b)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r)minus

ρ(t~r)2 (IV23)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r) (IV24)

[~ω(t~r)

ρ(t~r)

]= ~0 (IV25)

δ~(t)δ~(t+ δt)

~v(t~r

+ δ~ (t)) δt

~v(t~r)δt

[δ~(t)middot~nabla

]~v(t~r)δt

D~vδ~(t)Dt

= ~0 (IV26)

]~ω(~r)minus

]~v(~r)

]ρ(~r)

Remarks

partt+ ~nabla

[~nablaϕ(t~r)

+ Φ(t~r)

= minus 1

partt+ ~nabla

[~nablaϕ(t~r)

+P (t~r)

ρ+ Φ(t~r)

= ~0 (IV29)

or equivalently

minuspartϕ(t~r)

partt+

[~nablaϕ(t~r)

+P (t~r)

ρ+ Φ(t~r) = C(t) (IV30)

Laplace equation(t)

4ϕ(t~r) = 0 (IV31)

K(t) (IV32b)

Remarks

IV42 a

(IV33)

IV42 b

bull~r0

bull~r1

bull~r2

-~γ0rarr1

-~γ prime

0rarr1

6~γ0rarr2

6~γ primeprime

0rarr2

6~γ prime

0rarr2

Figure IV7

(1minus λ)

(IV35)

Remarks

IV43 a

party vy(x y) =

partψ(x y)

partx(IV36)

(~x(λ)

party= 0

4ψ(x y) = 0 (IV37a)

(lii)Stromfunktion

partφ(x y)

partx=partψ(x y)

[= minusvx(x y)

partφ(x y)

[= minusvy(x y)

] (IV38)

IV43 b

3333333

33333 33

33333333

333333

Figure IV8

~v(x y) =(

φ(z) = minus Q2π

w(z) =Q

~v(r θ) =Q

2πr~er (IV42c)

θ (IV42d)

φ(z) =iΓ

~v(r θ) =Γ

2πr2~eθ (IV43b)

ϕ(r θ) = minus Γ

2πθ ψ(r θ) =

2πlog r (IV43c)

micro eiα

w(z) =micro eiα

~v(r θ) =micro

ϕ(x y) =microx

x2 + y2 (IV44d)

φ(z) = limεrarr0

)minus log

)] (IV45)

w(z) =

infinsump=0

(IV46a)

infinsump=1

p aminuspminus1

2 z0 isin C (IV47a)

35 and 1

IV43 c

φ(z) = minusvinfin

) (IV48a)

(r minus R2

)sin θ (IV48b)

w(z) = vinfin

(1minus R2

) (IV49a)

~v(r θ) = vinfin

[(1minus R2

)cos θ~er minus

)sin θ

] (IV49b)

P (~r) +1

2ρv2infin

(liii)Staupunkte

]= Pinfin +

2πlog

R (IV50a)

)cos θ minus Γ

(r minus R2

)sin θ +

2πlog

R (IV50b)

w(z) = vinfin

(1minus R2

)minus iΓ

2πz (IV51a)

~v(r θ) = vinfin

[(1minus R2

)cos θ~er minus

r2minus Γ

2πrvinfin

)sin θ

] (IV51b)

IV43 d

Z = f (z) = z +R2

z(IV52)

)cosϑ+ i

(Rminus

CHAPTER V

ρ(t~r) = ρ0(t~r) + δρ(t~r) (V1a)

P (t~r) = P 0(t~r) + δP (t~r) (V1b)

~v(t~r) =~v0(t~r) + δ~v(t~r) (V1c)

V1 Sound waves 71

ρ(t~r) = ρ0 + δρ(t~r) (V2a)

P (t~r) = P 0 + δP (t~r) (V2b)

~v(t~r) = ~0 + δ~v(t~r) (V2c)

|δρ(t~r)| ρ0 |δP (t~r)| P 0 (V2d)

partδρ(t~r)

partt+ ρ0

]= 0 (V3a)

[ρ0 + δρ(t~r)

]partδ~v(t~r)

]δ~v(t~r)

partδρ(t~r)

partt+ ρ0

ρ0partδ~v(t~r)

)= P 0 +

(partPpartρ

(partPpartS

(partPpartN

(partPpartρ

c2s equiv

(partPpartρ

minusω ρ0

c2s~k 2 minusωρ0

)(δρ

~k middot δ~v

(partPpartρ

= minus V 2

(partPpartV

= minus V 2

[minusNkBT

V 2+NkB

(partT

V1 Sound waves 73

P = minus(partU

= minus5

(partT

ie NkB

(partT

= minus2P5

= minus2

leading to c2s =7

∣∣~k∣∣∣∣δ~v∣∣ = c2s

(~k middot δ~v

equation(25)

part2ρ(t~r)

partt2minus c2

s4ρ(t~r) = 0 (V9a)

part2ϕ(t~r)

partt2minus c2

s4ϕ(t~r) = 0

(lv)Verduumlnnung

V2 Shock waves 75

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (V10a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

]+partδP (t x)

partx= 0 (V10b)

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (V11a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

partx= 0 (V11b)

partv(t x)

partt=

dv(ρ)

partρ(t x)

parttresp

partv(t x)

partx=

dv(ρ)

partρ(t x)

[dv(ρ)

minus cs(ρ)2

partρ(t x)

partx= 0

dv(ρ)

dρ= plusmncs(ρ)

ρ (V12)

v(ρ) =

int ρ

cs(ρprime)

ρprimedρprime

partρ(t x)

partt+[v(ρ(t x)

(ρ(t x)

)]partρ(t x)

partx= 0 (V13)

t4 gt t3

t3 gt t2

t2 gt t1

t1 gt t0

V2 Shock waves 77

+minus gminus (V14)

where g+

ρ v1]]

= 0 (V15a)[[TTTi1]]

= 0 foralli = 1 2 3 (V15b)[[(1

2ρ~v2 + e+ P

]]= 0 (V15c)

[[v2]]

= 0 resp[[

ρ+minus 1

ρminus

) (V18)

]2=j21

=Pminusminus P +

ρminusminus ρ+

ρminusρ+

=Pminusminus P +

ρminusminus ρ+

ρminusρ+

Pminusminus P +

ρminusminus ρ+

ρminus

]2 (partPpartρ

ρminusρ+

=ρminusρ+c2s

[(v1)minus

]2 ρ+

ρminusc2s

2(v1)2 +

e+ Pρ

]]=j21

minus 1

ρ2minus

)+e+ + P +

ρminus= 0

Pminus minus P +

ρminus

ρ+(V19a)

Pminus + P +

ρ+minus 1

ρminus

V3 Gravity waves 79

minus partϕ(t x z)

partt+

[~nablaϕ(t x z)

+P (t x z)

partx2+part2

partz2

]ϕ(t x z) = 0 (V20b)

∣∣∣∣z=0

= 0 (V21a)

(lxi)Schwerewellen

minuspartϕ(t x z)

=Dδh(t x)

UsingD

Dt=part

partt+ vx

partx=part

parttminus partϕ

partx this gives[

partϕ(t x z)

partz+partδh(t x)

partϕ(t x z)

]z=h0+δh(tx)

= 0 (V21b)

P(t x z=h0+δh(t x)

)= P 0 (V21c)

partt+

[~nablaϕ(t x z)

]z=h0+δh(tx)

d2f(z)

dz2minus k2f(z) = 0

V3 Gravity waves 81

Linear waves

minus partϕ(t x z)

partt+

P (t x z)

ρ+ gz =

ρ+ gh0 (V24)

partϕ(t x z)

∣∣∣∣z=h0

+partδh(t x)

partt= 0 (V25a)

∣∣∣∣z=h0

partt2+ g

partϕ(t x z)

cϕ =ω

radicg

k and cg =

dω(k)

radicg

cϕ = cg =radicgh0

δh(t x) =1

partϕ(t x z)

∣∣∣∣z=h0

vx(t x z) =kg

cosh(kz)

sinh(kz)

x(t) = x0 +kgδh0

cosh(kz)

δh0 cosh(kz)

z(t) = z0 +kgδh0

sinh(kz)

δh0 sinh(kz)

[x(t)minus x0]2

cosh2(kz)+

[z(t)minus z0]2

sinh2(kz)=

[kgδh0

ω2 cosh(kh0)

sinh(kh0)

V3 Gravity waves 83

partt= P 0 + ρg

[h0 minus z + δh0

cosh(kz)

V32 Solitary waves

]~v(t~r) = minus1

vz(t x z=0) = 0 (V27c)

)=partδh(t x)

partt+ vx(t~r)

partδh(t x)

partx (V27d)

P(t x z=h0+δh(t x)

)= P 0

(t x z=h0+δh(t x)

tlowast equiv t

tc xlowast equiv x

Lc zlowast equiv z

δhc vlowastx equiv

vxδhctc

vlowastz equivvz

ρ δhcLct2c

= 0 (V28a)

partxlowast (V28b)

partzlowast (V28c)

partδhlowast

P lowast =gt2cLc

Fr equivradicLcg

P lowast =1

Velocity potential

partvxpartt

(vxpartvxpartx

+ vzpartvzpartx

partδh

partx= 0 (V29)

V3 Gravity waves 85

part2ϕ

partx2+part2ϕ

partz2= 0 (V30)

znϕn(t x) (V31)

infinsumn=0

zn[part2ϕn(t x)

partx2+ (n+ 1)(n+ 2)ϕn+2(t x)

(n+ 1)(n+ 2)

part2ϕn(t x)

part2ϕ0(t x)

partx2+z4

part4ϕ0(t x)

partx4+

partx+z2

part3ϕ0(t x)

partx3minus z4

part5ϕ0(t x)

partx5+

partx2minus z3

part4ϕ0(t x)

partx4+

part2u(t x)

partx+z3

part3u(t x)

partx (V34a)

partt= δ

partφ(t x)

partt (V34b)

partvx(t x)

partt+

partφ(t x)

partx= 0 (V34c)

part2u(t x)

partt2minus δ

part2u(t x)

partx2= 0 (V35)

radicgh0(Lctc)

partvx(t x z)

partt+ ε

[vx(t x z)

partvx(t x z)

partx+ vz(t x z)

partvz(t x z)

]+partφ(t x)

partx= 0 (V36)

instead of Eq (V29)

)= u(t x)minus δ2

part2u(t x)

partx2 (V37a)

)= minusδ

[1 + εφ(t x)

]partu(t x)

partx+δ3

part3u(t x)

partx3 (V37b)

partu(t x)

parttminus δ2

part3u(t x)

partu(t x)

partx+partφ(t x)

partx= 0 (V38)

partφ(t x)

partt+ δεvx

(t x z=δ(1 + εφ)

)partφ(t x)

partφ(t x)

partt+ εu(t x)

partφ(t x)

]partu(t x)

partxminus δ2

part3u(t x)

partx3= 0 (V39)

partu(t x)

partt+partφ(t x)

partx= 0

partφ(t x)

partt+partu(t x)

partx= 0

partu(t x)

partt+partu(t x)

partx= 0 (V40)

V3 Gravity waves 87

partφ

partt+partφ

partx+ ε

partu(ε)

partx+ δ2partu(δ)

partx+ 2εφ

partφ

partxminus δ2

part3φ

partx3= 0

partφ

partt+partφ

partx+ ε

partu(ε)

partt+ δ2partu(δ)

partt+ εφ

partφ

partxminus δ2

part3φ

partx2 partt= 0

[partu(ε)(t x)

partx+

2φ(t x)

partφ(t x)

]+ δ2

[partu(δ)(t x)

partxminus 1

part3φ(t x)

partx3

u(ε)(t x) = minus1

4φ(t x) + C(ε)(t) u(δ)(t x) =

part2φ(t x)

partx2+ C(δ)(t)

partφ(t x)

partt+partφ(t x)

partx+

2εφ(t x)

partφ(t x)

partx+

6δ2 part

3φ(t x)

partx3= 0 (V42)

partφ(τ ξ)

partτ+

2εφ(τ ξ)

partφ(τ ξ)

partξ+

6δ2 part

3φ(τ ξ)

partξ3= 0 (V43)

partφ(τ ξ)

partτ+ 6φ(τ ξ)

partφ(τ ξ)

partξ3= 0 (V44)

φ(τ ξ) =φ0

] (V45a)

δh(t x) =δhmax

radic3δhmax

[xminusradicgh0

δhmax

] (V45b)

δhmax=1 t = t0

δhmax= 025 t = t0

δhmax= 025 t = t1

δh(t x)

CHAPTER VI

partρ(t~r)

partt= 0 (VI1a)

parte(t~r)

] (VI1c)

]~v(~r) = minus1

partv(y)

partx= 0 (VI3a)

v(y)partv(y)

partx~ex = minus1

ρ~nablaP (~r) + ν

d2v(y)

dy2~ex (VI3b)

partP (~r)

partx= η

d2v(y)

dy2 (VI4)

d2v(y)

dy2= 0

v(y=0) = 0 v(y=h) = |~u|

~v(~r) =y

h~u for 0 le y le h

[partvi(~r)

partxj+partvj(~r)

partxi

0 0 minusPinfin

d2 ~Fs(~r)

d2S= ~Ts(~r) =

[ 3sumij=1

3sumij=1

)~ei =

η |~u|h

minusPinfin0

6yP 1 P 2

v(y) = minus 1

∆PLy2 + γy + δ

v(y=0) = 0 v(y=h) = 0

v(y) =1

[y(hminus y)

P 1 P 2-z

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= η

[part2v(r)

partx2+part2v(r))

party2

[d2v(r)

partP (~r)

partz= minus∆P

dr= minus∆P

ηL (VI7)

dχ(r)

dr+χ(r)

v(r) =∆P4ηL

(a2 minus r2

)for r le a (VI8)

∆P4ηL

(a2r minus r3

)dr = 2πρ

∆P4ηL

4=πρa4

∆PL (VI9)

〈v〉 equiv 1

0v(r) 2πr dr =

2v(r=0)

~rlowast equiv ~r

Lc ~vlowast equiv

vc tlowast equiv t

(VI10)

Re equiv ρvcLcη

=vcLcν

(VI12)

~v(t~r) = vc~f1

) P (t~r) = P 0 + ρv2

Fr equiv vcradicgLc

(VI14)

[minus 1

(~Ω0times~r

Ek equiv η

ρΩL2c

(VI15)

Ro equiv vcΩLc

(VI16)

Ro = Re middot Ek

VI31 a

Stokes equation

(~v middot ~nabla

[~nablatimes ~c(~r)

]= ~nabla

4P (~r) = 0 (VI20)

VI31 b

) This also holds

~vinfin ~er~eϕ

4[~nablatimes~u(~r)

]= ~0 (VI22a)

]= minus~nabla

[4f(r)

]times~vinfin

])times~vinfin = ~0

])only has a com-

(~nabla[4f(r)

])= ~nabla

(4[4f(r)]

) so that the

4[4f(r)] = 0

4 =part2

partr2+

partrminus `(`+ 1)

4f(r) =d2f(r)

r(VI24a)

2r (VI24b)

)middot~vinfin

]minus C

r~vinfin = minusB

rminus C

r~vinfin

(~er middot~vinfin

r3minus C

R3minus C

)~vinfin +

R3minus C

)(~er middot~vinfin

)~er = ~0

[~vinfin +

(~er middot~vinfin

)~er]minus R3

[~vinfin minus 3

(~er middot~vinfin

)~er] (VI25)

P (~r) =3

~er middot~vinfinr2

+ const

~v(~r) =~vinfin +R3

[~vinfin minus 3

(~er middot~vinfin

D = microkBT =kBT

6πRη

partv(t y)

partt= ν

part2v(t y)

party2 (VI27a)

limyrarrinfin

v(t y) = uf1

radicνt) ie write

v(t y) = uf2

2radicνt

v(t y) = uf

2radicνt

)(VI28)

f(ξ) = 0 (VI29b)

int ξ

0eminusυ

2dυ (VI31)

int infinξ

eminusυ2dυ (VI32)

v(t y) = u

[1minus erf

2radicνt

)] (VI33)

radicνω

partvxpartx

+partvyparty

= 0 (VI35a)

vxpart

partx+ vy

)vx = minus1

partPpartx

(part2

partx2+part2

party2

)vx (VI35b)(

vxpart

partx+ vy

)vy = minus1

partPparty

(part2

partx2+part2

party2

)vy (VI35c)

xlowast equiv x

Lc ylowast equiv y

δl(VI36)

δl Lc (VI37)

vlowasty equivvyu

u vinfin (VI39)

Re equiv Lcvinfinν

(VI40)

+Lcδl

vinfin

= 0 (VI41a)

+Lcδl

= minuspartP lowast

partxlowast+

) (VI41b)

+Lcδl

v2infin

= minusLcδ

partP lowast

partylowast+

vinfin

) (VI41c)

vinfin= 1 (VI42)

Re= 1 (VI43)

δl =LcradicRe

= 0 (VI45a)

= minuspartP lowast

(VI45b)

partP lowast

(~v middot ~nabla

)~v = 1

2~nabla(~v2)

part~ω(t~r)

]= minus

ρ(t~r)2+ ν4~ω(t~r) (VI46)

D~ω(t~r)

]~v(t~r)minus

]~ω(t~r)minus

ρ(t~r)2+ ν4~ω(t~r) (VI47a)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r)minus

ρ(t~r)2+ ν4~ω(t~r) (VI47b)

~ω(t=0~r) =Γ0

2πrδ(z)~ez (VI48)

(~ω middot ~nabla

partωz(t r)

[part2ωz(t r)

partr2+

partωz(t r)

] (VI50)

ωz(t r) =Γ0

νt(VI51)

f(ξ) = C eminusξ4

that isωz(t r) =

νtC eminusr

2(4νt) (VI53)

Γ(t R) =

~v(t~r) middot d~=

int 2π

2(4νt)]

ωz(t r) =Γ0

4πνteminusr

2(4νt) (VI55)

radicν

Γ(t R) = Γ0

[1minus eminusR

2(4νt)] (VI56)

~v(t~r) =Γ0

[1minus eminusr

2(4νt)]~eθr (VI57)

where |~eθ| = r

∣∣δ~v(t~r)∣∣ cs (VI58b)

c2s equiv

(partPpartρ

(VI58c)

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (VI59a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

partx+

3η + ζ

)part2δv(t x)

partx2 (VI59b)

partδρ(t x)

partt+ ρ0

partδv(t x)

partx= 0 (VI60a)

ρ0partδv(t x)

partx+

3η + ζ

)part2δv(t x)

partx2 (VI60b)

ν equiv 1

3η + ζ

) (VI61)

ρ0partδv(t x)

partt+ c2

partδρ(t x)

partx= ρ0ν

part2δv(t x)

partx2 (VI62)

part2δv(t x)

partt2minus c2

part2δv(t x)

partx2= ν

part3δv(t x)

part2δρ(t x)

partt2minus c2

part2δρ(t x)

partx2= ν

part3δρ(t x)

ω2 = c2sk

ω2r minus ω2

i + 2iωrωi = c2sk

ωi = minus1

2νk2 (VI66)

ω2r = c2

sk2 minus 1

4ν2k4 (VI67)

c2ϕ = c2

s minus1

4ν2k2 (VI68)

Remarks

s one finds

ω2 c2sk

4ω2τ2

)hArr cϕ equiv

8ω2τ2

ki νω2

(VI69)

((ωτν)2

CHAPTER VII

VII11 a

Q = minusπρa4

∆PL (VI9)

Ediss = minus1

∆PL〈v〉 =

8ν〈v〉2

a2(VII1)

〈v〉 =Q

πa2ρ= minusa

VII11 b

reads [29]

Nsumn=1

~v(n)(t~r) asymp 1

int t+T 2

(part~v(t~r)

]~v(t~r)

VII13 a

[part~v

)~vprime (VII9a)

partvi

)vi = minus1

dPdximinus

3sumj=1

dvprimeivprimej

dxj+ ν4vi (VII9b)

Dtequiv part

(ρ~v)

SSSij equiv 1

dximinus 2

) (VII12c)

(ρ~v2

[P~v +

(TTTR minus 2ηSSS

)middot~v]

+(TTTR minus 2ηSSS

) SSS (VII13)

ρP~v +

)middot~v]

) SSS (VII14a)

or component-wise

Dt= minus

3sumj=1

ρP vj +

3sumi=1

3sumij=1

)SSSij (VII14b)

condition (VII7)

VII13 b

3sumij=1

lang3sum

(VII15)

ρP~v +

)middot~v]

) SSS (VII16a)

or component-wise

Dt= minus

3sumj=1

ρP vj +

3sumi=1

(vprimeivprimej

3sumij=1

Ediss =

lang3sum

vprimeivprimejSSSij

rangsim v3

Lc (VII17)

part~vprime(t~r)

DTTTijRDt

dTTTijRdxk

)minus(TTTikR

dxk+TTTjkR

)minus2η

dvprimei

dxkdvprimej

part~v(t~r)

P (t~r)

[~vprime(t~r)]2

3sumij=1

SSSijSSSij

νeff =

minussumij

vprimeivprimejSSSij

2sumij

SSSijSSSij

2νsumij

SSSijSSSij

2sumij

SSSijSSSij= ν

∣∣∣∣ (VII22)

VII23 k-model

12 (VII23)

Ediss = Ckprime32`m

VII24 (k-ε)-model

Ediss(t~r) (VII24)

[part~vprime

)~vprime]

) (VII25a)

partvprimei

)vprimei = minus1

dP prime

)vi minus d

) (VII25b)

2~vprime2

Dkprime

Dt= minus

3sumj=1

ρP primevprimej +

3sumi=1

primeij)]minus

3sumij=1

(dvprimei

dvprimej

dximinus 2

minus3sum

3sumij=1

VII32 a

κ(n)i1i2in

κij(X) =XiXj

3sumk=1

εijkXk

partκij(X)

partXi= 0

dκ(X)

VII32 b

bull f(0) = 1

Xrarrinfinf(X) = 0

LI equivint infin

0f(X) dX (VII30)

f(X) Xrarr0

1minus 1

`2T =[vprime(t~r)

]2[dvprime(t~r)dx

]2 (VII32)

[vprime(t~r)]2

Xrarr0

[vprime(t~r)]2

prime(t~r)]

[δvprime(`K)]2

`2Kequiv lim

Xrarr0

[δvprime(X)]2

[dvprime(t~r)

(VII34)

[δvprime(X)]2

2[vprime(X)]2sim

Xrarr0

(K41-1)

[δvprime(X)]2 =

radicνEdiss Φ

)for X LI with `K =

(VII35)

(νEdiss

(K41-2)

B(2)x23

SE(k) = CK Ediss

I k `minus1K =

(VII37)

(EdissX

bull Frisch [35]

CHAPTER VIII

Dρ(t~r)

ρ(t~r)D~v(t~r)

[η(t~r)SSS(t~r)

]+ ~nabla

] (VIII1b)

ρ(t~r)D

[s(t~r)

ρ(t~r)

]= ~nabla middot

2η(t~r)

ζ(t~r)

T (t~r)

]2 (VIII1c)

part~v(t~r)

]~v(t~r) = minus 1

ρ(t~r)D

[s(t~r)

ρ(t~r)

)= T d

)minus P d

)= cP dT (VIII4)

α equiv κ

ρcP (VIII5)

DT (t~r)

Dt=partT (t~r)

]T (t~r) = α4T (t~r) (VIII6)

Pr equiv ν

α=ηcP

κ(VIII7)

α(V ) equiv minus1

(partρ

(VIII11)

minus 1

~nablaP (t~r)

[1 + α(V )Θ(t~r)

one finds

minus 1

~nablaP eff(t~r)

part~v(t~r)

]~v(t~r) = minus

~nablaP eff(t~r)

partΘ(t~r)

VIII21 a

Remarks

VIII21 b

τQ = CR2

δΘ =partΘ

partzδz =

partΘ

partzvτQ = C

α(V )∆T

minus 4π

3R3δρg =

3ρ0gv

α(V )∆T

d (VIII15)

3ρ0gv

α(V )∆T

dgt 6πRρ0νv hArr

α(V )∆T gR4

ανdgt

ναgt

C= Rac

partvzpartt

partδΘ

parttminus ∆T

dv0 hArr ∆T

dv0 minus

(γ + αk2

)δΘ0 = 0

)(γ + αk2

)minusα(V )∆T

dg = 0 (VIII18)

Rac =α(V )∆T g d

αν= π4

Raminus RacRac

α+ νk2

v prop(

Raminus RacRac

)βwith β =

2(VIII19)

of part of Ref [43]

CHAPTER IX

IX11 a

(c n(x)~N (x)

)(IX1)

Nmicro(x) =

(c n(x)~N (x)

Remarks

micro(x) = 0

IX11 b

Global formulation

Figure IX1

d3σmicro equiv1

bull 1

Remarks

P equivintΣ

= (c 0 0 0) (IX10)

umicro(x)∣∣R

(γ(x)c

γ(x)~v(x)

) (IX11)

with γ(x) = 1radic

(IX12)

P (x)∣∣LR(x)

= P (ε(x) n(x)) (IX13)

Remarks

cN0(x)

∣∣LR(x)

=N0(x)u0(x)

[u0(x)]2

∣∣∣∣LR(x)

=N0(x)u0(x)

g00(x)[u0(x)]2

∣∣∣∣LR(x)

=Nmicro(x)umicro(x)

uν(x)uν(x)

∣∣∣∣LR(x)

uν(x)uν(x)=

N(x) middot u(x)

[u(x)]2 (IX14)

[uρ(x)uρ(x)]2

∣∣∣∣LR(x)

N0(x)∣∣LR(x)

= cn(x) ~(x)∣∣LR(x)

= ~0 (IX16a)

T 00(x)∣∣LR(x)

= ε(x)

T ij(x)∣∣LR(x)

T i0(x)∣∣LR(x)

= T 0j(x)∣∣LR(x)

= 0 foralli j = 1 2 3

ε(x) 0 0 0

0 P (x) 0 00 0 P (x) 00 0 0 P (x)

(IX16c)

]umicro(x)uν(x)

c2(IX17b)

N(x) = n(x)u(x) (IX18a)

]u(x)otimes u(x)

c2 (IX18b)

Remarks

)-tensor

umicro(x)uν(x)

c2(IX19b)

ν(x)uν(x) = 0

P (x) =1

IX32 a

]= 0 (IX22)

n)umicrouν

c2= Pgmicroν +

N(numicro)

]uνc2

[T (sumicro)+micro

N(numicro)

]dmicrouνc2

]umicrouνc2

Ndmicro(numicro)

]uνc2

N(numicro)

]uν dmicrouν

c2+ (dmicroP )

umicrouνuνc2

+[T dmicro(sumicro)

]uνuνc2

IX32 b

)+~v middot ~nabla

γu middot d

u middot d(s

nu middot dsminus s

n2u middot dn =

(u middot dsminus s

nu middot dn

γu middot d

γ(x) sim|~v|c

~v(x)2

(~v(x)4

) (IX24)

umicro(x) sim|~v|c

) (IX25)

IX33 a

Nmicro sim|~v|c

(n cn~v

part(n c)partt

3sumi=1

part(n vi)

partxi=partnpartt

IX33 b

ρc2 + e+ ρ~v2 +O(~v2

) (IX27a)

T 0j = T j0 = γ2(ρc2 + e+ P )vj

csim|~v|c

(|~v|3

) (IX27b)

c2sim|~v|c

)= TTTij +O

) (IX27c)

part(ρcvj)

partt+

3sumi=1

partTTTij

partxi+O

)=part(ρvj)

partt+

3sumi=1

partTTTij

partxi+O

) (IX28)

0 =part(ρc)

partt+

3sumi=1

part(ρcvi)

partxi+O

2ρ~v2 +O

2ρ~v2

(|~v|5

)one finds

2ρ~v2 + e

3sumj=1

partxj

2ρ~v2 + e+ P

that ispart

2ρ~v2 + e

)+ ~nabla middot

2ρ~v2 + e+ P

]asymp 0 (IX29)

IX33 c

part(sc)

partt+

3sumi=1

part(svi)

partxi=parts

or equivalently

N(x) = n(x)u(x) + n(x) (IX33b)

Tensor algebra

$microν(x) =1

[∆micro

]T ρσ(x) (IX38b)

Remarks

a[microν] equiv 1

∆ (microρ ∆ν)

σ minus1

3∆microν∆ρσ

)aρσ

(IX39)

(IX40a)

or equivalently

∆ρν(x)dmicroT

]umicro(x)dmicrou

respuν(x)dmicroT

]dmicrou

[uν(x)πmicroν(x)

]minus[dmicrouν(x)

[dmicrouν(x)

]πmicroν(x)

]dmicrou

micro(x)

dmicro

T (x)nmicro(x)

nablamicro(x)uν(x)

[microN(x)

] (IX43a)

3∆microν

)equiv SSSmicroν +

3∆microν

dmicro

T (x)nmicro(x)

]= minus$

microν(x)

[microN(x)

] (IX43b)

3∆microν(x)

[nablaρ(x)uρ(x)

nmicro(x) = κ(x)

[n(x)T (x)

ε(x)+P (x)

]2nablamicro(x)

[microN(x)

](IX44c)

d middot S(x) =$$$(x) $$$(x)

2η(x)T (x)+

Π(x)2

ζ(x)T (x)+

[ε(x)+P (x)

n(x)T (x)

]2 n(x)2

κ(x)T (x) (IX45)

or equivalently

T (x)nmicro(x) +

T (x)n(x) +

T (x)q(x) +

T (x)Q(x) (IX47a)

T (x)nmicro(x) +

T (x)qmicro(x) +

) (IX47c)

(IX48a)where

n(x)n(x)

T (x)Π(x)qmicro

N(x)minusα1(x)

T (x)$micro

ρ(x)qρN

N(x) equivNsumk=1

intuk(τ)δ(4)

(xminusxk(τ)

)d(cτ) (IXA1a)

or component-wise

intumicrok(τ)δ(4)

(xνminusxνk(τ)

cN0(t~r) =

Nsumk=1

) (IXA2a)

N i(t~r) =

Nsumk=1

)(IXA2b)

N0(t~r) = c

Nsumk=1

)dtk(τ)

dτd(cτ) = c

Nsumk=1

intδ(tminustk(t)

ie N0(t~r) = c

Nsumk=1

δ(3)(~xminus~xk(t)

TTT(x) equivNsumk=1

(xminusxk(τ)

)d(cτ) (IXA3a)

(xλminusxλ(τ)

Tmicro0(t~r) =

Nsumk=1

) (IXA4a)

T 0j(t~r) =Nsumk=1

p0k(t)v

jk(t)δ

(3)(~r minus ~xk(t)

)(IXA4b)

T ij(t~r) =Nsumk=1

pik(t)vjk(t)δ

(3)(~r minus ~xk(t)

)(IXA4c)

CHAPTER X

X21 Landau flow

[53 54]

Appendices

APPENDIX A

To be written

de = d

VdU minus U

VdS minus P

VdN minus TS

V 2dV +

dV minus microN

V 2dV = T d

)+ microd

APPENDIX B

B11 Vectors

~c = ci~ei (B1)

B12 One-forms

B13 Tensors

bull the(

linear) n-forms

bull Eventually(

)-tensors

For example the(

(mprime

nprime

n+nprime

Let TTT be a(mn

(mminus1nminus1

) which is

Tensor coordinates

B14 Metric tensor

gij = g(~ei~ej

(~ei~ej

(~ei~ei

arbitrary basis

ci = gijcj (B8b)

(m∓1nplusmn1

) respectively

Remarks

kprimej = δij

cjprime

= Λjprimei ci TTTj

prime1j

primem = Λj

APPENDIX C

Tensor calculus

166 Tensor calculus

part~ei(P )

~c(P ) = ci(P )~ei(P ) (C2)

part~c(P )

partxk=

dci(P )

dxk~ei(P ) (C3a)

dci(P )

dxk=partci(P )

part~c(P )

partxk=partci(P )

partxk~ei(P ) + ci

part~ei(P )

partxk=partci(P )

part~c(P )

partxk=partci(P )

dci(P )

dxk~ei(P )

)-tensor field

Γilk(P ) =1

2gip(P )

[partgpl(P )

partxk+partgpk(P )

partxp

Polar coordinates

part~er(r θ)

partxr= ~0

part~er(r θ)

partxθ=

r~eθ(r θ)

part~eθ(r θ)

partxr=

r~eθ(r θ)

part~eθ(r θ)

168 Tensor calculus

dxr=partcr

partxr= 0

dxr=partcθ

partxr+ Γθθrc

θ =sin θ

(minus sin θ)

dxθ=partcr

partxθ+ Γrθθc

dxθ=partcθ

partxθ+ Γθrθc

r = minuscos θ

rcos θ = 0

Scalar fields

scalar field f(P )

df(P )

dxk=partf(P )

partxk (C9)

One-forms

dhj(P )

dxk=parthj(P )

to be completed

attempt (C12)

APPENDIX D

D11 Definitions

f (z)minus f (z0)

z minus z0(D1)

D12 Some properties

f (x+iy) = P (x y) + iQ(x y) (D2)

partP (x y)

partx=partQ(x y)

partyand

partP (x y)

partx(D3)

= 0 (D4)

where z = xminus iy

4P (x y) = 0 4Q(x y) = 0 (D5)

on the domain

Cf (z) dz =

af(γ(t)

)γprime(t) dt (D6)

f (z) dz = 0 (D7)

f(z0) =1

z minus z0dz (D8)

D31 Taylor series

f (z) =

infinsumn=0

f (n)(z0)

n(z minus z0)n (D9)

f (n)(z0) =n

Definitions

Laurent series

f (z) =

D33 Singular points

Bibliography

174 Bibliography

176 Bibliography

Contents

Introduction












Streamlines

Material derivative

Mechanical stress


Fluids






Strain rate tensor



Kinematic criteria

Physical criteria









Local formulation














Isentropic fluid



Bernoulli equation





Potential flows





Sound waves




Shock waves



Gravity waves


Solitary waves




Plane Couette flow




Reynolds number





Boundary layer














Turbulent viscosity

Mixing-length model

k-model

(k-epsilon)-model













Conservation laws










Local rest frames














Landau flow

Bjorken flow

Appendices




Vectors

One-forms

Tensors

Metric tensor

Change of basis

C Tensor calculus









Definitions

Some properties


Series expansions

Taylor series


Singular points

Conformal maps

Bibliography

Foreword

Contents

I33 Streamlines 10

I42 Fluids 14

VII23 k-model 118

B11 Vectors 160

B12 One-forms 160

B13 Tensors 161

D11 Definitions 170

Introduction

CHAPTER I

observation scale

measuredlocalq

uantity

at every time t

L sim=

|G(t~r)|

]minus1

Kn equiv`mfp

L 1 (I4)

~r = ~r(t ~R) (I5a)

G = G(t ~R) (I6)

~R = ~R(t~r) (I7)

partt (I8)

partt (I9)

~vt(t~r) = ~v(t ~R(t~r)

) (I13a)

) (I13b)

part~r(t ~R)

partt=~vt

(t~r(t ~R)

)~r(t0 ~R) = ~R

(I14a)

) (I14c)

I33 Streamlines

d~x(λ)

dλ= α(λ)~v

(t ~x(λ)

)(I15a)

(xi)Stromlinien

d~xtimes~v(t ~x(λ)

)= ~0 (I15b)

dx1(λ)

v1(t ~x(λ)

) =dx2(λ)

v2(t ~x(λ)

) =dx3(λ)

v3(t ~x(λ)

) (I15c)

v1(t~r)=

v2(t~r)=

v3(t~r)

d~v part~v(t~r)

parttdt+

part~v(t~r)

partx1dx1 +

part~v(t~r)

partx2dx2 +

part~v(t~r)

partx3dx3

partx1+ dx2 part

partx2+ dx3 part

partx3

(xii)Stromroumlhre

d~v part~v(t~r)

parttdt+

(d~r middot ~nabla

)~v(t~r) (I16)

~a(t) =part~v(t~r)

]~v(t~r) (I17)

Dtequiv part

~a(t) =D~v(t~r)

Dt (I19)

Remarks

d2S ~en

d2 ~Fs

Figure I2

~Ts equivd2 ~Fsd2S

T is =

3sumj=1

σσσij ejn (I21b)

tensor

I42 Fluids

Remarks

CHAPTER II

δ~(t)

δ~(t+ δt)

~v(t~r + δ~(t)

~v(t~r)

resp~v(t~r + δ~(t)

) That is one finds

δ~(t+ δt) = δ~(t) +[~v(t~r + δ~(t)

)minus~v(t~r)]δt+O

(δt2) (II1)

δvi 3sumj=1

partvi(t~r)

partxjδxj (II2a)

Introducing the(

DDD(t~r) equiv 1

]T) (II3b)

DDDij(t~r) =1

[partvi(t~r)

partxj+partvj(t~r)

partxi

] RRRij(t~r) =

[partvi(t~r)

partxi

] (II3c)

DDDij(t~r) =

[partvi(t~r)

partxj+partvj(t~r)

partxi

DDDij(t~r) =

2δikδlj

[partvk(t~r)

partxl+partvl(t~r)

partxk

2gik(t~r)glj(t~r)

[partvk(t~r)

partxl+partvl(t~r)

partxk

DDDij(t~r) =

2gik(t~r)glj(t~r)

[dvk(t~r)

dvl(t~r)

](II4a)

RRRij(t~r) =1

2gik(t~r)glj(t~r)

[dvk(t~r)

dxlminus dvl(t~r)

](II4b)

~v(t~r + δ~r

)=~v(t~r)

)(II5)

]δt+O

(δt2)

)middot

3sumjk=1

(xxiv)Wirbeltensor

equivalently reads

~Ω(t~r) =1

)middot

ω1(t~r)=

ω2(t~r)=

ω3(t~r) (II11b)

II13 a

Diagonal components

δ`i(t+ δt) = δ`i(t) +[vi(t~r + δ~(t)

)minus vi

(t~r)]δt+O

(δt2)

partvi(t~r)

partxjδ`j(t) δt

i(t~r) δ`i(t) δt

direction i

δL3(t)

=[DDD1

3(t~r)]δt

DDD11(t~r) +DDD2

2(t~r) +DDD33(t~r) =

partv1(t~r)

partx1+partv2(t~r)

partx2+partv3(t~r)

II13 b

v1 δt

(v1+δv1)δt

v2 δt

(v2+δv2)δtδα1

δv1 =partv1(t~r)

partx2δ`2 δv2 =

partv2(t~r)

partx1d`1

δα1 δv2 δt

δ`1and δα2 minus

δv1 δt

δα1 partv2(t~r)

partv1(t~r)

partx2δt

δt partv2(t~r)

]111+SSS(t~r) (II15a)

]T minus 2

) (II15b)

DDDij(t~r) =1

with [cf Eq (II4a)]

SSSij(t~r) equiv1

[gki(t~r)g

lj(t~r)

(dvk(t~r)

dvl(t~r)

)minus 2

]gij(t~r)

] (II15d)

Summary

bull a translation

Ma equiv v

cs(II16)

CHAPTER III

boundary of Σ

at time t+ δt

streamlines

(t~r) =dG(t~r)

dM(t~r) (III1)

G(t) =

(t~r) dM =

δG =(G1 + G2+

= δG1 + δG2

δG1 equiv(G1

)t+δtminus(G1

)t δG2 equiv

)t+δtminus(G2minus

δG1 =dG1(t)

dtδt =

[ intV

(t~r) ρ(t~r) d3~r

]δt =

-|~v| δt

mρd3V with

δG2 =

(t~r) ρ(t~r) d3~r]

Remarks

Dt= 0 (III6)

m(t~r) = 1 reads

[ intVρ(t~r) d3~r

[ρ(t~r)~v(t~r)

[ intV

n(t~r) d3~r

[n(t~r)~v(t~r)

partρ(t~r)

[ρ(t~r)~v(t~r)

]d3~r = 0

resp intV

partn(t~r)

[n(t~r)~v(t~r)

]d3~r = 0

partρ(t~r)

[ρ(t~r)~v(t~r)

]= 0 (III9)

resppartn(t~r)

[n(t~r)~v(t~r)

]= 0 (III10)

Remarks

]= 0 ie

D~P (t)

Dt= ~F (t) (III11)

D~P (t)

minus~v(t~r)

partρ(t~r)

partt+ ρ(t~r)

]~v(t~r)

d3~r (III13)

D~P (t)

intVρ(t~r)

part~v(t~r)

]~v(t~r)

d3~r =

intVρ(t~r)

D~v(t~r)

Dtd3~r (III14)

J i(t) =

]middot d2~S =

]= minusvi(t~r)

partρ(t~r)

partt+ ρ(t~r)

]vi(t~r)

III32 a

~F (t) =

ie the(

III32 b

Euler equation

part~v(t~r)

]~v(t~r)

d3~r =

ρ(t~r)

part~v(t~r)

]~v(t~r)

Remarks

III32 c

Boundary conditions

III32 d

dM(t~r)=

~fV (t~r)

ρ(t~r)

D~v(t~r)

Dt= minus 1

)= ~nabla

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

III32 e

Remarks

(III23)

[ρ(t~r) vi(t~r)

3sumj=1

dTTTij(t~r)

part(ρvi)

partt+

3sumj=1

dTTTij

dxj=partρ

parttvi + ρ

partvi

partt+

3sumj=1

gijdPdxj

3sumj=1

vid(ρvj)

3sumj=1

ρvjdvi

= vi[partρ

[partvi

[ρ(t~r)~v(t~r)

III33 a

dxj dxk

minus η(t~r)

[dvi(t~r)

dvj(t~r)

dximinus 2

](III26a)

[DDD(t~r)minus 1

]gminus1(t~r)

]minusζ(t~r)

]gminus1(t~r)

[DDDij(t~r)minus 1

]gij(t~r)

]minus ζ(t~r)

]gij(t~r) (III26d)

]gij(t~r) (III26f)

Remarks

III33 b

σij(t~r) =

minusP (t~r)+

[ζ(t~r)minus 2

3η(t~r)

gij(t~r)+η(t~r)

[dvi(t~r)

dvj(t~r)

] (III27c)

3sumij=1

[minusP (t~r) +

(ζ(t~r)minus 2

3η(t~r)

]gij(t~r)

+ η(t~r)

(dvi(t~r)

dvj(t~r)

= minusint

~nablaP (t~r) d3V +

= minusint

~nablaP (t~r) d3V +

~fvisc(t~r) =3sum

η(t~r)

[dvi(t~r)

dvj(t~r)

]~ej(t~r)

3η(t~r)

(III29b)

III33 c

ρ(t~r)

part~v(t~r)

]~v(t~r)

or component-wise

ρ(t~r)

partvi(t~r)

]vi(t~r)

=minusdP (t~r)

[ζ(t~r)minus 2

3η(t~r)

3sumj=1

η(t~r)

[dvi(t~r)

dvj(t~r)

]+[~fV (t~r)

]i(III30b)

ρ(t~r)

part~v(t~r)

]~v(t~r)

]+ ~fV (t~r)

part~v(t~r)

]~v(t~r) = minus1

III33 d

Boundary conditions

]+ ~nabla middot

]~v(t~r)

(III33)

Remarks

parts(t~r)

[s(t~r)~v(t~r)

]= 0 (III34)

sumj π

ij vj to the

2ρ(t~r)~v(t~r)2 + e(t~r)

]+ ~nabla middot

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r)

]~v(t~r)

minus η(t~r)

)~v(t~r) + ~nabla

(~v(t~r)2

(III35)

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r)

2ρ~v2

)+ ~nabla middot

2ρ~v2

ρ~v middot part~v

partt+

partρ

partt~v2 +

3sumi=1

[partvi

)vi] (III37a)

]= T~nablamiddot

)= T~nablamiddot

partt= T

partt+ micro

partnpartt

= Tparts

(n~v) (III37c)

3sumi=1

[partvi

+ Tparts

3sumij=1

partxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot~v

3sumi=1

partxi

[ζ(~nabla middot~v

) (III37d)

[partvi

+ vipartPpartxi

=3sumj=1

vipart

partxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot~v

)]+ vi

Tparts

= η3sum

partvipartxj

(partvi

partxj+partvj

partximinus 2

)2+ ~nabla middot

(κ~nablaT

) (III38)

3sumij=1

(partvi

partxj+partvj

partximinus2

3gij~nablamiddot~v

)(partvipartxj

+partvjpartximinus2

3gij~nablamiddot~v

3sumij=1

(partvi

partxj+partvj

partximinus2

3gij~nablamiddot~v

)partvjpartxi

(III39a)

)= T~nabla middot

(κ~nablaTT

(~nablaT)2 (III39b)

(κ~nablaTT

3sumij=1

(partvi

partxj+partvj

partximinus 2

)(partvipartxj

3gij~nabla middot~v

(~nabla middot~v

)2+ κ

(~nablaT)2

T 2 (III40a)

parts(t~r)

~nablaT (t~r)

T (t~r)

]2+ κ(t~r)

[~nablaT (t~r)

]2T (t~r)

(III40b)

One thus findsdS

CHAPTER IV

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= minusρg

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= minus mg

kBTP (~r)

P (~r) = P (z) = P 0 exp

(minusmgzkBT

)= T d

ρdP (IV4)

~nabla[w(~r)

ρ(~r)+ Φ(~r)

]= ~0 (IV5)

that isw(z)

enthalpy density

w = e+ P =5

2nkBT + nkBT =

That isw

dT (z)

dz= minus mg7

[w(~r)

ρ(~r)

[s(~r)

ρ(~r)

] (IV6)

fluid 1

fluid 2 S

fluid 1

fluid 2 solid body

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

[s(t~r)

ρ(t~r)

]minus ~nabla

[w(t~r)

ρ(t~r)

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

2+w(t~r)

ρ(t~r)+ Φ(t~r)

[s(t~r)

ρ(t~r)

] (IV9)

~v(~r)2

2+w(~r)

~nabla middot[(

ρ+ Φ

P (~r)

IV22 a

Figure IV2

2+ gzA =

2+ gzB

v2B = v2

A + 2gh

vB =radic

IV22 b

Venturi effect

s-v1 -v2

Figure IV3

P 1minusP 2

s2minus 1

IV22 c

Pitot tube

Obull bullIbull

OprimebullA

-manometer

-bullAprime

PO + ρ~v2

2= PAprime + ρ

~v2Aprime

PI minus PB = ρ~v2

Remarks

IV22 d

Magnus effect

~v0~ωC

Γ~γ(t) =

intS~γ

[~nablatimes~v(t~r)

]middot d2~S =

DΓ~γ(t)

Dt= 0 (IV14b)

DΓ~γDt

part~γ(t λ)

[part2~γ

part~γ

partλmiddot(part~v

partt+sumi

part~v

partxipartγi

DΓ~γDt

part~v

partλmiddot ~v +

part~γ

partλmiddot[part~v

DΓ~γDt

DΓ~γ(t)

Dt= minus

∮~γ

~nablaP (t~r)

~nablatimes(~nablaP

ρ2 (IV16)

part~ω(t~r)

]= minus

ρ(t~r)2 (IV20)

(~nabla middot ~ω

)~v minus

(~v middot ~nabla

)~ω minus

(~nabla middot~v

part~ω(t~r)

]~ω(t~r)minus

]~v(t~r) = minus

]~ω(t~r)minus

ρ(t~r)2

D~v ~ω(t~r)

]~ω(t~r)minus

]~v(t~r) (IV22a)

Dtequiv D~ω(t~r)

]~v(t~r) (IV22b)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r)minus

ρ(t~r)2 (IV23)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r) (IV24)

[~ω(t~r)

ρ(t~r)

]= ~0 (IV25)

δ~(t)δ~(t+ δt)

~v(t~r

+ δ~ (t)) δt

~v(t~r)δt

[δ~(t)middot~nabla

]~v(t~r)δt

D~vδ~(t)Dt

= ~0 (IV26)

]~ω(~r)minus

]~v(~r)

]ρ(~r)

Remarks

partt+ ~nabla

[~nablaϕ(t~r)

+ Φ(t~r)

= minus 1

partt+ ~nabla

[~nablaϕ(t~r)

+P (t~r)

ρ+ Φ(t~r)

= ~0 (IV29)

or equivalently

minuspartϕ(t~r)

partt+

[~nablaϕ(t~r)

+P (t~r)

ρ+ Φ(t~r) = C(t) (IV30)

Laplace equation(t)

4ϕ(t~r) = 0 (IV31)

K(t) (IV32b)

Remarks

IV42 a

(IV33)

IV42 b

bull~r0

bull~r1

bull~r2

-~γ0rarr1

-~γ prime

0rarr1

6~γ0rarr2

6~γ primeprime

0rarr2

6~γ prime

0rarr2

Figure IV7

(1minus λ)

(IV35)

Remarks

IV43 a

party vy(x y) =

partψ(x y)

partx(IV36)

(~x(λ)

party= 0

4ψ(x y) = 0 (IV37a)

(lii)Stromfunktion

partφ(x y)

partx=partψ(x y)

[= minusvx(x y)

partφ(x y)

[= minusvy(x y)

] (IV38)

IV43 b

3333333

33333 33

33333333

333333

Figure IV8

~v(x y) =(

φ(z) = minus Q2π

w(z) =Q

~v(r θ) =Q

2πr~er (IV42c)

θ (IV42d)

φ(z) =iΓ

~v(r θ) =Γ

2πr2~eθ (IV43b)

ϕ(r θ) = minus Γ

2πθ ψ(r θ) =

2πlog r (IV43c)

micro eiα

w(z) =micro eiα

~v(r θ) =micro

ϕ(x y) =microx

x2 + y2 (IV44d)

φ(z) = limεrarr0

)minus log

)] (IV45)

w(z) =

infinsump=0

(IV46a)

infinsump=1

p aminuspminus1

2 z0 isin C (IV47a)

35 and 1

IV43 c

φ(z) = minusvinfin

) (IV48a)

(r minus R2

)sin θ (IV48b)

w(z) = vinfin

(1minus R2

) (IV49a)

~v(r θ) = vinfin

[(1minus R2

)cos θ~er minus

)sin θ

] (IV49b)

P (~r) +1

2ρv2infin

(liii)Staupunkte

]= Pinfin +

2πlog

R (IV50a)

)cos θ minus Γ

(r minus R2

)sin θ +

2πlog

R (IV50b)

w(z) = vinfin

(1minus R2

)minus iΓ

2πz (IV51a)

~v(r θ) = vinfin

[(1minus R2

)cos θ~er minus

r2minus Γ

2πrvinfin

)sin θ

] (IV51b)

IV43 d

Z = f (z) = z +R2

z(IV52)

)cosϑ+ i

(Rminus

CHAPTER V

ρ(t~r) = ρ0(t~r) + δρ(t~r) (V1a)

P (t~r) = P 0(t~r) + δP (t~r) (V1b)

~v(t~r) =~v0(t~r) + δ~v(t~r) (V1c)

V1 Sound waves 71

ρ(t~r) = ρ0 + δρ(t~r) (V2a)

P (t~r) = P 0 + δP (t~r) (V2b)

~v(t~r) = ~0 + δ~v(t~r) (V2c)

|δρ(t~r)| ρ0 |δP (t~r)| P 0 (V2d)

partδρ(t~r)

partt+ ρ0

]= 0 (V3a)

[ρ0 + δρ(t~r)

]partδ~v(t~r)

]δ~v(t~r)

partδρ(t~r)

partt+ ρ0

ρ0partδ~v(t~r)

)= P 0 +

(partPpartρ

(partPpartS

(partPpartN

(partPpartρ

c2s equiv

(partPpartρ

minusω ρ0

c2s~k 2 minusωρ0

)(δρ

~k middot δ~v

(partPpartρ

= minus V 2

(partPpartV

= minus V 2

[minusNkBT

V 2+NkB

(partT

V1 Sound waves 73

P = minus(partU

= minus5

(partT

ie NkB

(partT

= minus2P5

= minus2

leading to c2s =7

∣∣~k∣∣∣∣δ~v∣∣ = c2s

(~k middot δ~v

equation(25)

part2ρ(t~r)

partt2minus c2

s4ρ(t~r) = 0 (V9a)

part2ϕ(t~r)

partt2minus c2

s4ϕ(t~r) = 0

(lv)Verduumlnnung

V2 Shock waves 75

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (V10a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

]+partδP (t x)

partx= 0 (V10b)

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (V11a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

partx= 0 (V11b)

partv(t x)

partt=

dv(ρ)

partρ(t x)

parttresp

partv(t x)

partx=

dv(ρ)

partρ(t x)

[dv(ρ)

minus cs(ρ)2

partρ(t x)

partx= 0

dv(ρ)

dρ= plusmncs(ρ)

ρ (V12)

v(ρ) =

int ρ

cs(ρprime)

ρprimedρprime

partρ(t x)

partt+[v(ρ(t x)

(ρ(t x)

)]partρ(t x)

partx= 0 (V13)

t4 gt t3

t3 gt t2

t2 gt t1

t1 gt t0

V2 Shock waves 77

+minus gminus (V14)

where g+

ρ v1]]

= 0 (V15a)[[TTTi1]]

= 0 foralli = 1 2 3 (V15b)[[(1

2ρ~v2 + e+ P

]]= 0 (V15c)

[[v2]]

= 0 resp[[

ρ+minus 1

ρminus

) (V18)

]2=j21

=Pminusminus P +

ρminusminus ρ+

ρminusρ+

=Pminusminus P +

ρminusminus ρ+

ρminusρ+

Pminusminus P +

ρminusminus ρ+

ρminus

]2 (partPpartρ

ρminusρ+

=ρminusρ+c2s

[(v1)minus

]2 ρ+

ρminusc2s

2(v1)2 +

e+ Pρ

]]=j21

minus 1

ρ2minus

)+e+ + P +

ρminus= 0

Pminus minus P +

ρminus

ρ+(V19a)

Pminus + P +

ρ+minus 1

ρminus

V3 Gravity waves 79

minus partϕ(t x z)

partt+

[~nablaϕ(t x z)

+P (t x z)

partx2+part2

partz2

]ϕ(t x z) = 0 (V20b)

∣∣∣∣z=0

= 0 (V21a)

(lxi)Schwerewellen

minuspartϕ(t x z)

=Dδh(t x)

UsingD

Dt=part

partt+ vx

partx=part

parttminus partϕ

partx this gives[

partϕ(t x z)

partz+partδh(t x)

partϕ(t x z)

]z=h0+δh(tx)

= 0 (V21b)

P(t x z=h0+δh(t x)

)= P 0 (V21c)

partt+

[~nablaϕ(t x z)

]z=h0+δh(tx)

d2f(z)

dz2minus k2f(z) = 0

V3 Gravity waves 81

Linear waves

minus partϕ(t x z)

partt+

P (t x z)

ρ+ gz =

ρ+ gh0 (V24)

partϕ(t x z)

∣∣∣∣z=h0

+partδh(t x)

partt= 0 (V25a)

∣∣∣∣z=h0

partt2+ g

partϕ(t x z)

cϕ =ω

radicg

k and cg =

dω(k)

radicg

cϕ = cg =radicgh0

δh(t x) =1

partϕ(t x z)

∣∣∣∣z=h0

vx(t x z) =kg

cosh(kz)

sinh(kz)

x(t) = x0 +kgδh0

cosh(kz)

δh0 cosh(kz)

z(t) = z0 +kgδh0

sinh(kz)

δh0 sinh(kz)

[x(t)minus x0]2

cosh2(kz)+

[z(t)minus z0]2

sinh2(kz)=

[kgδh0

ω2 cosh(kh0)

sinh(kh0)

V3 Gravity waves 83

partt= P 0 + ρg

[h0 minus z + δh0

cosh(kz)

V32 Solitary waves

]~v(t~r) = minus1

vz(t x z=0) = 0 (V27c)

)=partδh(t x)

partt+ vx(t~r)

partδh(t x)

partx (V27d)

P(t x z=h0+δh(t x)

)= P 0

(t x z=h0+δh(t x)

tlowast equiv t

tc xlowast equiv x

Lc zlowast equiv z

δhc vlowastx equiv

vxδhctc

vlowastz equivvz

ρ δhcLct2c

= 0 (V28a)

partxlowast (V28b)

partzlowast (V28c)

partδhlowast

P lowast =gt2cLc

Fr equivradicLcg

P lowast =1

Velocity potential

partvxpartt

(vxpartvxpartx

+ vzpartvzpartx

partδh

partx= 0 (V29)

V3 Gravity waves 85

part2ϕ

partx2+part2ϕ

partz2= 0 (V30)

znϕn(t x) (V31)

infinsumn=0

zn[part2ϕn(t x)

partx2+ (n+ 1)(n+ 2)ϕn+2(t x)

(n+ 1)(n+ 2)

part2ϕn(t x)

part2ϕ0(t x)

partx2+z4

part4ϕ0(t x)

partx4+

partx+z2

part3ϕ0(t x)

partx3minus z4

part5ϕ0(t x)

partx5+

partx2minus z3

part4ϕ0(t x)

partx4+

part2u(t x)

partx+z3

part3u(t x)

partx (V34a)

partt= δ

partφ(t x)

partt (V34b)

partvx(t x)

partt+

partφ(t x)

partx= 0 (V34c)

part2u(t x)

partt2minus δ

part2u(t x)

partx2= 0 (V35)

radicgh0(Lctc)

partvx(t x z)

partt+ ε

[vx(t x z)

partvx(t x z)

partx+ vz(t x z)

partvz(t x z)

]+partφ(t x)

partx= 0 (V36)

instead of Eq (V29)

)= u(t x)minus δ2

part2u(t x)

partx2 (V37a)

)= minusδ

[1 + εφ(t x)

]partu(t x)

partx+δ3

part3u(t x)

partx3 (V37b)

partu(t x)

parttminus δ2

part3u(t x)

partu(t x)

partx+partφ(t x)

partx= 0 (V38)

partφ(t x)

partt+ δεvx

(t x z=δ(1 + εφ)

)partφ(t x)

partφ(t x)

partt+ εu(t x)

partφ(t x)

]partu(t x)

partxminus δ2

part3u(t x)

partx3= 0 (V39)

partu(t x)

partt+partφ(t x)

partx= 0

partφ(t x)

partt+partu(t x)

partx= 0

partu(t x)

partt+partu(t x)

partx= 0 (V40)

V3 Gravity waves 87

partφ

partt+partφ

partx+ ε

partu(ε)

partx+ δ2partu(δ)

partx+ 2εφ

partφ

partxminus δ2

part3φ

partx3= 0

partφ

partt+partφ

partx+ ε

partu(ε)

partt+ δ2partu(δ)

partt+ εφ

partφ

partxminus δ2

part3φ

partx2 partt= 0

[partu(ε)(t x)

partx+

2φ(t x)

partφ(t x)

]+ δ2

[partu(δ)(t x)

partxminus 1

part3φ(t x)

partx3

u(ε)(t x) = minus1

4φ(t x) + C(ε)(t) u(δ)(t x) =

part2φ(t x)

partx2+ C(δ)(t)

partφ(t x)

partt+partφ(t x)

partx+

2εφ(t x)

partφ(t x)

partx+

6δ2 part

3φ(t x)

partx3= 0 (V42)

partφ(τ ξ)

partτ+

2εφ(τ ξ)

partφ(τ ξ)

partξ+

6δ2 part

3φ(τ ξ)

partξ3= 0 (V43)

partφ(τ ξ)

partτ+ 6φ(τ ξ)

partφ(τ ξ)

partξ3= 0 (V44)

φ(τ ξ) =φ0

] (V45a)

δh(t x) =δhmax

radic3δhmax

[xminusradicgh0

δhmax

] (V45b)

δhmax=1 t = t0

δhmax= 025 t = t0

δhmax= 025 t = t1

δh(t x)

CHAPTER VI

partρ(t~r)

partt= 0 (VI1a)

parte(t~r)

] (VI1c)

]~v(~r) = minus1

partv(y)

partx= 0 (VI3a)

v(y)partv(y)

partx~ex = minus1

ρ~nablaP (~r) + ν

d2v(y)

dy2~ex (VI3b)

partP (~r)

partx= η

d2v(y)

dy2 (VI4)

d2v(y)

dy2= 0

v(y=0) = 0 v(y=h) = |~u|

~v(~r) =y

h~u for 0 le y le h

[partvi(~r)

partxj+partvj(~r)

partxi

0 0 minusPinfin

d2 ~Fs(~r)

d2S= ~Ts(~r) =

[ 3sumij=1

3sumij=1

)~ei =

η |~u|h

minusPinfin0

6yP 1 P 2

v(y) = minus 1

∆PLy2 + γy + δ

v(y=0) = 0 v(y=h) = 0

v(y) =1

[y(hminus y)

P 1 P 2-z

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= η

[part2v(r)

partx2+part2v(r))

party2

[d2v(r)

partP (~r)

partz= minus∆P

dr= minus∆P

ηL (VI7)

dχ(r)

dr+χ(r)

v(r) =∆P4ηL

(a2 minus r2

)for r le a (VI8)

∆P4ηL

(a2r minus r3

)dr = 2πρ

∆P4ηL

4=πρa4

∆PL (VI9)

〈v〉 equiv 1

0v(r) 2πr dr =

2v(r=0)

~rlowast equiv ~r

Lc ~vlowast equiv

vc tlowast equiv t

(VI10)

Re equiv ρvcLcη

=vcLcν

(VI12)

~v(t~r) = vc~f1

) P (t~r) = P 0 + ρv2

Fr equiv vcradicgLc

(VI14)

[minus 1

(~Ω0times~r

Ek equiv η

ρΩL2c

(VI15)

Ro equiv vcΩLc

(VI16)

Ro = Re middot Ek

VI31 a

Stokes equation

(~v middot ~nabla

[~nablatimes ~c(~r)

]= ~nabla

4P (~r) = 0 (VI20)

VI31 b

) This also holds

~vinfin ~er~eϕ

4[~nablatimes~u(~r)

]= ~0 (VI22a)

]= minus~nabla

[4f(r)

]times~vinfin

])times~vinfin = ~0

])only has a com-

(~nabla[4f(r)

])= ~nabla

(4[4f(r)]

) so that the

4[4f(r)] = 0

4 =part2

partr2+

partrminus `(`+ 1)

4f(r) =d2f(r)

r(VI24a)

2r (VI24b)

)middot~vinfin

]minus C

r~vinfin = minusB

rminus C

r~vinfin

(~er middot~vinfin

r3minus C

R3minus C

)~vinfin +

R3minus C

)(~er middot~vinfin

)~er = ~0

[~vinfin +

(~er middot~vinfin

)~er]minus R3

[~vinfin minus 3

(~er middot~vinfin

)~er] (VI25)

P (~r) =3

~er middot~vinfinr2

+ const

~v(~r) =~vinfin +R3

[~vinfin minus 3

(~er middot~vinfin

D = microkBT =kBT

6πRη

partv(t y)

partt= ν

part2v(t y)

party2 (VI27a)

limyrarrinfin

v(t y) = uf1

radicνt) ie write

v(t y) = uf2

2radicνt

v(t y) = uf

2radicνt

)(VI28)

f(ξ) = 0 (VI29b)

int ξ

0eminusυ

2dυ (VI31)

int infinξ

eminusυ2dυ (VI32)

v(t y) = u

[1minus erf

2radicνt

)] (VI33)

radicνω

partvxpartx

+partvyparty

= 0 (VI35a)

vxpart

partx+ vy

)vx = minus1

partPpartx

(part2

partx2+part2

party2

)vx (VI35b)(

vxpart

partx+ vy

)vy = minus1

partPparty

(part2

partx2+part2

party2

)vy (VI35c)

xlowast equiv x

Lc ylowast equiv y

δl(VI36)

δl Lc (VI37)

vlowasty equivvyu

u vinfin (VI39)

Re equiv Lcvinfinν

(VI40)

+Lcδl

vinfin

= 0 (VI41a)

+Lcδl

= minuspartP lowast

partxlowast+

) (VI41b)

+Lcδl

v2infin

= minusLcδ

partP lowast

partylowast+

vinfin

) (VI41c)

vinfin= 1 (VI42)

Re= 1 (VI43)

δl =LcradicRe

= 0 (VI45a)

= minuspartP lowast

(VI45b)

partP lowast

(~v middot ~nabla

)~v = 1

2~nabla(~v2)

part~ω(t~r)

]= minus

ρ(t~r)2+ ν4~ω(t~r) (VI46)

D~ω(t~r)

]~v(t~r)minus

]~ω(t~r)minus

ρ(t~r)2+ ν4~ω(t~r) (VI47a)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r)minus

ρ(t~r)2+ ν4~ω(t~r) (VI47b)

~ω(t=0~r) =Γ0

2πrδ(z)~ez (VI48)

(~ω middot ~nabla

partωz(t r)

[part2ωz(t r)

partr2+

partωz(t r)

] (VI50)

ωz(t r) =Γ0

νt(VI51)

f(ξ) = C eminusξ4

that isωz(t r) =

νtC eminusr

2(4νt) (VI53)

Γ(t R) =

~v(t~r) middot d~=

int 2π

2(4νt)]

ωz(t r) =Γ0

4πνteminusr

2(4νt) (VI55)

radicν

Γ(t R) = Γ0

[1minus eminusR

2(4νt)] (VI56)

~v(t~r) =Γ0

[1minus eminusr

2(4νt)]~eθr (VI57)

where |~eθ| = r

∣∣δ~v(t~r)∣∣ cs (VI58b)

c2s equiv

(partPpartρ

(VI58c)

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (VI59a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

partx+

3η + ζ

)part2δv(t x)

partx2 (VI59b)

partδρ(t x)

partt+ ρ0

partδv(t x)

partx= 0 (VI60a)

ρ0partδv(t x)

partx+

3η + ζ

)part2δv(t x)

partx2 (VI60b)

ν equiv 1

3η + ζ

) (VI61)

ρ0partδv(t x)

partt+ c2

partδρ(t x)

partx= ρ0ν

part2δv(t x)

partx2 (VI62)

part2δv(t x)

partt2minus c2

part2δv(t x)

partx2= ν

part3δv(t x)

part2δρ(t x)

partt2minus c2

part2δρ(t x)

partx2= ν

part3δρ(t x)

ω2 = c2sk

ω2r minus ω2

i + 2iωrωi = c2sk

ωi = minus1

2νk2 (VI66)

ω2r = c2

sk2 minus 1

4ν2k4 (VI67)

c2ϕ = c2

s minus1

4ν2k2 (VI68)

Remarks

s one finds

ω2 c2sk

4ω2τ2

)hArr cϕ equiv

8ω2τ2

ki νω2

(VI69)

((ωτν)2

CHAPTER VII

VII11 a

Q = minusπρa4

∆PL (VI9)

Ediss = minus1

∆PL〈v〉 =

8ν〈v〉2

a2(VII1)

〈v〉 =Q

πa2ρ= minusa

VII11 b

reads [29]

Nsumn=1

~v(n)(t~r) asymp 1

int t+T 2

(part~v(t~r)

]~v(t~r)

VII13 a

[part~v

)~vprime (VII9a)

partvi

)vi = minus1

dPdximinus

3sumj=1

dvprimeivprimej

dxj+ ν4vi (VII9b)

Dtequiv part

(ρ~v)

SSSij equiv 1

dximinus 2

) (VII12c)

(ρ~v2

[P~v +

(TTTR minus 2ηSSS

)middot~v]

+(TTTR minus 2ηSSS

) SSS (VII13)

ρP~v +

)middot~v]

) SSS (VII14a)

or component-wise

Dt= minus

3sumj=1

ρP vj +

3sumi=1

3sumij=1

)SSSij (VII14b)

condition (VII7)

VII13 b

3sumij=1

lang3sum

(VII15)

ρP~v +

)middot~v]

) SSS (VII16a)

or component-wise

Dt= minus

3sumj=1

ρP vj +

3sumi=1

(vprimeivprimej

3sumij=1

Ediss =

lang3sum

vprimeivprimejSSSij

rangsim v3

Lc (VII17)

part~vprime(t~r)

DTTTijRDt

dTTTijRdxk

)minus(TTTikR

dxk+TTTjkR

)minus2η

dvprimei

dxkdvprimej

part~v(t~r)

P (t~r)

[~vprime(t~r)]2

3sumij=1

SSSijSSSij

νeff =

minussumij

vprimeivprimejSSSij

2sumij

SSSijSSSij

2νsumij

SSSijSSSij

2sumij

SSSijSSSij= ν

∣∣∣∣ (VII22)

VII23 k-model

12 (VII23)

Ediss = Ckprime32`m

VII24 (k-ε)-model

Ediss(t~r) (VII24)

[part~vprime

)~vprime]

) (VII25a)

partvprimei

)vprimei = minus1

dP prime

)vi minus d

) (VII25b)

2~vprime2

Dkprime

Dt= minus

3sumj=1

ρP primevprimej +

3sumi=1

primeij)]minus

3sumij=1

(dvprimei

dvprimej

dximinus 2

minus3sum

3sumij=1

VII32 a

κ(n)i1i2in

κij(X) =XiXj

3sumk=1

εijkXk

partκij(X)

partXi= 0

dκ(X)

VII32 b

bull f(0) = 1

Xrarrinfinf(X) = 0

LI equivint infin

0f(X) dX (VII30)

f(X) Xrarr0

1minus 1

`2T =[vprime(t~r)

]2[dvprime(t~r)dx

]2 (VII32)

[vprime(t~r)]2

Xrarr0

[vprime(t~r)]2

prime(t~r)]

[δvprime(`K)]2

`2Kequiv lim

Xrarr0

[δvprime(X)]2

[dvprime(t~r)

(VII34)

[δvprime(X)]2

2[vprime(X)]2sim

Xrarr0

(K41-1)

[δvprime(X)]2 =

radicνEdiss Φ

)for X LI with `K =

(VII35)

(νEdiss

(K41-2)

B(2)x23

SE(k) = CK Ediss

I k `minus1K =

(VII37)

(EdissX

bull Frisch [35]

CHAPTER VIII

Dρ(t~r)

ρ(t~r)D~v(t~r)

[η(t~r)SSS(t~r)

]+ ~nabla

] (VIII1b)

ρ(t~r)D

[s(t~r)

ρ(t~r)

]= ~nabla middot

2η(t~r)

ζ(t~r)

T (t~r)

]2 (VIII1c)

part~v(t~r)

]~v(t~r) = minus 1

ρ(t~r)D

[s(t~r)

ρ(t~r)

)= T d

)minus P d

)= cP dT (VIII4)

α equiv κ

ρcP (VIII5)

DT (t~r)

Dt=partT (t~r)

]T (t~r) = α4T (t~r) (VIII6)

Pr equiv ν

α=ηcP

κ(VIII7)

α(V ) equiv minus1

(partρ

(VIII11)

minus 1

~nablaP (t~r)

[1 + α(V )Θ(t~r)

one finds

minus 1

~nablaP eff(t~r)

part~v(t~r)

]~v(t~r) = minus

~nablaP eff(t~r)

partΘ(t~r)

VIII21 a

Remarks

VIII21 b

τQ = CR2

δΘ =partΘ

partzδz =

partΘ

partzvτQ = C

α(V )∆T

minus 4π

3R3δρg =

3ρ0gv

α(V )∆T

d (VIII15)

3ρ0gv

α(V )∆T

dgt 6πRρ0νv hArr

α(V )∆T gR4

ανdgt

ναgt

C= Rac

partvzpartt

partδΘ

parttminus ∆T

dv0 hArr ∆T

dv0 minus

(γ + αk2

)δΘ0 = 0

)(γ + αk2

)minusα(V )∆T

dg = 0 (VIII18)

Rac =α(V )∆T g d

αν= π4

Raminus RacRac

α+ νk2

v prop(

Raminus RacRac

)βwith β =

2(VIII19)

of part of Ref [43]

CHAPTER IX

IX11 a

(c n(x)~N (x)

)(IX1)

Nmicro(x) =

(c n(x)~N (x)

Remarks

micro(x) = 0

IX11 b

Global formulation

Figure IX1

d3σmicro equiv1

bull 1

Remarks

P equivintΣ

= (c 0 0 0) (IX10)

umicro(x)∣∣R

(γ(x)c

γ(x)~v(x)

) (IX11)

with γ(x) = 1radic

(IX12)

P (x)∣∣LR(x)

= P (ε(x) n(x)) (IX13)

Remarks

cN0(x)

∣∣LR(x)

=N0(x)u0(x)

[u0(x)]2

∣∣∣∣LR(x)

=N0(x)u0(x)

g00(x)[u0(x)]2

∣∣∣∣LR(x)

=Nmicro(x)umicro(x)

uν(x)uν(x)

∣∣∣∣LR(x)

uν(x)uν(x)=

N(x) middot u(x)

[u(x)]2 (IX14)

[uρ(x)uρ(x)]2

∣∣∣∣LR(x)

N0(x)∣∣LR(x)

= cn(x) ~(x)∣∣LR(x)

= ~0 (IX16a)

T 00(x)∣∣LR(x)

= ε(x)

T ij(x)∣∣LR(x)

T i0(x)∣∣LR(x)

= T 0j(x)∣∣LR(x)

= 0 foralli j = 1 2 3

ε(x) 0 0 0

0 P (x) 0 00 0 P (x) 00 0 0 P (x)

(IX16c)

]umicro(x)uν(x)

c2(IX17b)

N(x) = n(x)u(x) (IX18a)

]u(x)otimes u(x)

c2 (IX18b)

Remarks

)-tensor

umicro(x)uν(x)

c2(IX19b)

ν(x)uν(x) = 0

P (x) =1

IX32 a

]= 0 (IX22)

n)umicrouν

c2= Pgmicroν +

N(numicro)

]uνc2

[T (sumicro)+micro

N(numicro)

]dmicrouνc2

]umicrouνc2

Ndmicro(numicro)

]uνc2

N(numicro)

]uν dmicrouν

c2+ (dmicroP )

umicrouνuνc2

+[T dmicro(sumicro)

]uνuνc2

IX32 b

)+~v middot ~nabla

γu middot d

u middot d(s

nu middot dsminus s

n2u middot dn =

(u middot dsminus s

nu middot dn

γu middot d

γ(x) sim|~v|c

~v(x)2

(~v(x)4

) (IX24)

umicro(x) sim|~v|c

) (IX25)

IX33 a

Nmicro sim|~v|c

(n cn~v

part(n c)partt

3sumi=1

part(n vi)

partxi=partnpartt

IX33 b

ρc2 + e+ ρ~v2 +O(~v2

) (IX27a)

T 0j = T j0 = γ2(ρc2 + e+ P )vj

csim|~v|c

(|~v|3

) (IX27b)

c2sim|~v|c

)= TTTij +O

) (IX27c)

part(ρcvj)

partt+

3sumi=1

partTTTij

partxi+O

)=part(ρvj)

partt+

3sumi=1

partTTTij

partxi+O

) (IX28)

0 =part(ρc)

partt+

3sumi=1

part(ρcvi)

partxi+O

2ρ~v2 +O

2ρ~v2

(|~v|5

)one finds

2ρ~v2 + e

3sumj=1

partxj

2ρ~v2 + e+ P

that ispart

2ρ~v2 + e

)+ ~nabla middot

2ρ~v2 + e+ P

]asymp 0 (IX29)

IX33 c

part(sc)

partt+

3sumi=1

part(svi)

partxi=parts

or equivalently

N(x) = n(x)u(x) + n(x) (IX33b)

Tensor algebra

$microν(x) =1

[∆micro

]T ρσ(x) (IX38b)

Remarks

a[microν] equiv 1

∆ (microρ ∆ν)

σ minus1

3∆microν∆ρσ

)aρσ

(IX39)

(IX40a)

or equivalently

∆ρν(x)dmicroT

]umicro(x)dmicrou

respuν(x)dmicroT

]dmicrou

[uν(x)πmicroν(x)

]minus[dmicrouν(x)

[dmicrouν(x)

]πmicroν(x)

]dmicrou

micro(x)

dmicro

T (x)nmicro(x)

nablamicro(x)uν(x)

[microN(x)

] (IX43a)

3∆microν

)equiv SSSmicroν +

3∆microν

dmicro

T (x)nmicro(x)

]= minus$

microν(x)

[microN(x)

] (IX43b)

3∆microν(x)

[nablaρ(x)uρ(x)

nmicro(x) = κ(x)

[n(x)T (x)

ε(x)+P (x)

]2nablamicro(x)

[microN(x)

](IX44c)

d middot S(x) =$$$(x) $$$(x)

2η(x)T (x)+

Π(x)2

ζ(x)T (x)+

[ε(x)+P (x)

n(x)T (x)

]2 n(x)2

κ(x)T (x) (IX45)

or equivalently

T (x)nmicro(x) +

T (x)n(x) +

T (x)q(x) +

T (x)Q(x) (IX47a)

T (x)nmicro(x) +

T (x)qmicro(x) +

) (IX47c)

(IX48a)where

n(x)n(x)

T (x)Π(x)qmicro

N(x)minusα1(x)

T (x)$micro

ρ(x)qρN

N(x) equivNsumk=1

intuk(τ)δ(4)

(xminusxk(τ)

)d(cτ) (IXA1a)

or component-wise

intumicrok(τ)δ(4)

(xνminusxνk(τ)

cN0(t~r) =

Nsumk=1

) (IXA2a)

N i(t~r) =

Nsumk=1

)(IXA2b)

N0(t~r) = c

Nsumk=1

)dtk(τ)

dτd(cτ) = c

Nsumk=1

intδ(tminustk(t)

ie N0(t~r) = c

Nsumk=1

δ(3)(~xminus~xk(t)

TTT(x) equivNsumk=1

(xminusxk(τ)

)d(cτ) (IXA3a)

(xλminusxλ(τ)

Tmicro0(t~r) =

Nsumk=1

) (IXA4a)

T 0j(t~r) =Nsumk=1

p0k(t)v

jk(t)δ

(3)(~r minus ~xk(t)

)(IXA4b)

T ij(t~r) =Nsumk=1

pik(t)vjk(t)δ

(3)(~r minus ~xk(t)

)(IXA4c)

CHAPTER X

X21 Landau flow

[53 54]

Appendices

APPENDIX A

To be written

de = d

VdU minus U

VdS minus P

VdN minus TS

V 2dV +

dV minus microN

V 2dV = T d

)+ microd

APPENDIX B

B11 Vectors

~c = ci~ei (B1)

B12 One-forms

B13 Tensors

bull the(

linear) n-forms

bull Eventually(

)-tensors

For example the(

(mprime

nprime

n+nprime

Let TTT be a(mn

(mminus1nminus1

) which is

Tensor coordinates

B14 Metric tensor

gij = g(~ei~ej

(~ei~ej

(~ei~ei

arbitrary basis

ci = gijcj (B8b)

(m∓1nplusmn1

) respectively

Remarks

kprimej = δij

cjprime

= Λjprimei ci TTTj

prime1j

primem = Λj

APPENDIX C

Tensor calculus

166 Tensor calculus

part~ei(P )

~c(P ) = ci(P )~ei(P ) (C2)

part~c(P )

partxk=

dci(P )

dxk~ei(P ) (C3a)

dci(P )

dxk=partci(P )

part~c(P )

partxk=partci(P )

partxk~ei(P ) + ci

part~ei(P )

partxk=partci(P )

part~c(P )

partxk=partci(P )

dci(P )

dxk~ei(P )

)-tensor field

Γilk(P ) =1

2gip(P )

[partgpl(P )

partxk+partgpk(P )

partxp

Polar coordinates

part~er(r θ)

partxr= ~0

part~er(r θ)

partxθ=

r~eθ(r θ)

part~eθ(r θ)

partxr=

r~eθ(r θ)

part~eθ(r θ)

168 Tensor calculus

dxr=partcr

partxr= 0

dxr=partcθ

partxr+ Γθθrc

θ =sin θ

(minus sin θ)

dxθ=partcr

partxθ+ Γrθθc

dxθ=partcθ

partxθ+ Γθrθc

r = minuscos θ

rcos θ = 0

Scalar fields

scalar field f(P )

df(P )

dxk=partf(P )

partxk (C9)

One-forms

dhj(P )

dxk=parthj(P )

to be completed

attempt (C12)

APPENDIX D

D11 Definitions

f (z)minus f (z0)

z minus z0(D1)

D12 Some properties

f (x+iy) = P (x y) + iQ(x y) (D2)

partP (x y)

partx=partQ(x y)

partyand

partP (x y)

partx(D3)

= 0 (D4)

where z = xminus iy

4P (x y) = 0 4Q(x y) = 0 (D5)

on the domain

Cf (z) dz =

af(γ(t)

)γprime(t) dt (D6)

f (z) dz = 0 (D7)

f(z0) =1

z minus z0dz (D8)

D31 Taylor series

f (z) =

infinsumn=0

f (n)(z0)

n(z minus z0)n (D9)

f (n)(z0) =n

Definitions

Laurent series

f (z) =

D33 Singular points

Bibliography

174 Bibliography

176 Bibliography

Contents

Introduction












Streamlines

Material derivative

Mechanical stress


Fluids






Strain rate tensor



Kinematic criteria

Physical criteria









Local formulation














Isentropic fluid



Bernoulli equation





Potential flows





Sound waves




Shock waves



Gravity waves


Solitary waves




Plane Couette flow




Reynolds number





Boundary layer














Turbulent viscosity

Mixing-length model

k-model

(k-epsilon)-model













Conservation laws










Local rest frames














Landau flow

Bjorken flow

Appendices




Vectors

One-forms

Tensors

Metric tensor

Change of basis

C Tensor calculus









Definitions

Some properties


Series expansions

Taylor series


Singular points

Conformal maps

Bibliography

Contents

I33 Streamlines 10

I42 Fluids 14

VII23 k-model 118

B11 Vectors 160

B12 One-forms 160

B13 Tensors 161

D11 Definitions 170

Introduction

CHAPTER I

observation scale

measuredlocalq

uantity

at every time t

L sim=

|G(t~r)|

]minus1

Kn equiv`mfp

L 1 (I4)

~r = ~r(t ~R) (I5a)

G = G(t ~R) (I6)

~R = ~R(t~r) (I7)

partt (I8)

partt (I9)

~vt(t~r) = ~v(t ~R(t~r)

) (I13a)

) (I13b)

part~r(t ~R)

partt=~vt

(t~r(t ~R)

)~r(t0 ~R) = ~R

(I14a)

) (I14c)

I33 Streamlines

d~x(λ)

dλ= α(λ)~v

(t ~x(λ)

)(I15a)

(xi)Stromlinien

d~xtimes~v(t ~x(λ)

)= ~0 (I15b)

dx1(λ)

v1(t ~x(λ)

) =dx2(λ)

v2(t ~x(λ)

) =dx3(λ)

v3(t ~x(λ)

) (I15c)

v1(t~r)=

v2(t~r)=

v3(t~r)

d~v part~v(t~r)

parttdt+

part~v(t~r)

partx1dx1 +

part~v(t~r)

partx2dx2 +

part~v(t~r)

partx3dx3

partx1+ dx2 part

partx2+ dx3 part

partx3

(xii)Stromroumlhre

d~v part~v(t~r)

parttdt+

(d~r middot ~nabla

)~v(t~r) (I16)

~a(t) =part~v(t~r)

]~v(t~r) (I17)

Dtequiv part

~a(t) =D~v(t~r)

Dt (I19)

Remarks

d2S ~en

d2 ~Fs

Figure I2

~Ts equivd2 ~Fsd2S

T is =

3sumj=1

σσσij ejn (I21b)

tensor

I42 Fluids

Remarks

CHAPTER II

δ~(t)

δ~(t+ δt)

~v(t~r + δ~(t)

~v(t~r)

resp~v(t~r + δ~(t)

) That is one finds

δ~(t+ δt) = δ~(t) +[~v(t~r + δ~(t)

)minus~v(t~r)]δt+O

(δt2) (II1)

δvi 3sumj=1

partvi(t~r)

partxjδxj (II2a)

Introducing the(

DDD(t~r) equiv 1

]T) (II3b)

DDDij(t~r) =1

[partvi(t~r)

partxj+partvj(t~r)

partxi

] RRRij(t~r) =

[partvi(t~r)

partxi

] (II3c)

DDDij(t~r) =

[partvi(t~r)

partxj+partvj(t~r)

partxi

DDDij(t~r) =

2δikδlj

[partvk(t~r)

partxl+partvl(t~r)

partxk

2gik(t~r)glj(t~r)

[partvk(t~r)

partxl+partvl(t~r)

partxk

DDDij(t~r) =

2gik(t~r)glj(t~r)

[dvk(t~r)

dvl(t~r)

](II4a)

RRRij(t~r) =1

2gik(t~r)glj(t~r)

[dvk(t~r)

dxlminus dvl(t~r)

](II4b)

~v(t~r + δ~r

)=~v(t~r)

)(II5)

]δt+O

(δt2)

)middot

3sumjk=1

(xxiv)Wirbeltensor

equivalently reads

~Ω(t~r) =1

)middot

ω1(t~r)=

ω2(t~r)=

ω3(t~r) (II11b)

II13 a

Diagonal components

δ`i(t+ δt) = δ`i(t) +[vi(t~r + δ~(t)

)minus vi

(t~r)]δt+O

(δt2)

partvi(t~r)

partxjδ`j(t) δt

i(t~r) δ`i(t) δt

direction i

δL3(t)

=[DDD1

3(t~r)]δt

DDD11(t~r) +DDD2

2(t~r) +DDD33(t~r) =

partv1(t~r)

partx1+partv2(t~r)

partx2+partv3(t~r)

II13 b

v1 δt

(v1+δv1)δt

v2 δt

(v2+δv2)δtδα1

δv1 =partv1(t~r)

partx2δ`2 δv2 =

partv2(t~r)

partx1d`1

δα1 δv2 δt

δ`1and δα2 minus

δv1 δt

δα1 partv2(t~r)

partv1(t~r)

partx2δt

δt partv2(t~r)

]111+SSS(t~r) (II15a)

]T minus 2

) (II15b)

DDDij(t~r) =1

with [cf Eq (II4a)]

SSSij(t~r) equiv1

[gki(t~r)g

lj(t~r)

(dvk(t~r)

dvl(t~r)

)minus 2

]gij(t~r)

] (II15d)

Summary

bull a translation

Ma equiv v

cs(II16)

CHAPTER III

boundary of Σ

at time t+ δt

streamlines

(t~r) =dG(t~r)

dM(t~r) (III1)

G(t) =

(t~r) dM =

δG =(G1 + G2+

= δG1 + δG2

δG1 equiv(G1

)t+δtminus(G1

)t δG2 equiv

)t+δtminus(G2minus

δG1 =dG1(t)

dtδt =

[ intV

(t~r) ρ(t~r) d3~r

]δt =

-|~v| δt

mρd3V with

δG2 =

(t~r) ρ(t~r) d3~r]

Remarks

Dt= 0 (III6)

m(t~r) = 1 reads

[ intVρ(t~r) d3~r

[ρ(t~r)~v(t~r)

[ intV

n(t~r) d3~r

[n(t~r)~v(t~r)

partρ(t~r)

[ρ(t~r)~v(t~r)

]d3~r = 0

resp intV

partn(t~r)

[n(t~r)~v(t~r)

]d3~r = 0

partρ(t~r)

[ρ(t~r)~v(t~r)

]= 0 (III9)

resppartn(t~r)

[n(t~r)~v(t~r)

]= 0 (III10)

Remarks

]= 0 ie

D~P (t)

Dt= ~F (t) (III11)

D~P (t)

minus~v(t~r)

partρ(t~r)

partt+ ρ(t~r)

]~v(t~r)

d3~r (III13)

D~P (t)

intVρ(t~r)

part~v(t~r)

]~v(t~r)

d3~r =

intVρ(t~r)

D~v(t~r)

Dtd3~r (III14)

J i(t) =

]middot d2~S =

]= minusvi(t~r)

partρ(t~r)

partt+ ρ(t~r)

]vi(t~r)

III32 a

~F (t) =

ie the(

III32 b

Euler equation

part~v(t~r)

]~v(t~r)

d3~r =

ρ(t~r)

part~v(t~r)

]~v(t~r)

Remarks

III32 c

Boundary conditions

III32 d

dM(t~r)=

~fV (t~r)

ρ(t~r)

D~v(t~r)

Dt= minus 1

)= ~nabla

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

III32 e

Remarks

(III23)

[ρ(t~r) vi(t~r)

3sumj=1

dTTTij(t~r)

part(ρvi)

partt+

3sumj=1

dTTTij

dxj=partρ

parttvi + ρ

partvi

partt+

3sumj=1

gijdPdxj

3sumj=1

vid(ρvj)

3sumj=1

ρvjdvi

= vi[partρ

[partvi

[ρ(t~r)~v(t~r)

III33 a

dxj dxk

minus η(t~r)

[dvi(t~r)

dvj(t~r)

dximinus 2

](III26a)

[DDD(t~r)minus 1

]gminus1(t~r)

]minusζ(t~r)

]gminus1(t~r)

[DDDij(t~r)minus 1

]gij(t~r)

]minus ζ(t~r)

]gij(t~r) (III26d)

]gij(t~r) (III26f)

Remarks

III33 b

σij(t~r) =

minusP (t~r)+

[ζ(t~r)minus 2

3η(t~r)

gij(t~r)+η(t~r)

[dvi(t~r)

dvj(t~r)

] (III27c)

3sumij=1

[minusP (t~r) +

(ζ(t~r)minus 2

3η(t~r)

]gij(t~r)

+ η(t~r)

(dvi(t~r)

dvj(t~r)

= minusint

~nablaP (t~r) d3V +

= minusint

~nablaP (t~r) d3V +

~fvisc(t~r) =3sum

η(t~r)

[dvi(t~r)

dvj(t~r)

]~ej(t~r)

3η(t~r)

(III29b)

III33 c

ρ(t~r)

part~v(t~r)

]~v(t~r)

or component-wise

ρ(t~r)

partvi(t~r)

]vi(t~r)

=minusdP (t~r)

[ζ(t~r)minus 2

3η(t~r)

3sumj=1

η(t~r)

[dvi(t~r)

dvj(t~r)

]+[~fV (t~r)

]i(III30b)

ρ(t~r)

part~v(t~r)

]~v(t~r)

]+ ~fV (t~r)

part~v(t~r)

]~v(t~r) = minus1

III33 d

Boundary conditions

]+ ~nabla middot

]~v(t~r)

(III33)

Remarks

parts(t~r)

[s(t~r)~v(t~r)

]= 0 (III34)

sumj π

ij vj to the

2ρ(t~r)~v(t~r)2 + e(t~r)

]+ ~nabla middot

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r)

]~v(t~r)

minus η(t~r)

)~v(t~r) + ~nabla

(~v(t~r)2

(III35)

2ρ(t~r)~v(t~r)2 + e(t~r) + P (t~r)

2ρ~v2

)+ ~nabla middot

2ρ~v2

ρ~v middot part~v

partt+

partρ

partt~v2 +

3sumi=1

[partvi

)vi] (III37a)

]= T~nablamiddot

)= T~nablamiddot

partt= T

partt+ micro

partnpartt

= Tparts

(n~v) (III37c)

3sumi=1

[partvi

+ Tparts

3sumij=1

partxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot~v

3sumi=1

partxi

[ζ(~nabla middot~v

) (III37d)

[partvi

+ vipartPpartxi

=3sumj=1

vipart

partxj

(partvi

partxj+partvj

partximinus 2

3gij~nabla middot~v

)]+ vi

Tparts

= η3sum

partvipartxj

(partvi

partxj+partvj

partximinus 2

)2+ ~nabla middot

(κ~nablaT

) (III38)

3sumij=1

(partvi

partxj+partvj

partximinus2

3gij~nablamiddot~v

)(partvipartxj

+partvjpartximinus2

3gij~nablamiddot~v

3sumij=1

(partvi

partxj+partvj

partximinus2

3gij~nablamiddot~v

)partvjpartxi

(III39a)

)= T~nabla middot

(κ~nablaTT

(~nablaT)2 (III39b)

(κ~nablaTT

3sumij=1

(partvi

partxj+partvj

partximinus 2

)(partvipartxj

3gij~nabla middot~v

(~nabla middot~v

)2+ κ

(~nablaT)2

T 2 (III40a)

parts(t~r)

~nablaT (t~r)

T (t~r)

]2+ κ(t~r)

[~nablaT (t~r)

]2T (t~r)

(III40b)

One thus findsdS

CHAPTER IV

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= minusρg

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= minus mg

kBTP (~r)

P (~r) = P (z) = P 0 exp

(minusmgzkBT

)= T d

ρdP (IV4)

~nabla[w(~r)

ρ(~r)+ Φ(~r)

]= ~0 (IV5)

that isw(z)

enthalpy density

w = e+ P =5

2nkBT + nkBT =

That isw

dT (z)

dz= minus mg7

[w(~r)

ρ(~r)

[s(~r)

ρ(~r)

] (IV6)

fluid 1

fluid 2 S

fluid 1

fluid 2 solid body

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

[s(t~r)

ρ(t~r)

]minus ~nabla

[w(t~r)

ρ(t~r)

part~v(t~r)

partt+ ~nabla

[~v(t~r)2

2+w(t~r)

ρ(t~r)+ Φ(t~r)

[s(t~r)

ρ(t~r)

] (IV9)

~v(~r)2

2+w(~r)

~nabla middot[(

ρ+ Φ

P (~r)

IV22 a

Figure IV2

2+ gzA =

2+ gzB

v2B = v2

A + 2gh

vB =radic

IV22 b

Venturi effect

s-v1 -v2

Figure IV3

P 1minusP 2

s2minus 1

IV22 c

Pitot tube

Obull bullIbull

OprimebullA

-manometer

-bullAprime

PO + ρ~v2

2= PAprime + ρ

~v2Aprime

PI minus PB = ρ~v2

Remarks

IV22 d

Magnus effect

~v0~ωC

Γ~γ(t) =

intS~γ

[~nablatimes~v(t~r)

]middot d2~S =

DΓ~γ(t)

Dt= 0 (IV14b)

DΓ~γDt

part~γ(t λ)

[part2~γ

part~γ

partλmiddot(part~v

partt+sumi

part~v

partxipartγi

DΓ~γDt

part~v

partλmiddot ~v +

part~γ

partλmiddot[part~v

DΓ~γDt

DΓ~γ(t)

Dt= minus

∮~γ

~nablaP (t~r)

~nablatimes(~nablaP

ρ2 (IV16)

part~ω(t~r)

]= minus

ρ(t~r)2 (IV20)

(~nabla middot ~ω

)~v minus

(~v middot ~nabla

)~ω minus

(~nabla middot~v

part~ω(t~r)

]~ω(t~r)minus

]~v(t~r) = minus

]~ω(t~r)minus

ρ(t~r)2

D~v ~ω(t~r)

]~ω(t~r)minus

]~v(t~r) (IV22a)

Dtequiv D~ω(t~r)

]~v(t~r) (IV22b)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r)minus

ρ(t~r)2 (IV23)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r) (IV24)

[~ω(t~r)

ρ(t~r)

]= ~0 (IV25)

δ~(t)δ~(t+ δt)

~v(t~r

+ δ~ (t)) δt

~v(t~r)δt

[δ~(t)middot~nabla

]~v(t~r)δt

D~vδ~(t)Dt

= ~0 (IV26)

]~ω(~r)minus

]~v(~r)

]ρ(~r)

Remarks

partt+ ~nabla

[~nablaϕ(t~r)

+ Φ(t~r)

= minus 1

partt+ ~nabla

[~nablaϕ(t~r)

+P (t~r)

ρ+ Φ(t~r)

= ~0 (IV29)

or equivalently

minuspartϕ(t~r)

partt+

[~nablaϕ(t~r)

+P (t~r)

ρ+ Φ(t~r) = C(t) (IV30)

Laplace equation(t)

4ϕ(t~r) = 0 (IV31)

K(t) (IV32b)

Remarks

IV42 a

(IV33)

IV42 b

bull~r0

bull~r1

bull~r2

-~γ0rarr1

-~γ prime

0rarr1

6~γ0rarr2

6~γ primeprime

0rarr2

6~γ prime

0rarr2

Figure IV7

(1minus λ)

(IV35)

Remarks

IV43 a

party vy(x y) =

partψ(x y)

partx(IV36)

(~x(λ)

party= 0

4ψ(x y) = 0 (IV37a)

(lii)Stromfunktion

partφ(x y)

partx=partψ(x y)

[= minusvx(x y)

partφ(x y)

[= minusvy(x y)

] (IV38)

IV43 b

3333333

33333 33

33333333

333333

Figure IV8

~v(x y) =(

φ(z) = minus Q2π

w(z) =Q

~v(r θ) =Q

2πr~er (IV42c)

θ (IV42d)

φ(z) =iΓ

~v(r θ) =Γ

2πr2~eθ (IV43b)

ϕ(r θ) = minus Γ

2πθ ψ(r θ) =

2πlog r (IV43c)

micro eiα

w(z) =micro eiα

~v(r θ) =micro

ϕ(x y) =microx

x2 + y2 (IV44d)

φ(z) = limεrarr0

)minus log

)] (IV45)

w(z) =

infinsump=0

(IV46a)

infinsump=1

p aminuspminus1

2 z0 isin C (IV47a)

35 and 1

IV43 c

φ(z) = minusvinfin

) (IV48a)

(r minus R2

)sin θ (IV48b)

w(z) = vinfin

(1minus R2

) (IV49a)

~v(r θ) = vinfin

[(1minus R2

)cos θ~er minus

)sin θ

] (IV49b)

P (~r) +1

2ρv2infin

(liii)Staupunkte

]= Pinfin +

2πlog

R (IV50a)

)cos θ minus Γ

(r minus R2

)sin θ +

2πlog

R (IV50b)

w(z) = vinfin

(1minus R2

)minus iΓ

2πz (IV51a)

~v(r θ) = vinfin

[(1minus R2

)cos θ~er minus

r2minus Γ

2πrvinfin

)sin θ

] (IV51b)

IV43 d

Z = f (z) = z +R2

z(IV52)

)cosϑ+ i

(Rminus

CHAPTER V

ρ(t~r) = ρ0(t~r) + δρ(t~r) (V1a)

P (t~r) = P 0(t~r) + δP (t~r) (V1b)

~v(t~r) =~v0(t~r) + δ~v(t~r) (V1c)

V1 Sound waves 71

ρ(t~r) = ρ0 + δρ(t~r) (V2a)

P (t~r) = P 0 + δP (t~r) (V2b)

~v(t~r) = ~0 + δ~v(t~r) (V2c)

|δρ(t~r)| ρ0 |δP (t~r)| P 0 (V2d)

partδρ(t~r)

partt+ ρ0

]= 0 (V3a)

[ρ0 + δρ(t~r)

]partδ~v(t~r)

]δ~v(t~r)

partδρ(t~r)

partt+ ρ0

ρ0partδ~v(t~r)

)= P 0 +

(partPpartρ

(partPpartS

(partPpartN

(partPpartρ

c2s equiv

(partPpartρ

minusω ρ0

c2s~k 2 minusωρ0

)(δρ

~k middot δ~v

(partPpartρ

= minus V 2

(partPpartV

= minus V 2

[minusNkBT

V 2+NkB

(partT

V1 Sound waves 73

P = minus(partU

= minus5

(partT

ie NkB

(partT

= minus2P5

= minus2

leading to c2s =7

∣∣~k∣∣∣∣δ~v∣∣ = c2s

(~k middot δ~v

equation(25)

part2ρ(t~r)

partt2minus c2

s4ρ(t~r) = 0 (V9a)

part2ϕ(t~r)

partt2minus c2

s4ϕ(t~r) = 0

(lv)Verduumlnnung

V2 Shock waves 75

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (V10a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

]+partδP (t x)

partx= 0 (V10b)

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (V11a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

partx= 0 (V11b)

partv(t x)

partt=

dv(ρ)

partρ(t x)

parttresp

partv(t x)

partx=

dv(ρ)

partρ(t x)

[dv(ρ)

minus cs(ρ)2

partρ(t x)

partx= 0

dv(ρ)

dρ= plusmncs(ρ)

ρ (V12)

v(ρ) =

int ρ

cs(ρprime)

ρprimedρprime

partρ(t x)

partt+[v(ρ(t x)

(ρ(t x)

)]partρ(t x)

partx= 0 (V13)

t4 gt t3

t3 gt t2

t2 gt t1

t1 gt t0

V2 Shock waves 77

+minus gminus (V14)

where g+

ρ v1]]

= 0 (V15a)[[TTTi1]]

= 0 foralli = 1 2 3 (V15b)[[(1

2ρ~v2 + e+ P

]]= 0 (V15c)

[[v2]]

= 0 resp[[

ρ+minus 1

ρminus

) (V18)

]2=j21

=Pminusminus P +

ρminusminus ρ+

ρminusρ+

=Pminusminus P +

ρminusminus ρ+

ρminusρ+

Pminusminus P +

ρminusminus ρ+

ρminus

]2 (partPpartρ

ρminusρ+

=ρminusρ+c2s

[(v1)minus

]2 ρ+

ρminusc2s

2(v1)2 +

e+ Pρ

]]=j21

minus 1

ρ2minus

)+e+ + P +

ρminus= 0

Pminus minus P +

ρminus

ρ+(V19a)

Pminus + P +

ρ+minus 1

ρminus

V3 Gravity waves 79

minus partϕ(t x z)

partt+

[~nablaϕ(t x z)

+P (t x z)

partx2+part2

partz2

]ϕ(t x z) = 0 (V20b)

∣∣∣∣z=0

= 0 (V21a)

(lxi)Schwerewellen

minuspartϕ(t x z)

=Dδh(t x)

UsingD

Dt=part

partt+ vx

partx=part

parttminus partϕ

partx this gives[

partϕ(t x z)

partz+partδh(t x)

partϕ(t x z)

]z=h0+δh(tx)

= 0 (V21b)

P(t x z=h0+δh(t x)

)= P 0 (V21c)

partt+

[~nablaϕ(t x z)

]z=h0+δh(tx)

d2f(z)

dz2minus k2f(z) = 0

V3 Gravity waves 81

Linear waves

minus partϕ(t x z)

partt+

P (t x z)

ρ+ gz =

ρ+ gh0 (V24)

partϕ(t x z)

∣∣∣∣z=h0

+partδh(t x)

partt= 0 (V25a)

∣∣∣∣z=h0

partt2+ g

partϕ(t x z)

cϕ =ω

radicg

k and cg =

dω(k)

radicg

cϕ = cg =radicgh0

δh(t x) =1

partϕ(t x z)

∣∣∣∣z=h0

vx(t x z) =kg

cosh(kz)

sinh(kz)

x(t) = x0 +kgδh0

cosh(kz)

δh0 cosh(kz)

z(t) = z0 +kgδh0

sinh(kz)

δh0 sinh(kz)

[x(t)minus x0]2

cosh2(kz)+

[z(t)minus z0]2

sinh2(kz)=

[kgδh0

ω2 cosh(kh0)

sinh(kh0)

V3 Gravity waves 83

partt= P 0 + ρg

[h0 minus z + δh0

cosh(kz)

V32 Solitary waves

]~v(t~r) = minus1

vz(t x z=0) = 0 (V27c)

)=partδh(t x)

partt+ vx(t~r)

partδh(t x)

partx (V27d)

P(t x z=h0+δh(t x)

)= P 0

(t x z=h0+δh(t x)

tlowast equiv t

tc xlowast equiv x

Lc zlowast equiv z

δhc vlowastx equiv

vxδhctc

vlowastz equivvz

ρ δhcLct2c

= 0 (V28a)

partxlowast (V28b)

partzlowast (V28c)

partδhlowast

P lowast =gt2cLc

Fr equivradicLcg

P lowast =1

Velocity potential

partvxpartt

(vxpartvxpartx

+ vzpartvzpartx

partδh

partx= 0 (V29)

V3 Gravity waves 85

part2ϕ

partx2+part2ϕ

partz2= 0 (V30)

znϕn(t x) (V31)

infinsumn=0

zn[part2ϕn(t x)

partx2+ (n+ 1)(n+ 2)ϕn+2(t x)

(n+ 1)(n+ 2)

part2ϕn(t x)

part2ϕ0(t x)

partx2+z4

part4ϕ0(t x)

partx4+

partx+z2

part3ϕ0(t x)

partx3minus z4

part5ϕ0(t x)

partx5+

partx2minus z3

part4ϕ0(t x)

partx4+

part2u(t x)

partx+z3

part3u(t x)

partx (V34a)

partt= δ

partφ(t x)

partt (V34b)

partvx(t x)

partt+

partφ(t x)

partx= 0 (V34c)

part2u(t x)

partt2minus δ

part2u(t x)

partx2= 0 (V35)

radicgh0(Lctc)

partvx(t x z)

partt+ ε

[vx(t x z)

partvx(t x z)

partx+ vz(t x z)

partvz(t x z)

]+partφ(t x)

partx= 0 (V36)

instead of Eq (V29)

)= u(t x)minus δ2

part2u(t x)

partx2 (V37a)

)= minusδ

[1 + εφ(t x)

]partu(t x)

partx+δ3

part3u(t x)

partx3 (V37b)

partu(t x)

parttminus δ2

part3u(t x)

partu(t x)

partx+partφ(t x)

partx= 0 (V38)

partφ(t x)

partt+ δεvx

(t x z=δ(1 + εφ)

)partφ(t x)

partφ(t x)

partt+ εu(t x)

partφ(t x)

]partu(t x)

partxminus δ2

part3u(t x)

partx3= 0 (V39)

partu(t x)

partt+partφ(t x)

partx= 0

partφ(t x)

partt+partu(t x)

partx= 0

partu(t x)

partt+partu(t x)

partx= 0 (V40)

V3 Gravity waves 87

partφ

partt+partφ

partx+ ε

partu(ε)

partx+ δ2partu(δ)

partx+ 2εφ

partφ

partxminus δ2

part3φ

partx3= 0

partφ

partt+partφ

partx+ ε

partu(ε)

partt+ δ2partu(δ)

partt+ εφ

partφ

partxminus δ2

part3φ

partx2 partt= 0

[partu(ε)(t x)

partx+

2φ(t x)

partφ(t x)

]+ δ2

[partu(δ)(t x)

partxminus 1

part3φ(t x)

partx3

u(ε)(t x) = minus1

4φ(t x) + C(ε)(t) u(δ)(t x) =

part2φ(t x)

partx2+ C(δ)(t)

partφ(t x)

partt+partφ(t x)

partx+

2εφ(t x)

partφ(t x)

partx+

6δ2 part

3φ(t x)

partx3= 0 (V42)

partφ(τ ξ)

partτ+

2εφ(τ ξ)

partφ(τ ξ)

partξ+

6δ2 part

3φ(τ ξ)

partξ3= 0 (V43)

partφ(τ ξ)

partτ+ 6φ(τ ξ)

partφ(τ ξ)

partξ3= 0 (V44)

φ(τ ξ) =φ0

] (V45a)

δh(t x) =δhmax

radic3δhmax

[xminusradicgh0

δhmax

] (V45b)

δhmax=1 t = t0

δhmax= 025 t = t0

δhmax= 025 t = t1

δh(t x)

CHAPTER VI

partρ(t~r)

partt= 0 (VI1a)

parte(t~r)

] (VI1c)

]~v(~r) = minus1

partv(y)

partx= 0 (VI3a)

v(y)partv(y)

partx~ex = minus1

ρ~nablaP (~r) + ν

d2v(y)

dy2~ex (VI3b)

partP (~r)

partx= η

d2v(y)

dy2 (VI4)

d2v(y)

dy2= 0

v(y=0) = 0 v(y=h) = |~u|

~v(~r) =y

h~u for 0 le y le h

[partvi(~r)

partxj+partvj(~r)

partxi

0 0 minusPinfin

d2 ~Fs(~r)

d2S= ~Ts(~r) =

[ 3sumij=1

3sumij=1

)~ei =

η |~u|h

minusPinfin0

6yP 1 P 2

v(y) = minus 1

∆PLy2 + γy + δ

v(y=0) = 0 v(y=h) = 0

v(y) =1

[y(hminus y)

P 1 P 2-z

partP (~r)

partx=partP (~r)

party= 0

partP (~r)

partz= η

[part2v(r)

partx2+part2v(r))

party2

[d2v(r)

partP (~r)

partz= minus∆P

dr= minus∆P

ηL (VI7)

dχ(r)

dr+χ(r)

v(r) =∆P4ηL

(a2 minus r2

)for r le a (VI8)

∆P4ηL

(a2r minus r3

)dr = 2πρ

∆P4ηL

4=πρa4

∆PL (VI9)

〈v〉 equiv 1

0v(r) 2πr dr =

2v(r=0)

~rlowast equiv ~r

Lc ~vlowast equiv

vc tlowast equiv t

(VI10)

Re equiv ρvcLcη

=vcLcν

(VI12)

~v(t~r) = vc~f1

) P (t~r) = P 0 + ρv2

Fr equiv vcradicgLc

(VI14)

[minus 1

(~Ω0times~r

Ek equiv η

ρΩL2c

(VI15)

Ro equiv vcΩLc

(VI16)

Ro = Re middot Ek

VI31 a

Stokes equation

(~v middot ~nabla

[~nablatimes ~c(~r)

]= ~nabla

4P (~r) = 0 (VI20)

VI31 b

) This also holds

~vinfin ~er~eϕ

4[~nablatimes~u(~r)

]= ~0 (VI22a)

]= minus~nabla

[4f(r)

]times~vinfin

])times~vinfin = ~0

])only has a com-

(~nabla[4f(r)

])= ~nabla

(4[4f(r)]

) so that the

4[4f(r)] = 0

4 =part2

partr2+

partrminus `(`+ 1)

4f(r) =d2f(r)

r(VI24a)

2r (VI24b)

)middot~vinfin

]minus C

r~vinfin = minusB

rminus C

r~vinfin

(~er middot~vinfin

r3minus C

R3minus C

)~vinfin +

R3minus C

)(~er middot~vinfin

)~er = ~0

[~vinfin +

(~er middot~vinfin

)~er]minus R3

[~vinfin minus 3

(~er middot~vinfin

)~er] (VI25)

P (~r) =3

~er middot~vinfinr2

+ const

~v(~r) =~vinfin +R3

[~vinfin minus 3

(~er middot~vinfin

D = microkBT =kBT

6πRη

partv(t y)

partt= ν

part2v(t y)

party2 (VI27a)

limyrarrinfin

v(t y) = uf1

radicνt) ie write

v(t y) = uf2

2radicνt

v(t y) = uf

2radicνt

)(VI28)

f(ξ) = 0 (VI29b)

int ξ

0eminusυ

2dυ (VI31)

int infinξ

eminusυ2dυ (VI32)

v(t y) = u

[1minus erf

2radicνt

)] (VI33)

radicνω

partvxpartx

+partvyparty

= 0 (VI35a)

vxpart

partx+ vy

)vx = minus1

partPpartx

(part2

partx2+part2

party2

)vx (VI35b)(

vxpart

partx+ vy

)vy = minus1

partPparty

(part2

partx2+part2

party2

)vy (VI35c)

xlowast equiv x

Lc ylowast equiv y

δl(VI36)

δl Lc (VI37)

vlowasty equivvyu

u vinfin (VI39)

Re equiv Lcvinfinν

(VI40)

+Lcδl

vinfin

= 0 (VI41a)

+Lcδl

= minuspartP lowast

partxlowast+

) (VI41b)

+Lcδl

v2infin

= minusLcδ

partP lowast

partylowast+

vinfin

) (VI41c)

vinfin= 1 (VI42)

Re= 1 (VI43)

δl =LcradicRe

= 0 (VI45a)

= minuspartP lowast

(VI45b)

partP lowast

(~v middot ~nabla

)~v = 1

2~nabla(~v2)

part~ω(t~r)

]= minus

ρ(t~r)2+ ν4~ω(t~r) (VI46)

D~ω(t~r)

]~v(t~r)minus

]~ω(t~r)minus

ρ(t~r)2+ ν4~ω(t~r) (VI47a)

D~v ~ω(t~r)

Dt= minus

]~ω(t~r)minus

ρ(t~r)2+ ν4~ω(t~r) (VI47b)

~ω(t=0~r) =Γ0

2πrδ(z)~ez (VI48)

(~ω middot ~nabla

partωz(t r)

[part2ωz(t r)

partr2+

partωz(t r)

] (VI50)

ωz(t r) =Γ0

νt(VI51)

f(ξ) = C eminusξ4

that isωz(t r) =

νtC eminusr

2(4νt) (VI53)

Γ(t R) =

~v(t~r) middot d~=

int 2π

2(4νt)]

ωz(t r) =Γ0

4πνteminusr

2(4νt) (VI55)

radicν

Γ(t R) = Γ0

[1minus eminusR

2(4νt)] (VI56)

~v(t~r) =Γ0

[1minus eminusr

2(4νt)]~eθr (VI57)

where |~eθ| = r

∣∣δ~v(t~r)∣∣ cs (VI58b)

c2s equiv

(partPpartρ

(VI58c)

partρ(t x)

partt+ ρ(t x)

partδv(t x)

partx+ δv(t x)

partρ(t x)

partx= 0 (VI59a)

ρ(t x)

[partδv(t x)

partt+ δv(t x)

partδv(t x)

partx+

3η + ζ

)part2δv(t x)

partx2 (VI59b)

partδρ(t x)

partt+ ρ0

partδv(t x)

partx= 0 (VI60a)

ρ0partδv(t x)

partx+

3η + ζ

)part2δv(t x)

partx2 (VI60b)

ν equiv 1

3η + ζ

) (VI61)

ρ0partδv(t x)

partt+ c2

partδρ(t x)

partx= ρ0ν

part2δv(t x)

partx2 (VI62)

part2δv(t x)

partt2minus c2

part2δv(t x)

partx2= ν

part3δv(t x)

part2δρ(t x)

partt2minus c2

part2δρ(t x)

partx2= ν

part3δρ(t x)

ω2 = c2sk

ω2r minus ω2

i + 2iωrωi = c2sk

ωi = minus1

2νk2 (VI66)

ω2r = c2

sk2 minus 1

4ν2k4 (VI67)

c2ϕ = c2

s minus1

4ν2k2 (VI68)

Remarks

s one finds

ω2 c2sk

4ω2τ2

)hArr cϕ equiv

8ω2τ2

ki νω2

(VI69)

((ωτν)2

CHAPTER VII

VII11 a

Q = minusπρa4

∆PL (VI9)

Ediss = minus1

∆PL〈v〉 =

8ν〈v〉2

a2(VII1)

〈v〉 =Q

πa2ρ= minusa

VII11 b

reads [29]

Nsumn=1

~v(n)(t~r) asymp 1

int t+T 2

(part~v(t~r)

]~v(t~r)

VII13 a

[part~v

)~vprime (VII9a)

partvi

)vi = minus1

dPdximinus

3sumj=1

dvprimeivprimej

dxj+ ν4vi (VII9b)

Dtequiv part

(ρ~v)

SSSij equiv 1

dximinus 2

) (VII12c)

(ρ~v2

[P~v +

(TTTR minus 2ηSSS

)middot~v]

+(TTTR minus 2ηSSS

) SSS (VII13)

ρP~v +

)middot~v]

) SSS (VII14a)

or component-wise

Dt= minus

3sumj=1

ρP vj +

3sumi=1

3sumij=1

)SSSij (VII14b)

condition (VII7)

VII13 b

3sumij=1

lang3sum

(VII15)

ρP~v +

)middot~v]

) SSS (VII16a)

or component-wise

Dt= minus

3sumj=1

ρP vj +

3sumi=1

(vprimeivprimej

3sumij=1

Ediss =

lang3sum

vprimeivprimejSSSij

rangsim v3

Lc (VII17)

part~vprime(t~r)

DTTTijRDt

dTTTijRdxk

)minus(TTTikR

dxk+TTTjkR

)minus2η

dvprimei

dxkdvprimej

part~v(t~r)

P (t~r)

[~vprime(t~r)]2

3sumij=1

SSSijSSSij

νeff =

minussumij

vprimeivprimejSSSij

2sumij

SSSijSSSij

2νsumij

SSSijSSSij

2sumij

SSSijSSSij= ν

∣∣∣∣ (VII22)

VII23 k-model

12 (VII23)

Ediss = Ckprime32`m

VII24 (k-ε)-model

Ediss(t~r) (VII24)

[part~vprime

)~vprime]

) (VII25a)

partvprimei

)vprimei = minus1

dP prime

)vi minus d

) (VII25b)

2~vprime2

Dkprime

Dt= minus

3sumj=1

ρP primevprimej +

3sumi=1

primeij)]minus

3sumij=1

(dvprimei

dvprimej

dximinus 2

minus3sum

3sumij=1

VII32 a

κ(n)i1i2in

κij(X) =XiXj

3sumk=1

εijkXk

partκij(X)

partXi= 0

dκ(X)

VII32 b

bull f(0) = 1

Xrarrinfinf(X) = 0

LI equivint infin

0f(X) dX (VII30)

f(X) Xrarr0

1minus 1

`2T =[vprime(t~r)

]2[dvprime(t~r)dx

]2 (VII32)

[vprime(t~r)]2

Xrarr0

[vprime(t~r)]2

prime(t~r)]

[δvprime(`K)]2

`2Kequiv lim

Xrarr0

[δvprime(X)]2

[dvprime(t~r)

(VII34)

[δvprime(X)]2

2[vprime(X)]2sim

Xrarr0

(K41-1)

[δvprime(X)]2 =

radicνEdiss Φ

)for X LI with `K =

(VII35)

(νEdiss

(K41-2)

B(2)x23

SE(k) = CK Ediss

I k `minus1K =

(VII37)

(EdissX

bull Frisch [35]

CHAPTER VIII

Dρ(t~r)

ρ(t~r)D~v(t~r)

[η(t~r)SSS(t~r)

]+ ~nabla

] (VIII1b)

ρ(t~r)D

[s(t~r)

ρ(t~r)

]= ~nabla middot

2η(t~r)

ζ(t~r)

T (t~r)

]2 (VIII1c)

part~v(t~r)

]~v(t~r) = minus 1

ρ(t~r)D

[s(t~r)

ρ(t~r)

)= T d

)minus P d

)= cP dT (VIII4)

α equiv κ

ρcP (VIII5)

DT (t~r)

Dt=partT (t~r)

]T (t~r) = α4T (t~r) (VIII6)

Pr equiv ν

α=ηcP

κ(VIII7)

α(V ) equiv minus1

(partρ

(VIII11)

minus 1

~nablaP (t~r)

[1 + α(V )Θ(t~r)

one finds

minus 1

~nablaP eff(t~r)

part~v(t~r)

]~v(t~r) = minus

~nablaP eff(t~r)

partΘ(t~r)

VIII21 a

Remarks

VIII21 b

τQ = CR2

δΘ =partΘ

partzδz =

partΘ

partzvτQ = C

α(V )∆T

minus 4π

3R3δρg =

3ρ0gv

α(V )∆T

d (VIII15)

3ρ0gv

α(V )∆T

dgt 6πRρ0νv hArr

α(V )∆T gR4

ανdgt

ναgt

C= Rac

partvzpartt

partδΘ

parttminus ∆T

dv0 hArr ∆T

dv0 minus

(γ + αk2

)δΘ0 = 0

)(γ + αk2

)minusα(V )∆T

dg = 0 (VIII18)

Rac =α(V )∆T g d

αν= π4

Raminus RacRac

α+ νk2

v prop(

Raminus RacRac

)βwith β =

2(VIII19)

of part of Ref [43]

CHAPTER IX

IX11 a

(c n(x)~N (x)

)(IX1)

Nmicro(x) =

(c n(x)~N (x)

Remarks

micro(x) = 0

IX11 b

Global formulation

Figure IX1

d3σmicro equiv1

bull 1

Remarks

P equivintΣ

= (c 0 0 0) (IX10)

umicro(x)∣∣R

(γ(x)c

γ(x)~v(x)

) (IX11)

with γ(x) = 1radic

(IX12)

P (x)∣∣LR(x)

= P (ε(x) n(x)) (IX13)

Remarks

cN0(x)

∣∣LR(x)

=N0(x)u0(x)

[u0(x)]2

∣∣∣∣LR(x)

=N0(x)u0(x)

g00(x)[u0(x)]2

∣∣∣∣LR(x)

=Nmicro(x)umicro(x)

uν(x)uν(x)

∣∣∣∣LR(x)

uν(x)uν(x)=

N(x) middot u(x)

[u(x)]2 (IX14)

[uρ(x)uρ(x)]2

∣∣∣∣LR(x)

N0(x)∣∣LR(x)

= cn(x) ~(x)∣∣LR(x)

= ~0 (IX16a)

T 00(x)∣∣LR(x)

= ε(x)

T ij(x)∣∣LR(x)

T i0(x)∣∣LR(x)

= T 0j(x)∣∣LR(x)

= 0 foralli j = 1 2 3

ε(x) 0 0 0

0 P (x) 0 00 0 P (x) 00 0 0 P (x)

(IX16c)

]umicro(x)uν(x)

c2(IX17b)

N(x) = n(x)u(x) (IX18a)

]u(x)otimes u(x)

c2 (IX18b)

Remarks

)-tensor

umicro(x)uν(x)

c2(IX19b)

ν(x)uν(x) = 0

P (x) =1

IX32 a

]= 0 (IX22)

n)umicrouν

c2= Pgmicroν +

N(numicro)

]uνc2

[T (sumicro)+micro

N(numicro)

]dmicrouνc2

]umicrouνc2

Ndmicro(numicro)

]uνc2

N(numicro)

]uν dmicrouν

c2+ (dmicroP )

umicrouνuνc2

+[T dmicro(sumicro)

]uνuνc2

IX32 b

)+~v middot ~nabla

γu middot d

u middot d(s

nu middot dsminus s

n2u middot dn =

(u middot dsminus s

nu middot dn

γu middot d

γ(x) sim|~v|c

~v(x)2

(~v(x)4

) (IX24)

umicro(x) sim|~v|c

) (IX25)

IX33 a

Nmicro sim|~v|c

(n cn~v

part(n c)partt

3sumi=1

part(n vi)

partxi=partnpartt

IX33 b

ρc2 + e+ ρ~v2 +O(~v2

) (IX27a)

T 0j = T j0 = γ2(ρc2 + e+ P )vj

csim|~v|c

(|~v|3

) (IX27b)

c2sim|~v|c

)= TTTij +O

) (IX27c)

part(ρcvj)

partt+

3sumi=1

partTTTij

partxi+O

)=part(ρvj)

partt+

3sumi=1

partTTTij

partxi+O

) (IX28)

0 =part(ρc)

partt+

3sumi=1

part(ρcvi)

partxi+O

2ρ~v2 +O

2ρ~v2

(|~v|5

)one finds

2ρ~v2 + e

3sumj=1

partxj

2ρ~v2 + e+ P

that ispart

2ρ~v2 + e

)+ ~nabla middot

2ρ~v2 + e+ P

]asymp 0 (IX29)

IX33 c

part(sc)

partt+

3sumi=1

part(svi)

partxi=parts

or equivalently

N(x) = n(x)u(x) + n(x) (IX33b)

Tensor algebra

$microν(x) =1

[∆micro

]T ρσ(x) (IX38b)

Remarks

a[microν] equiv 1

∆ (microρ ∆ν)

σ minus1

3∆microν∆ρσ

)aρσ

(IX39)

(IX40a)

or equivalently

∆ρν(x)dmicroT

]umicro(x)dmicrou

respuν(x)dmicroT

]dmicrou

[uν(x)πmicroν(x)

]minus[dmicrouν(x)

[dmicrouν(x)

]πmicroν(x)

]dmicrou

micro(x)

dmicro

T (x)nmicro(x)

nablamicro(x)uν(x)

[microN(x)

] (IX43a)

3∆microν

)equiv SSSmicroν +

3∆microν

dmicro

T (x)nmicro(x)

]= minus$

microν(x)

[microN(x)

] (IX43b)

3∆microν(x)

[nablaρ(x)uρ(x)

nmicro(x) = κ(x)

[n(x)T (x)

ε(x)+P (x)

]2nablamicro(x)

[microN(x)

](IX44c)

d middot S(x) =$$$(x) $$$(x)

2η(x)T (x)+

Π(x)2

ζ(x)T (x)+

[ε(x)+P (x)

n(x)T (x)

]2 n(x)2

κ(x)T (x) (IX45)

or equivalently

T (x)nmicro(x) +

T (x)n(x) +

T (x)q(x) +

T (x)Q(x) (IX47a)

T (x)nmicro(x) +

T (x)qmicro(x) +

) (IX47c)

(IX48a)where

n(x)n(x)

T (x)Π(x)qmicro

N(x)minusα1(x)

T (x)$micro

ρ(x)qρN

N(x) equivNsumk=1

intuk(τ)δ(4)

(xminusxk(τ)

)d(cτ) (IXA1a)

or component-wise

intumicrok(τ)δ(4)

(xνminusxνk(τ)

cN0(t~r) =

Nsumk=1

) (IXA2a)

N i(t~r) =

Nsumk=1

)(IXA2b)

N0(t~r) = c

Nsumk=1

)dtk(τ)

dτd(cτ) = c

Nsumk=1

intδ(tminustk(t)

ie N0(t~r) = c

Nsumk=1

δ(3)(~xminus~xk(t)

TTT(x) equivNsumk=1

(xminusxk(τ)

)d(cτ) (IXA3a)

(xλminusxλ(τ)

Tmicro0(t~r) =

Nsumk=1

) (IXA4a)

T 0j(t~r) =Nsumk=1

p0k(t)v

jk(t)δ

(3)(~r minus ~xk(t)

)(IXA4b)

T ij(t~r) =Nsumk=1

pik(t)vjk(t)δ

(3)(~r minus ~xk(t)

)(IXA4c)

CHAPTER X

X21 Landau flow

[53 54]

Appendices

APPENDIX A

To be written

de = d

VdU minus U

VdS minus P

VdN minus TS

V 2dV +

dV minus microN

V 2dV = T d

)+ microd

APPENDIX B

B11 Vectors

~c = ci~ei (B1)

B12 One-forms

B13 Tensors

bull the(

linear) n-forms

bull Eventually(

)-tensors

For example the(

(mprime

nprime

n+nprime

Let TTT be a(mn

(mminus1nminus1

) which is

Tensor coordinates

B14 Metric tensor

gij = g(~ei~ej

(~ei~ej

(~ei~ei

arbitrary basis

ci = gijcj (B8b)

(m∓1nplusmn1

) respectively

Remarks

kprimej = δij

cjprime

= Λjprimei ci TTTj

prime1j

primem = Λj

APPENDIX C

Tensor calculus

166 Tensor calculus

part~ei(P )

~c(P ) = ci(P )~ei(P ) (C2)

part~c(P )

partxk=

dci(P )

dxk~ei(P ) (C3a)

dci(P )

dxk=partci(P )

part~c(P )

partxk=partci(P )

partxk~ei(P ) + ci

part~ei(P )

partxk=partci(P )

part~c(P )

partxk=partci(P )

dci(P )

dxk~ei(P )

)-tensor field

Γilk(P ) =1

2gip(P )

[partgpl(P )

partxk+partgpk(P )

partxp

Polar coordinates

part~er(r θ)

partxr= ~0

part~er(r θ)

partxθ=

r~eθ(r θ)

part~eθ(r θ)

partxr=

r~eθ(r θ)

part~eθ(r θ)

168 Tensor calculus

dxr=partcr

partxr= 0

dxr=partcθ

partxr+ Γθθrc

θ =sin θ

(minus sin θ)

dxθ=partcr

partxθ+ Γrθθc

dxθ=partcθ

partxθ+ Γθrθc

r = minuscos θ

rcos θ = 0

Scalar fields

scalar field f(P )

df(P )

dxk=partf(P )

partxk (C9)

One-forms

dhj(P )

dxk=parthj(P )

to be completed

attempt (C12)

APPENDIX D

D11 Definitions

f (z)minus f (z0)

z minus z0(D1)

D12 Some properties

f (x+iy) = P (x y) + iQ(x y) (D2)

partP (x y)

partx=partQ(x y)

partyand

partP (x y)

partx(D3)

= 0 (D4)

where z = xminus iy

4P (x y) = 0 4Q(x y) = 0 (D5)

on the domain

Cf (z) dz =

af(γ(t)

)γprime(t) dt (D6)

f (z) dz = 0 (D7)

f(z0) =1

z minus z0dz (D8)

D31 Taylor series

f (z) =

infinsumn=0

f (n)(z0)

n(z minus z0)n (D9)

f (n)(z0) =n

Definitions

Laurent series

f (z) =

D33 Singular points

Bibliography

174 Bibliography

176 Bibliography

Contents

Introduction












Streamlines

Material derivative

Mechanical stress


Fluids






Strain rate tensor



Kinematic criteria

Physical criteria









Local formulation














Isentropic fluid



Bernoulli equation





Potential flows





Sound waves




Shock waves



Gravity waves


Solitary waves




Plane Couette flow




Reynolds number





Boundary layer














Turbulent viscosity

Mixing-length model

k-model

(k-epsilon)-model













Conservation laws










Local rest frames














Landau flow

Bjorken flow

Appendices




Vectors

One-forms

Tensors

Metric tensor

Change of basis

C Tensor calculus









Definitions

Some properties


Series expansions

Taylor series


Singular points

Conformal maps

Bibliography

Documents

Elements of Hydrodynamics - uni-bielefeld.de