Lecture Notes on Non-Newtonian Fluids

Part I: Inelastic Fluids

Professor A. M. RobertsonDepartment of Mechanical EngineeringUniversity of Pittsburgh

Fall 2005

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Contents

1 Background Material on Continuum Mechanics 51.1 Description of Motion of Material Points in a Body . . . . . . . . . . . . . . . . . . . 51.2 Referential and Spatial Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Deformation Gradient of the Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Deformation Measures. Strain Measures . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6.1 Right Cauchy Green Deformation Tensor . . . . . . . . . . . . . . . . . . . . 101.6.2 Relative Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Velocity Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7.1 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8 Special Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8.1 Rigid Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8.2 Isochoric Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.3 Irrotational Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.4 Simple Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.5 Uniaxial Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.9 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.2 Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.9.3 Balance of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.4 Mechanical Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.10 Invariance Under Superposed Rigid Body Motions . . . . . . . . . . . . . . . . . . . 221.10.1 Relationship between kinematic variables in the original and superposed mo-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.10.2 Requirement: Invariance under a superposed rigid body motion . . . . . . . . 28

2 Non-Linear Viscous Fluids 312.1 Constitutive Assumption T = T (ρ, L) . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.1 Incompressible Inelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 Reiner-Rivlin Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.1 Thermodynamic Restrictions on φ1 and φ2 for Reiner-Rivlin Fluids . . . . . . 352.2.2 Restrictions on φ1, φ2 based on behavior of real fluids in simple shear . . . . 36

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2.3 Generalized Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.1 Power-Law Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.2 Prandtl-Eyring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.3 Powell-Eyring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.4 Cross Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.5 Yasuda Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.6 Bingham Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.7 Modified Bingham Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4 Steady, Fully Developed Flow of a Generalized Newtonian Fluid in a Straight Pipe . 452.4.1 Power-Law Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Bingham Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.3 Q as a function of κ and η(S) for unspecified generalized Newtonian fluid . . 482.4.4 Determination of η(S) from measured values of driving pressure drop and

flow rate in a straight pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.5 Cone and Plate Flow for a Generalized Newtonian Fluid . . . . . . . . . . . . . . . . 50

2.5.1 Small Angle Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.5.2 Finite Cone Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A The Relationship between Coordinate Free Notation and Indicial Notation 61

B Isotropic Tensors 63

C Isotropic Tensor Functions 67C.1 Scalar valued isotropic tensor functions . . . . . . . . . . . . . . . . . . . . . . . . . . 67C.2 Symmetric Isotropic Tensor Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 68

D Material Line Elements, Area Elements, Volume Elements 69

E Rectangular Coordinates 71

F Cylindrical Coordinates 75

G Spherical Coordinates 79

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Chapter 1

Background Material on ContinuumMechanics

1.1 Description of Motion of Material Points in a Body

We will first consider a physical body, for example a body of fluid, which we identify by the symbolB. We use the symbol Y to identify an arbitrary material particle in the body. The motion ofarbitrary material particles Y can then be described through a function,

x = χ(Y, t) (1.1)

where x is the vector identifying the spatial position at time t of the material particle identifiedby Y . In continuum mechanics, we are interested in describing the position in time of an uncount-able number of points, which cannot be identified using letters or numbers. With this in mind,we identify material points in the body by their position in some chosen reference configuration.Namely, we assume the body is smooth and can be embedded in a three dimensional Euclideanspace. The mapping of each material particle of the body to Euclidean 3-space at an arbitrary timet will be called the configuration of B at time t, denoted by κ(t). This mapping is assumed to beone-to-one, invertible and differentiable as many times as necessary for all time. The configuration,κ0, is a special configuration, chosen for the purpose of identifying material particles. For example,κ0 could be the configuration of the body at time zero.

We denote the region occupied by the entire body in this configuration as R0 with closedboundary ∂R0. An arbitrary material region within B in the configuration κ0 will be denoted asV0 (V0 ⊆ R0) with boundary ∂V0, Fig. 1.1. We denote the corresponding volume in the currentconfiguration κ(t) as V (V ⊆ R) with boundary ∂V .

An arbitrary material point identified by Y may also be identified by its position, denoted byX, in the reference configuration κ0. It is not necessary that the configuration of the body everhave coincided with the reference configuration. The motion of material points in the body B maythen be described through the relationship,

x = χκ0(X, t). (1.2)

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Figure 1.1: Schematic of notation used to identify regions of an arbitrary body B in the referenceand current configurations .

Figure 1.2: Schematic of position vector in reference and current configurations for a fixed materialpoint of an arbitrary body B.

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The vector x is the position at time t of an arbitrary material particle will then be identified byits position X in reference configuration κ. We will assume that the transformation (1.2) possessesan inverse,

X = χ−1κ0

(x, t). (1.3)

For our current purposes it suffices to define one reference configuration for the body, so in furtherdiscussions we will drop the subscript κ0 and it will be understood that the functions χ(X, t) andχ−1(x, t) depend on the choice of reference configuration. In the discussion of viscoelastic fluidswith fading memory, we will use the current configuration as the reference configuration. Therelevant kinematics will be discussed later in the text.

1.2 Referential and Spatial Descriptions

Field variables such as density, ρ, may be written as a function of X and t,

ρ = ρ(X, t). (1.4)

The function, (1.4) is called the referential or Lagrangian description of the density field. Whereas,(1.4) is sufficient to describe the density of all material points in the body, it is frequently of interestin classical fluid mechanics to have knowledge of field variables as a function of spatial positions xand time,

ρ = ρ(x, t). (1.5)

The function in (1.5) is called the spatial or Eulerian description of density and is independent ofinformation about the position of individual material particles. Rather, the function given in (1.5)gives the value of the density at a specified position in space at a given time t. The representations(1.4) and (1.5) are related through,

ρ = ρ(x, t) = ρ(χ(X, t), t) = ρ(X, t). (1.6)

1.3 Velocity and Acceleration

The velocity of a material particle is defined as

v =∂χ(X, t)

∂t(1.7)

and the acceleration,

a =∂2χ(X, t)

∂t2. (1.8)

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Figure 1.3: Relationship between Eulerian and Lagrangian representation of density

As for the density field, the relationship between the Eulerian representation and Lagrangian rep-resentation of the velocity field can be obtained using (1.2),

v = v(x, t) = v(χ(X, t), t) = v(X, t). (1.9)

1.4 Material Derivative

The material derivative of a field variable such as density, is defined as the partial derivative of thefunction with respect to time holding the material point fixed,

Dρ

Dt=

∂ρ(X, t)∂t

. (1.10)

Frequently, in fluid mechanics, the spatial description of the field variables is of interest rather thanthe material description. Using the chain rule, the material derivative can be written for the spatialformulation of a field variable. For example, in the case of the density field,

Dρ

Dt=

∂ρ(x, t)∂t

+ vi∂ρ(x, t)∂xi

. (1.11)

Using (1.11), we may obtain the acceleration (1.8) from the spatial representation of the velocityfield,

ai =∂vi(x, t)

∂t+ vj

∂vi(x, t)∂xj

. (1.12)

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1.5 Deformation Gradient of the Motion

The deformation gradient or displacement gradient of the motion relative to the reference configu-ration κ0 is denoted by F . It is a second order tensor defined by,

F =∂χ(X, t)∂X

. (1.13)

The components of F are then,

FiA =∂χi(X, t)∂XA

. (1.14)

When the deformation gradient is given as a second order tensor field defined on R, then thedeformation field can be determined from (1.14) which is a system of nine partial differentialequations for three unknowns xi. In order that there exists a solution, the integrability condition,

∂FiA∂XB

=∂FiB∂XA

(1.15)

must be satisfied. The deformation can then be determined to within an arbitrary constant, corre-sponding to an arbitrary rotation,

xi =∫

FiAdXA. (1.16)

To understand the physical significance of F it is helpful to consider the transformation of twomaterial points during the motion (1.2), which occupy positions Xo and X in the reference config-uration and positions xo and x in the current configuration, respectively. Using a Taylor’s seriesexpansion, we see that,

xi = xoi +∂χi(X, t)∂XA

|Xo(XA −XoA) + higher order terms. (1.17)

or in index free notation,

x = xo + F (Xo, t) · (X −Xo) + higher order terms. (1.18)

In which the distance between X and Xo goes to zero,

dx = F · dX, dxi = FiAdXA, (1.19)

where dx and dX are the infinitesimal vectors (x− xo) and (X −Xo), respectively in the limit ofinfinitesimal distance between points

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1.6 Deformation Measures. Strain Measures

1.6.1 Right Cauchy Green Deformation Tensor

We now consider the change in magnitude dx during the deformation. We’ll denote the magnitudeof dx and dX as ds and dS, respectively. Let m and M be the unit vectors tangent to dx and dX,respectively. The square of the magnitude of the infinitesimal vector dx is related to the square ofthe magnitude of dX through,

ds2 = |dx|2 = dx · dx = dxidxi

= FiAFiBMAMB dS2

= CAB(X, t)MAMB dS2,

(1.20)

where CAB are the components of the right Cauchy-Green deformation tensor:

C ≡ F T · F and CAB ≡ FiA FiB . (1.21)

Note that C is symmetric.

Physical Significance of Diagonal elements of C

Consider an infinitesimal element represented by dX in the reference configuration and dx in thecurrent configuration. If we now consider the special case where dX = dSe1, then it follows from(1.20), that

ds2

dS2 = C11. (1.22)

The ratio ds2/dS2 is typically called the stretch ratio and denoted by λ. We can describe the result(1.22) in words as,

C11 is equal to the stretch ratio squared of an infinitesimal material element

which was aligned with the e1 axis in the reference configuration.

The other diagonal elements can be interpreted in a similar way.

Physical significance of off-diagonal elements of C

Now consider two infinitesimal material elements corresponding to dX(1) = dS(1)M (1) and dX(2) =dS(2)M (2), where M (1) and M (2) are unit vectors. In the current configuration, these infinitesimalelements are denoted as dx(1) = ds(1)m(1) and dx(2) = ds(2)m(2), respectively. The angle betweenthese same material elements in the current configuration will be denoted as α. Therefore,

cosα =dx(1) · dx(2)

ds(1)ds(2)=

FiAFiBM(1)A M

(2)B dS(1)dS(2)

ds(1)ds(2)=

CABM(1)A M

(2)B

λ(1) λ(2). (1.23)

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Now consider the special case, where the infinitesimal elements are perpendicular in the referenceconfiguration, for example, M (1) = e1 and M (2) = e2, in which case, (1.23) reduces to,

cosα =C12√C11C22

. (1.24)

We can describe in words, the physical significance of

C12 provides information about the angle between two material elements

which were parallel to e1 and e2 in the reference configuration.

The other off-diagonal elements can be interpreted in a similar way. For example, if the anglebetween the elements remains unchanged (still 90o), then C12 = 0. If the angle decreases from 90o,then C12 will be a positive number. If the α ∈ (90o, 270o), then C12 will be negative.

Jacobian of the transformation

In future discussions, we denote the the Jacobian of the transformation (1.2) by J. Namely, theJacobian is equivalent to the determinant of F ,

J ≡ det∂χi(X, t)∂XA

= det (F ) . (1.25)

Since χ(X, t) is assumed to have an inverse,

0 < J < ∞. (1.26)

Recall from continuum mechanics that the relationship between the infinitesimal volume occupiedby material points in the current configuration dv is related to the infinitesimal volume occupiedby the same material points in the reference configuration dV is,

dv = J dV. (1.27)

It will be useful in the discussion of the governing equations to note the following result for thematerial derivative of the Jacobian of the transformation (1.2),

DJ

Dt= J divv, (1.28)

where v is the velocity vector.

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1.6.2 Relative Strain Tensor

When the motion is rigid, C is equal to the identity tensor. It is useful to define a relative measureof strain which vanishes for rigid motions:

E ≡ 12(C − I), (1.29)

where E, is often referred to as the Lagrangian strain. A second measure of relative strain, e,defined as

e ≡ 12(I − B−1) (1.30)

is referred to as the Eulerian strain where B is the left Cauchy-Green tensor

B ≡ F · F T and Bij ≡ FiA FjA. (1.31)

1.7 Velocity Gradient

We denote L as the gradient of the spatial form of the velocity vector, so that the components ofL with respect to rectangular coordinates are 1

Lij =∂vi(x, t)∂xj

. (1.32)

Recall that any second order tensor can be decomposed into the sum of a symmetric and skewsymmetric second order tensor. We can represent L in this way,

GLij =12

(∂vi∂xj

+∂vj∂xi

)+

12

(∂vi∂xj

− ∂vj∂xi

). (1.33)

We define the rate of deformation tensor, D, as the symmetric part of L and the vorticity tensor,W , as the skew-symmetric part of L,

Dij =12

(∂vi∂xj

+∂vj∂xi

), Wij =

12

(∂vi∂xj

− ∂vj∂xi

). (1.34)

Physical Significance of Diagonal elements of D

The physical significance of D can be studied by considering the rate of change in magnitude of aninfinitesimal material element dx of length ds. We first consider the rate of chance of ds2 which isequal to

D(ds2)Dt

= 2 dxiDdxiDt

. (1.35)

1Note that in some books, the gradient of a second order tensor is defined as the transpose of that used here.

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The rate of change in the infinitesimal material element dx is,

DdxiDt

=D

DtFiAdXA

=∂FiA∂t

dXA

=∂2χi(X, t)∂XA∂ t

dXA

=∂vi∂XA

dXA

=∂vi∂xj

FjA dXA

= Lijdxj

(1.36)

and so from (1.35) and (1.36) we find,

D(ds2)Dt

= 2Dijdxidxj. (1.37)

Thus,

Dds

Dt=

Dijdxidxjds

. (1.38)

We see from (1.38) that the rate of change of magnitude of an infinitesimal element which at timet is parallel to the e1 axis is

Dds

Dt= D11ds. (1.39)

D11 is the rate of change of ds divided by ds of a material element which at

time t is aligned with the e1 axis.

The other diagonal elements can be interpreted in a similar way.

Physical Significance of the off diagonal elements of D

Now consider two infinitesimal material elements x and y which intersect at angle α with lengths|dx| and |dy| respectively. Then,

cosα =dx · dy|dx| |dy| (1.40)

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and therefore,

D

Dtcosα =

D

Dt

(dx · dy|dx| |dy|

)

=D

Dt

(dx · dy) 1

|dx| |dy| − dx · dy|dx| |dy|

D

Dt

(|dx| |dy|)

=Dijdxidyj|dx| |dx| − dx · dy

|dx| |dy|D

Dt

(|dx| |dy|)(1.41)

In particular, consider (1.41) for two infinitesimal material elements, dx, dy which at time t areparallel to the base vectors e1 and e2. We see that the rate of change of the angle between thesetwo vectors at time t is

Dα

Dt= − D12. (1.42)

Therefore,

D12(x, t) is the rate of change of angle between the two infinitesimal vectors which

at time t are located at position x and parallel to base vectors, e1 and e2.

The other off-diagonal elements can be interpreted in a similar way. Notice that this interpretationof the components of D does not require knowledge of the behavior of specific material elements.Rather D(x, t) is related to the rate of change of material elements which at time t are located atposition x.

1.7.1 Vorticity

Recall the definition of the vorticity, ω is the curl of the velocity vector,

ω ≡ ∇× v, (1.43)

or in indicial notation 2,

ωi ≡ εijk∂vk∂xj

. (1.44)

The components of the vorticity vector are related to the components of the vorticity tensor through,

ωi = − εijkWjk, (1.45)

and

Wij = − 12εijk ωk. (1.46)

2Note that alternate definitions of vorticity are sometimes used. For example, sometimes the vorticity is definedas the negative of that given in (1.43). In other cases, the vorticity is taken to be twice that in (1.43).

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1.8 Special Motions

1.8.1 Rigid Motions

A rigid motion is one in which the distance between points remains constant. Therefore, for rigidmotions, the material derivative of ds is zero for all points in the body for all time during the rigidmotion. We see from (1.38) that for rigid motions, D must be identically zero at all points in thebody for all time. Alternatively, we see from (1.38) that motions in which D is zero at all pointsin the body for all time are rigid motions.

One can show that the most general rigid motion can be written as,

x = xo(t) + Q(t) ·X, (1.47)

where Q is a proper orthogonal second order tensor. Namely,

Q ·QT = I, and therefore, QT ·Q = I. (1.48)

In indicial notation,QijQkj = δik, and therefore, QjiQjk = δik. (1.49)

In preparation for considering the form of the velocity field for a rigid body motion, we will firstconsider the time derivative of Q. If we take the derivative with respect to time of equation, (1.48),we find,

dQ

dt·QT + Q · dQ

T

dt= 0, (1.50)

and so,dQ

dt·QT = − (

dQ

dt·QT )T (1.51)

We see from (1.51) that the quantity dQ/dt · QT is a skew symmetric tensor, which we denote asΩ, so

dQ

dt= Ω ·Q. (1.52)

Since Ω is skew symmetric there exists a vector which we will denote as c, where,

Ωij = εijkck. (1.53)

The velocity field for a rigid motion can be obtained by taking the material derivative of (1.47),

v = vo(t) + c× (x− xo), (1.54)

where vo is the time derivative of xo. If we take the curl of (1.54), we find that,

ω = 2 c. (1.55)

Therefore, the velocity field for the most general rigid body motion can be written as,

v = vo(t) +12ω × (x− xo). (1.56)

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1.8.2 Isochoric Motions

Isochoric motions are those in which the volume occupied by fixed material particles is unchangedduring the motion. A material does not have to be incompressible to undergo isochoric motions.As will be discussed in the next section, all motions undergone by incompressible fluids must beisochoric. We see from (1.27) that if a motion is isochoric then the value of J is one throughoutthe motion. In this case, we have from, (1.28) that the divergence of v is equal to zero,

∇ · v = 0 (1.57)

and hence the trace of the rate of deformation tensor is zero.

1.8.3 Irrotational Motions

Irrotational motions are those for which the vorticity is zero. From (1.46), the vorticity tensor Wis zero for irrotational motions.

1.8.4 Simple Shear

A flow field that is of great significance in fluid mechanics is simple shear flow, sometimes calledlineal Couette flow. In this flow, the velocity field is steady, fully developed and uni-directional.The magnitude of the velocity depends linearly on the spatial component with axis perpendicularto the direction of flow. For example, if rectangular coordinates xi are chosen such that the flowdirections is parallel to the x1 axis, then the orientation and origen of the coordinate system canbe chosen such that the velocity field can be written as,

v = γ x2 e1. (1.58)

Clearly simple shear is an example of an isochoric motion. It is possible to show (see later chapters)that simple shear flow can be generated between parallel plates for a wide class of fluids called simplefluids (see, e.g. [8]). As such, it has great value in the design of rheometers. For example, considersimple shear flow between two parallel plates separated by a distance h, when the upper plate ismoving with speed U in the e1 direction and the origin of the coordinate system coincides witha point on the bottom plate, Figure 1.4. Using the no-slip boundary conditions at the top andbottom surface it can be shown that γ = U/h. Using the result (1.7) and the initial conditionx = X at t = 0, we can show that the motion corresponding to (1.58) is,

x1 = X1 + γ X2 t, x2 = X1, x3 = X3. (1.59)

It is helpful for discussions in later chapters, to calculate several kinematic variables for simple shear.In particular, using the results (1.58) and (1.59) for simple shear, it follows that the correspondingvalues of D,W are,

[D] =γ

2

0 1 0

1 0 00 0 0

, [W ] =

γ

2

0 1 0

−1 0 00 0 0

. (1.60)

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Figure 1.4: Schematic of velocity field in simple shear flow.

while F, C and E are

[F ] =

1 γt 0

0 1 00 0 1

, [C] =

1 γt 0γt 1 + γ2t2 00 0 1

, [E] =

12

0 γt 0γt γ2t2 00 0 0

.(1.61)

1.8.5 Uniaxial Extension

Another relatively simple flow field which is of particular interest for viscoelastic fluids is uniaxialextension. The Eulerian representation for the velocity field for this motion is,

v1 = ε x1, v2 = − ε

2x2, v3 = − ε

2x3. (1.62)

As can be seen from (1.62), uniaxial extension is an isochoric motion. Using the result (1.7) and theinitial condition x = X at t = 0, the position of material particles in this motion can be describedthrough,

x1 = X1 eεt, x2 = X2 e

−εt/2, x3 = X3 e−εt/2. (1.63)

Using the results (1.62) and (1.63), the corresponding values of D,W are,

[D] = ε

1 0 0

1 −1/2 00 0 −1/2

, [W ] =

ε

2

0 0 0

0 0 00 0 0

. (1.64)

while F, C and E are

[F ] =

eεt 0 0

0 e−εt/2 00 0 e−εt/2

, [C] =

e2εt 0 0

0 e−εt 00 0 e−ε

, (1.65)

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and

[E] =12

e2εt − 1 0 0

0 e−εt − 1 00 0 e−ε − 1

. (1.66)

1.9 Governing Equations

1.9.1 Conservation of Mass

The mass M of a fixed subset of material particles of the body occupying a region V(t) of R(t) is

M =∫V(t)

ρ dv, (1.67)

where ρ is the mass density of the fluid. The volume V(t) is a material volume of the fluid: thevolume occupied by a fixed set of material particles in the body, that may change in time. Theprinciple of conservation of mass is the postulate that the mass of this fixed set of material particlesdoes not change in time,

d

dt

∫V(t)

ρ dv = 0. (1.68)

Making use of the Transport Theorem, we can write the principle of conservation of mass withrespect to the spatial (Eulerian) representation of the field variables,

0 =∫V(t)

(∂ρ

∂t+ v · ∇ρ + ρ∇ · v

)dv, (1.69)

where v is the velocity vector. Making suitable assumptions about continuity of the field variablesand making use of the arbitrariness of the part of the body V(t), we obtain the local form of (1.69),

∂ρ

∂t+ v · (∇ρ) + ρ∇ · v = 0 (1.70)

or in indicial notation,

0 =∂ρ(x, t)∂t

+ vi∂ρ(x, t)∂xi

+ ρ∂vi(x, t)∂xi

. (1.71)

Implications of Conservation of Mass for Incompressible FluidsThe motion of an incompressible fluid is constrained to be isochoric, in which case the divergenceof the velocity vector must be zero for all motions. In this case, the conservation of mass reducesto,

∂ρ

∂t+ v · (∇ρ) = 0. (1.72)

18

From (1.72) we see that for an incompressible fluid, the material derivative of the density is alwayszero,

Dρ

Dt= 0. (1.73)

Note that the even thought the material derivative of the density it is zero, it is not necessarythat the density be constant in space. For example, a stratified fluid with density distribution,ρ = ρ0 + αy, where α is a constant, can experience simple shear, Section 1.8.4. It is easily seenthat this motion is isochoric, a necessary condition for an incompressible fluid to experience thismotion. In addition,

Dρ

Dt=

∂ρ

∂t+

∂ρ

∂xivi = 0 (1.74)

for the stratified fluid and given motion and therefore conservation of mass is satisfied.

1.9.2 Balance of Linear Momentum

The postulate of balance of linear momentum is the statement that the rate of change of linearmomentum of a fixed mass of the body is equal to the sum of the forces acting on the body. Theseforces can show up as body forces, or as forces due to stress vectors acting on the surface of thebody,

d

dt

∫V(t)

ρv dv =∫V(t)

ρ bdv +∫dV(t)

t da (1.75)

where b is the body force per unit mass, t(x, n, t) is the surface force acting on the body in thecurrent configuration per unit area of δP and n is the unit normal to the surface δP at x at timet. Recall that the stress vector is in general a function of the surface unit normal n. The first andsecond integrals on the right hand side of (1.75) represent the contributions due to body forcesand to surface forces respectively. Note that stress vector depends on position, time and the unitnormal to the surface at x. The first and second integrals on the right hand side of (1.75) representthe contributions due to body forces and to surface forces respectively. Recall, Cauchy’s Lemmawhich follows from suitable smoothness assumptions on the function t(x, n, t) and the balance oflinear momentum

t(x, n, t) = −t(x,−n, t). (1.76)

In addition, the existence of a second order tensor, T can be shown, where,

t = t(x, n, t) = T (x, t)n. (1.77)

The second order tensor T is called the Cauchy stress tensor. Significantly, T is independent of then.

Making use of the divergence theorem as well as the relationship between the stress tensor andthe stress vector, (1.75) can be written as,

d

dt

∫V(t)

ρv dv =∫V(t)

(ρb + T · n) dv. (1.78)

19

The local form of the equation of linear momentum can be obtained form (1.78) by using thetransport theorem and making suitable assumptions about the continuity of the field variables,

ρ

(∂v

∂t+ (∇v) · v

)= ∇ · T + ρb, (1.79)

or, using index notation,

ρ

(∂vi∂t

+ vj∂vi∂xj

)=

∂Tij∂xj

+ ρbi. (1.80)

It is often convenient to represent T as the sum of a deviatoric and spherical part.

T = τ + t I, (1.81)

where,

τii = 0, and t =13Tkk. (1.82)

The tensor τ is often referred to as the deviatoric part of T and tI as the spherical part. When theCauchy stress tensor is decomposed in this way, −t is often called the pressure and denoted by p.Using the decomposition, (1.81), the balance of linear momentum can be written as,

ρ

(∂v

∂t+ (∇v) · v

)= −∇p + ∇ · τ + ρb. (1.83)

Alternatively,

ρ

(∂vi∂t

+ vj∂vi∂xj

)= − ∂p

∂xi+

∂τij∂xj

+ ρbi. (1.84)

For compressible fluids, p is a thermodynamic pressure. An equation of state relating pressure toother thermodynamic variables such as mass density and temperature will be necessary. For in-compressible fluids, p is a mechanical pressure arising from the constraint of incompressibility. Noequation of state is necessary, rather, p will be determined as part of the solution to the governingequations and boundary conditions.

1.9.3 Balance of Angular Momentum

In the absence of body couples, the integral form of the balance of angular momentum is

d

dt

∫V(t)

ρx× v dv =∫V(t)

ρ x× b dv +∫dV(t)

x× t da. (1.85)

Making use of the balance of linear momentum, it can be shown that (1.85) reduces to the require-ment that the Cauchy stress tensor must be symmetric,

Tij = Tji. (1.86)

20

1.9.4 Mechanical Energy Equation

It is sometimes useful to consider the Mechanical Energy Equation which is not derived from theequation of balance of energy, rather it is obtained from the equation of linear momentum makinguse of the equation of conservation of mass. If we take the inner product of the velocity vector andthe equation of linear momentum (1.80) we obtain,

12ρD

Dt(vivi) =

∂Tij∂xj

vi + ρbivi. (1.87)

It is sometimes useful to have an integral form of this equation. This can be obtained by integrating(1.87) over a fixed part of the fluid (same material particles) occupying region V(t) with surface∂V(t), to obtain,

12d

dt

∫V(t)

ρv · vdv =∫∂V(t)

t · vda −∫V(t)

tr(T ·DT )dv +∫V(t)

ρb · vdv. (1.88)

where we have made use of the divergence theorem and the conservation of mass. For futurereference, we introduce the following notation for terms in (1.88):

K ≡ 12

∫V(t)

ρv · vdv = kinetic energy in V(t)

Rc ≡∫δV(t)

t · vda = rate of work done by surface forces on the boundary δV(t)

Rb ≡∫V(t)

ρb · vdv = rate of work done on the material volume V(t) by body forces

(1.89)The scalar, tr(T · DT ) = TijDij is the rate of work by stresses per unit volume of the body andis called the stress power. If we consider a part of the body V(t) in the current configuration, wecan hypothesize the existence of a scalar called the specific internal energy, u = u(x, t) (internalenergy per unit mass). The internal energy for the part V(t) of the body, will then be

∫V(t)

ρudv. (1.90)

Recall that the kinetic energy of the part V(t)of the body is

∫V(t)

12ρv · vdv. (1.91)

Heat may enter the body through the surface dV(t) of the body with outward unit normal n. Itcan be shown, that this heat flux can be represented as the scalar product of a vector q and thenormal to the surface n. Where q · n positive is associated with heat leaving the surface and q · n

21

negative is associated with heat entering the surface. In addition, heat may enter the body as aspecific heat supply, r = r(x, t): the heat entering the body per unit mass per unit time. Thereforethe rate of heat entering the part V(t) of the body is

−∫dV(t)

q · nda +∫V(t)

ρrdv. (1.92)

The balance of energy is a statement that the rate of increase in internal energy and kinetic energyin the part V(t) of the body is equal to the rate of work by body forces and contact forces plusenergies due to heat entering the body per unit time. We can write this statement as

d

dt

∫V(t)

ρ(u+12v · v)dv =

∫dV(t)

t · vda +∫V(t)

ρb · vdv −∫dV(t)

q · nda +∫V(t)

ρrdv (1.93)

where the first integral on the left hand side is the rate of work by contact forces, the secondintegral is the rate of work by body forces. Making suitable assumptions about continuity of thefield variables, we can obtain the local form of (1.93),

ρ

(Du

Dt+ v · Dv

Dt

)= T : D + v · (∇ · T ) + ρv · b−∇ · q + ρr. (1.94)

Using results from the mechanical energy equation, we can rewrite (1.94) as

ρDu

Dt= T : D − ∇ · q + ρr, (1.95)

or in indicial form

ρDu

Dt= TijDji − ∂qi

∂xi+ ρr. (1.96)

1.10 Invariance Under Superposed Rigid Body Motions

A variety of criteria have been used to assess whether a proposed constitutive equation is ‘rea-sonable’. In addition to assessing how well a constitutive equation models the behavior of a realfluid for a chosen flow, criteria such as invariance of the constitutive equation under a superposedrigid motion (or, alternatively, material frame indifference3) [30, 14]; thermodynamic considera-tions [28, 29, 18]; stability of the rest state and stability of some steady flows [10, 17, 16, 13]; andwell-posedeness of classes of initial value problems [6, 15, 12] have been used. For example, basedon thermodynamic restrictions, we require that the viscosity of a Newtonian fluid be non-negative.

In this section we focus on the assumption that a constitutive equation be invariant under asuperposed rigid body motion to be physically reasonable . In some works, authors apply the re-lated but different physical requirement that the form of the constitutive equation be independent

3Truesdell and Noll [30] include an interesting discussion of the history of these two principles.

22

of observer. This assumption is often referred to as the “principle of material frame indifference”.In both cases, an assumption is necessary to define how the stress vector (or stress tensor) in theoriginal and superposed motion (or between the two observers) are related. In the following subsec-tion, we discuss the kinematics associated with a general superposed motion without introducinginvariance assumptions. In the subsequent subsection, we provide a mathematical statement of therequirement that a constitutive equation be invariant under a superposed rigid body motion. Thediscussion of invariance under superposed rigid body motion follows that of Green and Naghdi, [14]and Naghdi (1972,pp. 484-486) [20]).

Superposed Rigid Body Motions

In this section, we consider two motions of a body: an arbitrary motion

x = χ(X, t) (1.97)

and a second motion which differs from the first by a superposed rigid body motion

x+ = χ+(X, t). (1.98)

Namely, the position of a material point which at time t is at position x in the first motion, is atposition x+ in the second motion, Figure 1.5. The spatial description of (1.98) is then,

x+ = χ+(x, t). (1.99)

Significantly, the motion (1.99) is not an arbitrary second motion. Since the second motion differs

Figure 1.5: Identification of points in the body in the current and superposed configurations.

from the first motion by a rigid body motion, the function χ+(x, t) must belong to a restricted class

23

of motions. In a rigid motion the distances between material points in the body are kept constant.In this section, we shown that the most general form for this function is,

x+ = χ+(x, t) = c(t) + Q(t) · x, (1.100)

where Q(t) is a proper orthogonal second order tensor,

Q ·QT = I, detQ = +1. (1.101)

First consider, two material points which we can identify as P and Q. These points can also beidentified by their respective positions, X and Y , in the reference configuration. The positions ofpoints P and Q in the current configuration, denoted by x and y respectively, are

x = χ(X, t), y = χ(Y , t). (1.102)

The positions occupied by these materials points at time t in the superposed motion, (1.99), arethen,

x+ = χ+(x, t). y+ = χ+(y, t). (1.103)

By defintion, the distance between two arbitrary points in the superposed motion must be the sameas the distance between those in the motion, (1.97),

(x+ − y+) · (x+ − y+) = (x− y) · (x− y), (1.104)

namely,(χ+(x, t)− χ+(y, t)) · (χ+(x, t)− χ+(y, t)) = (x− y) · (x− y). (1.105)

Equation (1.105) can be written in indicial notation as,

(χ+i (x, t)− χ+

i (y, t))(χ+i (x, t)− χ+

i (y, t)) = (xi − yi)(xi − yi). (1.106)

Recalling that x and y are independent, we can differentiate (1.106) with respect to xj and thenwith respect to yk to obtain,

∂χ+i (x, t)∂xj

∂χ+i (y, t)∂yk

= δjk. (1.107)

Equation (1.107) can be written in coordinate free notation as,

(∂χ+(x, t)

∂x

)T

· ∂χ+(y, t)∂y

= I. (1.108)

Equivalently, (∂χ+(x, t)

∂x

)T

=

(∂χ+(y, t)

∂y

)−1

. (1.109)

24

Equation (1.109) must hold for arbitrary choices of x and y in the body. Since the left side ofequation (1.109) is indpendent of y and the right side is independent of x, the second order tensorson the left and right hand side of (1.109) must be functions of t only, which we denote as thetranspose of Q(t),

Q(t) =

(∂χ+(x, t)

∂x

), or Qij(t) =

∂χ+i (x, t)∂xj

. (1.110)

Since (1.110) must hold for all x in R,

Q(t) =

(∂χ+(y, t)

∂x

)(1.111)

and therefore from (1.108)

QT ·Q = I, and detQ = ±1, (1.112)

orQliQlj = δij . (1.113)

We see from (1.112) that Q is an orthogonal tensor. Each superposed motion, (1.99), must includethe particular case in which χ+(x, t) = x. It can be seen from (1.110), that for this case, Q = Iand detQ = 1. Since the motions under consideration are continuous, we must always have,

detQ = 1. (1.114)

The equation (1.110), may be integrated with respect to x to yield,

x+ = χ+(x, t) = c(t) + Q(t) · x, or x+i = χ+

i (x, t) = ci(t) + Qik(t)xk, (1.115)

where c is a vector function of time. Therefore, the most general superposed rigid body motion isdescribed by (1.100). From (1.115), we also have

x = QT · (x+ − c), xk = Qlk(x+l − cl). (1.116)

So far, we have shown that necessarily a superposed rigid body motion is represented by (1.100).We now show that it is also sufficient, namely, that all motions of the form (1.100) preserve distancebetween points.

|x+ − y+|2 = (x+ − y+) · (x+ − y+) =(Q · (x− y)

) · (Q · (x− y))

= (x− y) · (QT ·Q · (x− y))

= (x− y) · (x− y)

= |x− y|2.

(1.117)

25

1.10.1 Relationship between kinematic variables in the original and superposedmotions

We now consider other kinematical quantities under the superposed rigid body motion, (1.100),Figure 1.6. The velocity vector v, transforms as,

v+ =∂χ+(X, t)

∂t=

dc(t)dt

+dQ

dt· x + Q · v. (1.118)

We define Ω as

Ω =dQ

dt·QT . (1.119)

Recall, that Q ·QT = I, so

dQ

dt·QT + Q · dQ

T

dt= 0 (1.120)

and hence, using (1.119) and (1.120), we find,

Ω + ΩT = 0 (1.121)

and therefore, Ω is skew-symmetric. Using the definition of Ω, (1.119), we can write the expressionfor v+ as,

v+ =dc(t)dt

+ Ω ·Q · x + Q · v, (1.122)

and

v+i =

dci(t)dt

+ ΩilQlmxm + Qilvl. (1.123)

For future use, we now derive the relationship between L+ and L,

L+ij =

∂v+i

∂x+j

=∂v+

i

∂xk

∂xk

∂x+j

=[ΩilQlmδmk + Qil

∂vl∂xk

]∂xk

∂x+j

= [ΩilQlk + QilLlk]Qjk

= Ωij + QilLlkQjk

(1.124)

or in coordinate free notation,

L+ = Ω + Q · L ·QT . (1.125)

Using (1.125), it can be shown that

26

Figure 1.6: Representative kinematic variables for the original motion (F,D) and the superposedmotion (F+, D+) as well as the relationship between these kinematic variables.

D+ = Q ·D ·QT ,

W+ = Ω + Q ·W ·QT .(1.126)

Recalling (1.14) and (1.98), it is clear that

F+ =∂χ+(X, t)

∂X, or F+

iA =∂χ+

i (X, t)∂XA

, (1.127)

Using (1.127) and (1.100),

F+iA =

∂χ+i (x, t)∂xj

∂xj∂XA

,

= Qij FjA

(1.128)

and therefore,F+ = Q · F. (1.129)

Using (1.129) it can also be shown that,

J+ = J. (1.130)

Additionally, the density ρ and the normal to a surface n, can be shown to transform as,

ρ+ = ρ

n+ = Q · n. (1.131)

27

A scaler, vector or second order tensor which transform as (1.131)1, (1.131)2 and (1.126)1, respec-tively are called objective.

1.10.2 Requirement: Invariance under a superposed rigid body motion

In this section, we consider the mechanical behavior of a fixed material undergoing two differentmotions: x = χ(X, t) and a second motion which is related to that at any arbitrary time t differsfrom the first by only a superposed rigid body motion. Since the material is the same for bothmotions, the form of the constitutive equation will remain the same. For example, suppose thematerial is linear in the symmetric part of the velocity gradient,

T = αI + βD. (1.132)

Then for the superposed rigid body motion,

T+ = αI + βD+. (1.133)

More generally, if T = T (D), thenT+ = T (D+). (1.134)

Note that the functional form does not change.In this section, we are going to impose a requirement on the relationship between t and t+ and

hence on T and T+. In particular, we might expect that the magnitude of the stress vector isunchanged by a superposed rigid body motion ( t+ and t to have the same magnitude) and that t+

and t to have the same orientation relative to n+ and n,respectively. Based on these expectations,we introduce the following assumption

t+ = Q · t. (1.135)

Using (1.135) and (1.131), it can be shown that,

|t+| = |t|, t+ · n+ = t · n. (1.136)

The next step is to determine the relationship between T and T+. Recall that,

t = t(x, t; n) = T (x, t) · n. (1.137)

Therefore,t+ = t+(x+, t; n+) = T+(x+, t) · n+. (1.138)

In addition, recall the relationship between the outward unit normal n to a material surface inthe current configuration and the corresponding outward unit normal in the superposed motion,n+ = Q · n. Therefore,

t+ = T+ · n+ = T+ ·Q · n. t+i = T+ij n

+j = = T+

ijQjknk. (1.139)

28

Using the assumption (1.135) and (1.137), it then follows that,

t+ = Q · t = Q · T · n, and t+i = Qijtj = QijTjknk. (1.140)

Combining (1.139) and (1.140), we have that

(T+ ·Q−Q · T ) · n = 0, and

(T+ijQjk −QijTjk

)nk = 0. (1.141)

Since (1.141) must hold for all n and the expression in brackets is independent of n, we can concludethat the expression in brackets is equal to zero. Hence,

0 = T+ ·Q−Q · T= T+ ·Q ·QT −Q · T ·QT

= T+ −Q · T ·QT

or

0 = T+ijQjk −QijTjk

= T+ijQjkQlk −QijTjkQlk

= T+il −QijTjkQlk

(1.142)

and therefore,

T+ = Q · T ·QT , and T+

il = QijTjkQlk . (1.143)

29

30

Chapter 2

Non-Linear Viscous Fluids

In this chapter, we start with the constitutive assumption that the stress tensor depends on thefluid mass density and the velocity gradient L.

T = T (ρ, L). (2.1)

Inherent in the form of the stress tensor, (2.1) is the assumption that the current state of stressdepends only on the velocity gradient at the current time and not on any previous deformation thefluid might have undergone. Using invariance of the stress tensor under a superposed rigid bodymotion and a representation theorem for symmetric isotropic tensor functions, it is shown that themost general form of (2.1), which does not violate invariance requirements is

T = φ0I + φ1D + φ2D2 (2.2)

where in general φ0, φ1, φ2 depend on ρ and the three principal invariants of D. 1

Similarly, we can start by considering incompressible fluids for which the extra stress tensoris a function of the velocity gradient. After imposing invariance requirements and using the samerepresentation theorem, we can deduce that the most general form of the Cauchy stress tensor thatsatisfies invariance can be written as,

T = αI + φ1D + φ2D2 (2.3)

where α is a Lagrange multiplier arising from the incompressibility constraint and φ1 and φ2 dependonly on the second and third principal invariants of D. Such incompressible fluids are typicallycalled Reiner-Rivlin fluids. The Navier-Stokes fluid is a special Reiner-Rivlin fluid with φ2 equalto zero and φ1 constant. The behavior of Reiner-Rivlin fluids with non-zero values of φ2 in simpleshear does not match experimental results on real fluids [1]. In addition, the dependence on thevalue of IIID is often considered negligible [1]. For this reason, attention is typically confined toa special class of Reiner-Rivlin fluids called generalized Newtonian fluids. These are Reiner-Rivlinfluids with φ2 zero and φ1 dependent only on the second principal invariant of D. In the latter partof this chapter, specific generalized Newtonian fluids are considered and the behavior is contrastedand compared with experimental results.

1Recall the principal invariants of a second order tensor A are IA = tr A, IIA = 1/2((tr A)2 − tr (A2)), IIIA =

detA.

31

2.1 Constitutive Assumption T = T (ρ, L)

In this section, we start with the constitutive assumption that the stress tensor depends on thefluid mass density and the velocity gradient L. Namely, we assume, T = T (ρ, L). We then obtainthe most general function which satisfies invariance requirements. Using the relationship

L = W + D (2.4)

we can rewrite the constitutive assumption as

T = T (ρ, D,W ). (2.5)

If the constitutive equation (2.5) is a valid one, then it must hold for all motions, in particular, itmust hold for a superposed rigid motion,

T+ = T (ρ+, D+, W+). (2.6)

Recalling that,T+ = Q · T ·QT , (2.7)

and using (2.6)T (ρ+, D+, W+) = Q · T (ρ, D,W ) ·QT . (2.8)

Recalling,ρ+ = ρ, D+ = Q ·D ·QT , W+ = Q ·W ·QT + Ω, (2.9)

we can write (2.8) as

T (ρ+, Q ·D ·QT , Q ·W ·QT + Ω) = Q · T (ρ, D,W) ·QT . (2.10)

Now consider a superposed motion, for which at some arbitrary time t, Q = I, and Ω = 0. In thiscase, (2.10) becomes,

T (ρ, D,W +Ω) = T (ρ, D,W), (2.11)

for arbitrary choices of the skew-symmetric tensor Ω. It then follows that the response function T

cannot depend on W , and so (2.5) is reduced to

T = T (ρ, D), (2.12)

32

and must satisfyT (ρ, Q ·D ·QT ) = Q · T (ρ, D) ·QT , (2.13)

for all proper orthogonal Q. Then, from (2.13), we can conclude that T (ρ, D) must be a symmetricisotropic tensor function of D, (see, for example, Appendix C). A representation theorem fortensors (Appendix C), states that the most general second order isotropic tensor function T (ρ, D)for symmetric second order tensor D, has the form

T = φ0 I + φ1D + φ2D2, (2.14)

where in general φ0, φ1, φ2 are functions of the three principal invariants of D as well as the massdensity ρ.

2.1.1 Incompressible Inelastic Fluids

In the case, where the fluid is incompressible, the density of a fixed material point is constant. Thestress tensor is therefore independent of the density ρ,

T = T (D). (2.15)

The Cauchy stress tensor is determined by D only to within a stress N which does no work inany motion satisfying the constraint of incompressibility. We can rewrite (2.15) as the sum of theindeterminant stress N and an extra stress τ , (see, for example, Truesdell [31], pp. 70-73),

T = N + τ(D). (2.16)

As discussed previously in Section 1.9, the expression for the rate at which stresses do work perunit volume, P, is

P = tr(T ·DT ). (2.17)

Recall, the scalar quantity tr(A · BT ) = AijBij satisfies the requirements for an inner product oftwo arbitrary second order tensors, A,B. We will use the notation (A,B) to denote this innerproduct. Namely,

(A,B) ≡ tr(A ·BT ). (2.18)

Using this notation, the condition of incompressibility can be written as (D, I) = 0. Since theindeterminate stress does no work on the constrained motions,

0 = (N,D) for all motions in which(D, I) = 0. (2.19)

We can state the conditions on N given in (2.19) as, “N must be orthogonal to all D which areorthogonal to I”. It therefore follows, that N must be a scalar multiple of I,

N = λI, (2.20)

33

where λ is a Lagrange multiplier. Therefore the Cauchy Stress, for an incompressible fluid takesthe form,

T = λI + τ(D). (2.21)

The most general form for τ(D) can be deduced in essentially the same way we deduced theform of the stress tensor for a compressible fluid. For an incompressible fluid, the most generalrepresentation for the extra stress, τ(D) is

τ = φ0 I + φ1D + φ2D2. (2.22)

The first principal invariant of D is 0 for isochoric motions, so φ0, φ1, φ2 are functions of only IIDand IIID. Without loss in generality, we can absorb φ0 into the Lagrange multiplier λ and write(2.22) as

Reiner Rivlin Fluids

T = αI + φ1D + φ2D2

where φ1 = φ1(IID, IIID) and φ2 = φ2(IID, IIID)

α is a Lagrange multiplier arising from the incompressibilityconstraint.

(2.23)

It remains to specify the two material functions φ1(IID, IIID) and φ2(IID, IIID). The Lagrangemultiplier α will be determined from solving the governing equations subject to the boundary con-ditions. Incompressible fluids for which the stress tensor is related to the velocity gradient through(2.23) are called Reiner-Rivlin fluids.

2.2 Reiner-Rivlin Fluids

As just stated, Reiner-Rivlin fluids are incompressible fluids with constitutive equation,

T = α I + φ1D + φ2 D2, (2.24)

where φ1(IID, IIID) and φ2(IID, IIID). We have not yet considered specific forms for the materialfunctions, φ1(IID, IIID) or φ2(IID, IIID). In this section, we discuss some restrictions on thesematerial functions which can be motivated by thermodynamic considerations. In addition, we willcompare the behavior of the Reiner-Rivlin fluid in simple shear with the behavior of real fluids tomotivate restricting our attention in the remainder of this chapter to a special case of Reiner-Rivlinfluids with φ2 equal to zero and φ1 a function only of IID. Fluids in this subclass of Reiner-Rivlinfluids are called generalized Newtonian fluids.

34

2.2.1 Thermodynamic Restrictions on φ1 and φ2 for Reiner-Rivlin Fluids

From the Second Law of Thermodynamics we require that the stress power be non-negative,

T : D ≥ 0. (2.25)

This requirement can be used to deduce restrictions on the choices of the material functions φ1, φ2,[28, 29, 7, 18]. For a Reiner-Rivlin fluid,

T : D ≡ tr (T ·D) = tr(αD + φ1 D

2 + φ2 D3),

= tr(αD + φ1 D

2 + φ2 (−IIDD + IIIDI)),

= φ1 tr (D2) + 3φ2IIID,

= φ1 tr (D2) + 3φ2detD,

(2.26)

where we have made use of the Cayley-Hamilton Theorem and the condition of incompressibility.Therefore from thermodynamics, we require,

0 ≤ φ1 tr (D2) + 3φ2detD. (2.27)

The first term on the right hand side of (2.27) is non-negative while the second term may bepositive, negative or zero. In order to deduce further information from (2.27), we consider somespecial cases of Reiner-Rivlin fluids.

Case I: φ2 = 0, φ1 not necessarily constant.For the special case of a Reiner-Rivlin fluid with φ2 equal to zero, we see from (2.27) that necessarilyφ1 ≥ 0.

Case II: φ1, φ2 both constant.We now consider the particular case where φ1 and φ2 are both constants. In this case, φ1 and φ2

will be independent of the invariants of D, so any conclusions we make based on specific choices ofD will hold for all D.

First, consider the following choice for the rectangular components of D,

[D] =γ

2

0 1 0

1 0 00 0 0

. (2.28)

For the choice of D defined in (2.28), tr (D2) = γ2/2 and detD = 0. Therefore the inequality(2.27) takes the form

0 ≤ γ2

2φ1, (2.29)

35

and therefore for this case φ1 must be greater than or equal to zero. Since φ1 is a constant, thisresult holds not only for the choice (2.28) for D but for all D. Now, consider the following twochoices for the rectangular components of D,

[D] = γ

2 0 0

0 −1 00 0 −1

, [D] = γ

−2 0 0

0 1 00 0 1

. (2.30)

For both cases, tr (D)2 = 6γ2. For the first case, detD = 2γ3 while for the second case detD =−2γ3. Therefore the inequality (2.27) must satisfy both of the following,

0 ≤ φ1 + φ2γ,

0 ≤ φ1 − φ2γ.(2.31)

Since φ1 is always constant and γ is arbitrary, we see from that (2.31) that φ2 must be zero cor-responding to an incompressible Newtonian fluid with kinematic viscosity equal to 1/2φ1. If φ2 iszero, then from (2.27), φ1 must be greater than zero. This is equivalent to the familiar requirementthat the kinematic viscosity be positive. Therefore, we can conclude that the only physically rea-sonable case with φ1 and φ2 both constant is the case of an incompressible Newtonian fluid.

2.2.2 Restrictions on φ1, φ2 based on behavior of real fluids in simple shear

Consider flow of an incompressible Reiner-Rivlin fluid between two infinite parallel plates which isdriven by the motion of the upper plate moving at a constant speed U , Fig. 1.4. We have selecteda rectangular coordinate system with the x1 axis parallel to the plates and x2 perpendicular to theplates. The location of the x1 axis is chosen such that x2 = 0 is coincident with the bottom plate.As discussed in Section 1.58, we look for solutions of the form,

v = v(x2) = γ x2 e1 (2.32)

where γ is a constant. Recall that for (2.32) to be a solution it must satisfy the boundary con-ditions for the problem, the incompressibility condition, the equation of linear momentum, andthe constitutive equation for the stress tensor. The boundary conditions for this problem are theno-slip condition at the two plate surfaces,

v(x2 = 0) · e1 = 0, v(x2 = h) · e1 = U, (2.33)

and the condition of impermeability of both plates,

v(x2 = 0) · e2 = 0, v(x2 = h) · e2 = 0. (2.34)

36

We see that boundary conditions, (2.33) and (2.34) are met when γ = U/h. For a velocity fieldof the form, (2.32), the components of the rate of deformation tensor D are

[Dij] =[12(∂vi∂xj

+∂vj∂xi

)]

=γ

2

0 1 0

1 0 0

0 0 0

ij

. (2.35)

It is clear from (2.35) that for the motion, (2.32), the deformation tensor D is a constant secondorder tensor. The components of D2 are,

[D2

]= [DijDjk] =

γ2

4

1 0 0

0 1 0

0 0 0

, (2.36)

and the three principal invariants of D are,

ID = tr D = 0,

IID =12((tr D)2 − (tr (D)2)

)= − γ2

4,

IIID = detD = 0.

(2.37)

Using (2.37), we see that the incompressibility condition

0 =∂vi∂xi

= tr D (2.38)

is identically satisfied.The incompressible Reiner-Rivlin constitutive equation (2.24) can be split into the constraint

response and response that does work,

T = α I + τ(D), where τ(D) = φ1(IID, IIID)D + φ2(IID, IIID)D2. (2.39)

Using the results (2.36) with (2.39), it follows that,

[τ ] = = φ1γ

2

0 1 0

1 0 0

0 0 0

+ φ2

γ2

4

1 0 0

0 1 0

0 0 0

, (2.40)

Therefore, the only non-zero components of τ are,

τ12 = τ21 = φ1γ

2τ11 = τ22 = φ2

γ2

4. (2.41)

37

where φ1 and φ2 are both even functions of γ,

φ1 = φ1(−14γ2, 0), φ2 = φ2(−1

4γ2, 0). (2.42)

It follows from (2.41) and (2.22), that the only non-zero components of T are,

T12 = T21 = φ1γ

2, T11 = T22 = α + φ2

γ2

4, T33 = 0. (2.43)

We see from (2.43) that T12 is an odd function of γ while T11 and T22 are even functions of γ.It follows from (2.43) that to fully determine the solution for the Cauchy stress tensor, we need

to solve for α, which will in general require consideration of the balance of linear momentum as wellas the boundary conditions. We will consider this issue at a later time. Here we focus attentionon the viscometric functions. As will be discussed in more detail in the following sections, thefollowing three functions, determine the behavior of a certain category of fluids in class of flowscalled viscometric flows:

τ(γ) = T12

N1(γ) = T11 − T22

N2(γ) = T22 − T33

(2.44)

where τ, N1, N2 are referred to as the shear stress, first normal stress and second normal stress.Three material functions commonly used in the literature are defined relative to τ, N1, N2,

η(γ) = T12/γ

ψ1(γ) = N1/γ2 = (T11 − T22)/γ2

ψ1(γ) = N2/γ2 = (T22 − T33)/γ2

(2.45)

where η, ψ1, ψ2 are called the viscometric functions, referred to as the viscosity, first normal stresscoefficient and second normal stress coefficient. Most rheometers are based on viscometric flowsand are directed at measuring the viscometric functions.

It follows from (2.43) and (2.45), that for a Reiner-Rivlin fluid,

η =φ1

2, ψ1 = 0, ψ2 =

14φ2. (2.46)

and therefore,φ1 = 2η, φ2 = 4ψ2. (2.47)

However, there is no evidence of real fluids exhibiting a zero value for ψ1 and a non-zero value forψ2 [1]. For this reason, attention is typically confined to Reiner-Rivlin fluids with φ2 equal to zero.Therefore, no constitutive equation of the form T = T (D) will be suitable for describing real fluidswith nonzero normal stress coefficients. In summary, based on invariance requirements, thermo-dynamic considerations and the behavior of real fluids, the most general “reasonable” constitutiveequation of the form T = T (D) is,

T = αI + φ1(IID, IIID)D where φ1 ≥ 0. (2.48)

38

Recalling that the mechanical pressure is defined as p = −tr(T )/3, it follows from (2.48), that

α = −p. (2.49)

In addition, in simple shear (and other viscometric flows), the quantity IIID is identically zeroand therefore the functional dependence of φ1 on this variable cannot be determined from mostrheometers. In the remainder of this chapter, we will only consider fluids for which φ1 is assumedindependent of IIID. As discussed in [1], page 54, there is some evidence that this may be reasonablefor real fluids. After imposing this restriction on (2.48), we obtain the equation for a generalizedNewtonian Fluid (incompressible),

T = −pI + φ1(IID)D where φ1 ≥ 0. (2.50)

If we rewrite this last results in terms of the viscometric functions we obtain,

T = −pI + 2η(IID)D where η ≥ 0. (2.51)

Recalling that IID = −1/2tr(D2), it is clear that it will be convenient to replace IID with anon-negative quantity such as,

S = −4IID (2.52)

where the factor of two is introduced in order that S = γ2 in simple shear. Using this new notation,

Incompressible, generalized Newtonian Fluid

T = −pI + 2η(S)D

where S ≡ −4IID, p is the mechanical pressure,

From Thermodynamics: η ≥ 0.

(2.53)

2.3 Generalized Newtonian Fluids

In this section we first consider some commonly used examples of generalized Newtonian fluids.We then consider the behavior of these models in several important flow fields: (i) steady, fullydeveloped flow in a straight pipe and (ii) steady cone and plate flow and (iii) steady flow betweenconcentric cylinders.

39

2.3.1 Power-Law Fluid

The Power-Law Fluid is an example of one of the simpler generalized Newtonian fluids. It isparticularly popular because of the number of exact solutions which can be obtained for this model.The Cauchy Stress tensor for a Power-Law fluid is defined as,

Tij = −pδij + 2K |4IID|(n−1)/2Dij (2.54)

and therefore, η(S) is defined asη(S) = K S(n−1)/2 (2.55)

where n is called the power law index and K is called the consistency. When n is equal to one, theNavier-Stokes constitutive equation is recovered. When n is less than one, the constitutive equation

Power Law Fluids for n less than or equalto 1 (shear thinning or Newtonian)

Power Law Fluids for n greater than orequal to 1 (shear thickening or Newtonian)

is shear thinning while when n is greater than one the behavior is shear thickening. The viscosityin the limit of vanishing S and in the limit of S tending to infinity are,

• n < 1 : limS→0

η(S) = ∞, limS→∞

η(S) = 0,

• n > 1 : limS→0

η(S) = 0, limS→∞

η(S) = ∞.(2.56)

The unboundedness of the viscosity function and the lack of a non-zero viscosity at zero shear ratedoes not match experimental results for real fluids and so limits the applicability of the Power-Lawmodel.

40

2.3.2 Prandtl-Eyring Model

An alternative to the Power-Law model is the Prandtl-Eyring Model which tends to a constantviscosity in the limit of γ going to zero. However, the viscosity function tends to zero as γ tendsto infinity,

η(S) = η0sinh−1(

√Sλ)√

Sλ(2.57)

where η0 and λ are material constants and

limS→0

η(S) = η0, limS→∞

η(S) = 0. (2.58)

2.3.3 Powell-Eyring Model

The Powell-Eyring Model is a three constant model which displays a non-zero bounded viscosityat both the upper and lower limits,

η(S) = η∞ + (ηo − η∞)sinh−1(

√Sλ)√

Sλ(2.59)

where η0, η∞ and λ are material constants and

limS→0

η(S) = η0, limS→∞

η(S) = η∞. (2.60)

It is convenient to write rewrite (2.59) as,

η − η∞η0 − η∞

=sinh−1(

√Sλ)√

Sλ(2.61)

2.3.4 Cross Model

The Cross Model [9] is a four constant model which also displays a non-zero bounded viscosity atboth the upper and lower limits

η − η∞η0 − η∞

=1

1 + (K2|S|)(1−n)/2(2.62)

The notation used in (2.62) is chosen to fit the shear thinning case where n < 1,

limS→0

η(S) = η0, limS→∞

η(S) = η∞. (2.63)

This model is also suited for shear thickening, though in this case, the notation should be modified,since, while for n > 1,

limS→0

η(S) = η∞, limS→∞

η(S) = η0. (2.64)

41

Ellis Model

Frequently, the flow regimes of interest are such that it is not essential to specify a nonzero viscosityηinfty. Setting η∞ to zero in (2.62) yields the Ellis model [?],

η =η0

1 + (K2|S|)(1−n)/2. (2.65)

A major advantage of the Ellis model is that it is possible to obtain analytical solutions for somesimple flows (e.g. [3], [27]).

2.3.5 Yasuda Model

Yasuda proposed a model similar to the Cross model but with an extra material constant a withwhich to fit the data [32],

η − η∞η0 − η∞

=1

[1 + λaS](1−n)/a. (2.66)

Several special cases of the Yasuda model are notable including the Carreau model.

Carreau Model

If a is set to 2 in (2.66), the four constant Carreau model is obtained [4]. It also displays a non-zerobounded viscosity at both upper and lower limits,

η − η∞η0 − η∞

=1(

1 + λ2 S)(1−n)/2

. (2.67)

Comments regarding the limiting behavior for the Cross model hold for the Yasuda model aswell.

2.3.6 Bingham Fluids

There are many real fluids which if tested in simple shear, would appear not to flow at all untilthe magnitude of the applied shear stress surpassed a fixed finite value termed the yield stress.E. C. Bingham (1916) observed this type of behavior in paints. Many other real fluids show anapparent yield stress as well. One type of constitutive equation which is used to describe fluidswhich appear to display a yield stress is the Bingham model (see, e.g. [2] the review articleby Barnes, for a discussion of experimental evidence supporting the lack of existence of a trueyield stress in real materials.). In simple shear of a Bingham fluid, there is no motion unless themagnitude of the applied shear stress is above a threshold, τ

Y, in which case, the material behaves

like an incompressible Newtonian fluid, Fig. (2.1). It is a simple matter to provide a mathematicalstatement of this response,

τ12 < τY

=⇒ γ = 0

τ12 ≥ τY

=⇒ γ =1µ(τ12 − τY ).

(2.68)

42

Figure 2.1: Shear stress τ12 versus shear rate γ in simple shear for (i) Newtonian fluids (ii) Binghamfluids with yield stress τY .

When the flow is more complex than simple shear, a more general measure of the magnitudeof the shear stress is used. Namely, we cannot simply use τ12 = τ

Yas a criterion. Note that

this expression is even incorrect for simple shear under a change of basis. The condition, mustbe rephrased in terms of an invariant of the stress tensor. For a Bingham fluid, the more generalmeasure of the shear stress is the second invariant of the extra stress tensor, IIτ . The threedimensional representation of the Bingham model can be represented as (see, e.g. [11],[25]),√|IIτ | < τ

Y=⇒ Dij = 0

√|IIτ | ≥ τY =⇒

Dij =12µ

1− τY√

|IIτ |

τij

τij = 2

µ+

τY

2√|IID|

Dij

(2.69)

where τY and µ are material constants. Recall that in simple shear, Iτ , IIIτ are zero for a Binghamfluid. In simple shear, 2.69 can be used to obtain .

Early studies of this constitutive equation were carried out by [22, 23, 26, 25]. Other materi-als which seem to be well modeled by the Bingham constitutive model include pastes, margarine,mayonnaise, ketchup. and concentrated suspensions of particles. In simple shear, the shear stressis constant throughout the flow field. In more complex flows, such as fully developed flow in astraight pipe or flow between two coaxial cylinders driven by the relative rotation of the cylinders,we will see that the shear stress is not constant through out the flow field, and as a result, theremay be regions where the yield criterion is reached ( and the fluid is flowing) while in other regionsthe value of IIτ is below τY and the fluid does not flow (D = 0 in that region). Note that D = 0does not necessarily imply v = 0. A collection of analytical solutions for the Bingham model and

43

extensive reference list can be found in a review article by Bird, Dai and Yarusso [5].

2.3.7 Modified Bingham Model

The Bingham fluid equation is important because it appears to model some real fluids well andyet it is possible, in some cases, to obtain analytical solutions for the flow field, (for example,Poiseuille and Couette flow). However, this constitutive equation is in general challenging to modelnumerically due to the difficulty in tracking the yield surfaces in the flow field. Papanastasiou [24]modified the Bingham model (2.69) to eliminate the yield stress. Recall, that for Bingham fluids, atzero shear rate, the viscosity function η(IID) abruptly changes from an infinite value to a constant(finite) viscosity µ. Papanastasiou defined the viscosity function as an increasing function of IID,as

η(S) = µ +τ

Y

(1− e−nS/2

)S

, (2.70)

where n, µ, τY are material constants. The value of n determines the rate of exponential increase

Figure 2.2: Shear stress versus shear rate for simple shear of a Bingham fluid and modified Binghamfluid.

to the constant viscosity µ. For simple shear, it follows from (2.70),

τ12 =

µ+

τY

(1− e−n

√γ)

γ

γ. (2.71)

In the limit of n tending to infinity the shear stress for the Bingham fluid is obtained.

44

2.4 Steady, Fully Developed Flow of a Generalized Newtonian

Fluid in a Straight Pipe

In this section, we consider steady, fully developed flow in a straight pipe of circular cross section ofradius R driven by a pressure drop per unit length in the axial direction equal to κ. In particular,we look for velocity fields of the form,

v = w(r)ez, (2.72)

where we have used standard cylindrical coordinates in (2.72) such that the z-axis is aligned withthe pipe centerline. The volumetric flow rate, Q, for velocity fields of the form (2.72) is

Q = 2 π∫ R

0w(r)rdr. (2.73)

In many engineering applications, it is of interest to obtain a relationship between Q and ∂p/∂z

(denoted as -κ). If a closed form solution for the velocity field can be calculated for a givenviscosity function η(S), then it is relatively straightforward to calculate the corresponding flow rateas a function of κ from (2.73). In the first part of this subsection, we consider the Power-Law,Cross and Bingham fluids for which it is possible to obtain closed form solutions for w(r). For mostGeneralized Newtonian fluids, it is not possible to obtain a closed form solution for the velocityfield for steady, fully developed flow in a straight pipe. In the second part of this subsection weconsider the more general case and obtain an equation relating Q as a function of κ and η(S).In the last part of this subsection, we discuss the inverse problem of using measured relationshipsbetween driving pressure drop and volumetric flow rate to obtain the viscosity function η(S) for ageneralized Newtonian fluid.

Velocity fields of the form (2.72) identically satisfy the incompressibility condition and thecomponents of D and D2 in cylindrical coordinates are,

[D] = − γ

2

0 0 1

0 0 0

1 0 0

, [

D2]=

γ2

4

1 0 0

0 0 0

0 0 1

, (2.74)

where we have denoted −dw(r)/dr by γ. It follows from (2.74) that the invariants of D are,

ID = 0, IID = −γ2/4, IIID = 0. (2.75)

From these results and (2.53), we see that the only nonzero components of the extra stress tensorfor a Generalized Newtonian Fluid in simple shear are,

τrz = τzr = −η(S)γ where S ≡ −4 IID = γ2. (2.76)

45

Note that γ and S depend at most on r. If we use the results (2.76) in the equation of linearmomentum in cylindrical coordinates we find,

0 = −∂p

∂r

0 = −1r

∂p

∂θ

0 = −∂p

∂z+

1r

∂(rτrz)∂r

.

(2.77)

It follows directly from (2.77), that p is independent of both r and θ and ∂p/∂z is a constant whichwe will denote as −κ. Hence, the pressure is p = −κz + f(t). We can now integrate (2.77)3 withrespect to r to obtain, τrz = −κ r/2+ constant/r. In order that the shear stress be bounded at thecenter of the pipe, we set the constant to zero.

τrz = −κ r

2. (2.78)

Combining this with the result (2.74), we obtain

η(γ2)γ =κ r

2. (2.79)

We cannot proceed further without specifying the viscosity function.

2.4.1 Power-Law Model

From (2.79) and (2.55), we obtain,

Kγn =κr

2(2.80)

and upon integration and use of the no-slip boundary condition at the wall, we have,

w(r) =(κR

2K

)1/n (R

1 + 1/n

) [1− (

r

R)1+1/n

]. (2.81)

After substituting (2.83) in (2.73) and integrating, we obtain the volumetric flow rate as a functionof the axial components of the pressure gradient κ, the materials constants K and n and the piperadius, R.

Q =(R κ

2K

)1/n R3 π

3 + 1/n. (2.82)

Using (2.82), we can write the axial velocity (2.83) as,

w(r) =Q

πR2

3 + 1/n1 + 1/n

[1− (

r

R)1+1/n

]. (2.83)

For the same flow rate, the velocity field in the shear thinning power-law model is flatter than forthe constant viscosity case, Fig. 2.4.1.

46

Figure 2.3: Nondimensional axial velocity profile, w(r)/(Q/πR2, as a function of nondimensionalradius r/R for steady, fully developed flow of power-law fluids in a straight pipe. The profiles shownare for the same flow rate but with different values of n.

2.4.2 Bingham Fluid

Recall that for a Bingham Fluid, D is zero at all points where√|IIτ | is less than the yield stress.

For the flow under consideration√|IIτ | is equal to the magnitude of τrz which is equal to κr/2. It is

clear from the form of the solution for τrz that the magnitude of√|IIτ | will be largest at the largest

radial position: r = R. When κR/2 < τY , the yield criterion will not be met anywhere in the fluidand hence D will be zero throughout the fluid. From the boundary conditions, it then follows thatthe velocity is zero everywhere. If the driving pressure gradient increased so that κR/2 > τY , therewill be a region r ∈ [0, 2τY /κ], where the yield criterion is not met, and r ∈ (2τY /κ, R] where it ismet. We will denote this critical radius as ry ≡ 2τY /κ. From (2.69),

Drz =

0 for r ≤ rY

−κr

4µ+

τY

2µfor r ≥ r

Y

(2.84)

As previously discussed, for the velocity field under consideration, Drz is equal to 1/2 dw/dr andso, from (2.84), we can obtain the axial velocity as a function of the radial variable r,

w(r) =

C1 for r ≤ rY

−κr2

4µ+

τY r

µ+ C2 for r ≥ rY

(2.85)

47

where C1 and C2 are constants of integration. If we require that the value of w(r) be zero at thepipe boundary and be continuous throughout the flow field (including r equal to r

Y), then

w(r) =

κR2

4µ

(1− rY

R

)2for r ≤ r

Y

κR2

4µ

(1− r2

R2

)− τ

YR

µ

(1− r

R

)for r ≥ r

Y

(2.86)

The flow rate as a function of κ can be calculated directly from (2.73) and (2.86) and is left as anexercise.

2.4.3 Q as a function of κ and η(S) for unspecified generalized Newtonian fluid

For most viscosity functions it will be difficult, if not impossible to obtain a closed form solutionfor the velocity field. In this section, we consider the possibility of directly obtaining an expressionfor the flow rate as a function of κ without first obtaining an explicity expression for the velocityfield. While in most cases we will still need to use numerics to obtain the relationship, we will beable to avoid the intermediate step of first calculating the velocity field.

Beginning with (2.73) and the definition of γ and performing integration by parts twice, we canwrite the volumetric flow rate as

Q =π

3

(R3 γw −

∫ γw

0r(γ)3 dγ

), (2.87)

where γw denotes the value of γ at the pipe wall. Eq. (2.79) provides and expression for r as afunction of γ,

r =2κγ η(γ2). (2.88)

Using the relationship (2.88) in (2.87), we obtain

Q =π

3

(R3 γw − 8

κ3

∫ γw

0γ3 η(γ2)3 dγ

). (2.89)

We can obtain γw as a function of κ and R by solving the equation resulting from evaluating (2.88)at the wall,

γw η(γ2w) =

κ

2R. (2.90)

Once this value is obtained, (2.89) can be integrated to obtain volumetric flow rate as a functionof κ.

Exercise: Use (2.89) to obtain the flowrate as a function of κ for the Power-Law model.

Exercise: Is it possible to evaluate (2.89) in closed form for the Carreau model?

48

2.4.4 Determination of η(S) from measured values of driving pressure drop andflow rate in a straight pipe

Capillary viscometers are commonly used to measure the viscosity function η(S). These rheometersare designed to reproduce, with some level of experimental error, the velocity field for steady fullydeveloped flow in a pipe of constant radius v = w(r)ez. The general idea is that using results frombalance of linear momentum for this velocity field, and measurements of flow rate and pressure dropat different flow rates, we can back out and expression for η(S). Note that we want this expressionto be for an unspecified generalized Newtonian fluid (in fact, we will show in later chapters that thecapillary viscometer can in principal be used to measure η(S) for all simple fluids). The argumentsin this section, follow that of [1].

As will be clear at the end of this section, it is useful to rephrase the problem as one of obtainingη as a function of a characteristic shear rate γa and the shear stress at the wall, τw. We will defineγa as,

γa ≡ 4QπR3 , (2.91)

which is the shear rate at the pipe wall if the fluid is Newtonian. Using measured values of Q it isa simple matter to obtain γa. Using the definition of flow rate, integration by parts and (2.91), wecan obtain an expression for the characteristic shear rate

γa =4R

(us +

1R2

∫ R

0r2 γ dr

), (2.92)

where the notation us denotes the slip velocity at the wall of the pipe. Here we focus attentionon fluids for which the slip velocity is assumed to be zero. However, there is evidence that thisassumption is not appropriate for some real fluids. See [1] for further discussion of using capillaryrheometers in cases where us is not negligible.

Recall, that for a generalized Newtonian fluid undergoing a velocity field v = vz(r)ez, thatnecessarily

τrz = −κr

2, η(γ2) = τrz/γ, (2.93)

based on balance of linear momentum and the constitutive equation. Evaluating (2.93)1 at thewall,

τw ≡ τrz|r=R = −12κR. (2.94)

It follows that τw can easily be calculated from measured values of κ. As mentioned above, we areinterested in obtaining an expression relating τw and γa. We can do so, by rewrite the integral in(2.92) over r as an integral over τrz. It follows from (2.93)1 and (2.94) that

r = τrzR/τw. (2.95)

When (2.93)2 is invertible,γ = γ(τrz). (2.96)

49

After using (2.95) and (2.96) in (2.92), we find

γa =4τ3w

∫ τw

0

τ2rz γ(τrz) dτrz. (2.97)

If we take the derivative of the expression, (2.97) with respect to τw, we find,

dγadτw

= −12τ4w

∫ τw

0τ2rz γ(τrz) dτrz +

4τw

γ(τw),

= −3γaτw

+4 γ(τw)τw

.

(2.98)

Rearranging (2.98) and using the chain rule, we obtain

4γwγa

=[3 +

∂ ln γa∂ ln τw

](2.99)

where γw = γ(τw). Defining,

n′ =∂ ln τw∂ ln γa

(2.100)

we can rewrite (2.99) as,

γw = γa3n′ + 14n′

. (2.101)

Equation (2.101) is the Mooney-Rabinowitsch equation. Experiments can be run at different pres-sure drops to obtain a curve of γa as a function of τw (recall, (2.91) and (2.94)). If these resultsare plotted on a log scale, the value n′ can be obtained from the slope of the curve, which will ingeneral be a function of τw. From the definition of the viscosity function, applied at the wall,

η(γw)2 = τw/γw. (2.102)

Using (2.101) with this last result, enables us to determine results for η(S) for this flow. Astaritaand Marrucci provide the details of applying this result to a specific data set (page 73, [1]).

2.5 Cone and Plate Flow for a Generalized Newtonian Fluid

The cone and plate rheometer is one of the most commonly used instruments for measuring viscosityand normal stress coefficients. In particular, it can be used to measure the first normal stresscoefficient ψ1 in addition to the viscosity function η 2. The implementation of the rheometer isbased on an idealized flow of fluid between a cone and plate, driven by the rotation of the cone aboutthe z-axis, Figure 2.5. In this section, we focus on the derivation of the fundamental relationships

2While the focus in this chapter is on generalized Newtonian fluids, this rheometer can also be used for viscoelasticfluids. Recall that ψ1 is zero for generalized Newtonian fluids.

50

for this rheometer including the implementation of the small angle approximation at the basis ofcone and plate rheometers. A detailed discussion of the assumptions involved in the use of thisrheometer can be found in [19], pages 205-217.

It is convenient to use standard spherical coordinates for which the the solid walls of the rheome-ter are coordinate surfaces. The angle between the cone and the plate is denoted by β and themagnitude of the angular velocity by Ω. The corresponding applied torque necessary to generatethis angular velocity as m = mzez. Using this notation, the boundary conditions on the velocityfield are,

v|φ=π/2 = 0, and v|φ=(π/2−β) = Ω r cos βeθ. (2.103)

The boundary condition on the stress vector at the free surface, denoted by Γ1, is

t = −patm n, on Γ1 (2.104)

where n is the normal to the free surface and patm denotes the atmospheric pressure. At this point,

Figure 2.4: Schematic of cone and plate geometry with spherical coordinates. Note: The angle βis typically less that 6 degrees for cone and plate rheometers.

we have not specified the shape of the free surface, which in general should be determined as partof the solution. Motivated by the boundary conditions (2.103) and (2.104), we look for a velocityfield of the form,

v = vθ(r, φ) eθ. (2.105)

Moreover, the boundary conditions will be met if vθ depends linearly on r, so we look for vθ of theparticular form,

v = r ω(φ) eθ. (2.106)

51

From the assumed form of the velocity field, (2.106), it is straightforward to calculate severalkinematic variables (see, e.g. Appendix G),

[D] = − γ

2

0 0 0

0 0 10 1 0

, [

D2]=

γ2

4

0 0 0

0 1 00 0 1

, (2.107)

where

γ ≡ − sinφd

dφ

(ω

sinφ

). (2.108)

For this flow field, we can evaluate (2.52) to show S = γ2. Therefore, the only non-zero componentsof the extra stress tensor for a Generalized Newtonian fluid are (2.53),

τφθ = τθφ = −η(γ2) γ. (2.109)

Since γ is only a function of φ, τφθ is also only a function of φ. From (2.106) and (2.109) theequation of linear momentum reduces to,

−ρ rω2 = −∂p

∂r,

−ρ ω2 r2 cotφ = −∂p

∂φ,

0 = − 1sinφ

∂p

∂θ+

1sin2 φ

∂

∂φ(τφθ sin2 φ).

(2.110)

From (2.110) and because τφθ depends only on φ, we can conclude that ∂p/∂θ is a constant.Furthermore, in order that the boundary condition (2.104) be met for all θ, necessarily, this constantis zero and hence p is at most dependent on r and φ.

Neglect Inertial Effects

By differentiating (2.110)1 with respect to φ and (2.110)2 with respect to r we obtain,

∂

∂φ(∂p

∂r) = 2 ρ rω

dω

dφ

∂

∂r(∂p

∂φ) = 2 ρ rω2 cot φ.

(2.111)

It then follows that either ω = 0 or ω = A sinφ where A is a constant of integration. Neither of thesesolutions can satisfy both boundary conditions. Therefore, our assumed form of the velocity fieldis only meaningful in the limit of negligible inertial effects in which case, the first two componentsof the balance of linear momentum reduce to the condition that p is independent of r and φ andhence p is constant throughout the flow field. It then follows from the condition at the free surface(2.104), the fact that τrφ = τφr are the only non-zero components of the extra stress, the freesurface must be a surface of constant radius and p = patm everywhere in the fluid.

52

Relationship between τrφ and mz

.It follows from the third component of linear momentum (2.111)3, that

τφθ = − C1

sin2 φ, (2.112)

where C1 is a constant of integration. It is convenient to write this constant in terms of measurablequantities, in particular the torque to the fluid by the cone. m = mzez. Recall, that in general,the applied torque on a surface with outward unit normal n is related to the stress vector t actingon that surface through

m =∫Ax× t|Γ1 da (2.113)

where x is the lever arm, da is the elemental area. At the fluid surface adjacent to the cone, φ isequal to π/2− β and

x = r cosβ(sinβeφ + cos βer),da = r cos β dr dθ,n = −eφt|Γ1 = T · n

= peφ − τφθ|Γ1eθ

(2.114)

and therefore,x× t|Γ1 = p r cos2 βeθ − τφθ|Γ1 r cos βez, (2.115)

where the base vector,ez = sinβer − cos βeφ (2.116)

is constant in space and therefore simplifies the evaluation of the integral in (2.113). Noting that,

τφθ|Γ1 = − C1

cos2 β(2.117)

if follows from (2.113) -(2.117) that

m =C12πR3

o

3ez. (2.118)

Using the condition on the applied torque m, we can determine C1 from (2.118),

C1 =3mz

2πR3o

. (2.119)

Therefore,

τφθ = − 3mz

2πR3o sin

2 φ. (2.120)

We now have an expression for the shear stress in terms of the applied torque and geometry. Toobtain a relationship between the function η(S), m z and Ω it remains to use (2.120) in conjunctionwith the expression for τφθ for a particular constitutive equation.

53

2.5.1 Small Angle Approximation

In this section, we consider flow between a cone and plate for cone angle β much less than one.Using this approximation, we will be able to obtain a function form for η(mz, R0, β,Ω) withoutspecifying a particular form for the constitutive equation. This result is the theoretical basis forthe use of the cone and plate geometry in rheometers.

In considering the form of the governing equations under the small angle approximation, it isconvenient to introduce an angle α defined relative to φ through the relation,

α =π

2− φ. (2.121)

In the fluid domain, α is always less than or equal to β which is much less than one. Using thedefinition (2.121), we see that,

cosφ = sinα = α + 0(β3) sinφ = cosα = 1+ 0(β2). (2.122)

Therefore,

τφθ = − C1

sin2 φ= − C1

cos2 α= −C1 + 0(β2). (2.123)

For small angles, the shear rate (2.108) becomes,

γ =(1 + 0(α2)

) ∂

∂α

(ω(1 + 0(α2))

). (2.124)

Recalling that η(γ2) is equal to −τφθ/γ which is a constant. It follows from (2.123) that for smallβ either η is proportional to 1/γ or γ is a constant. It then follows from (2.124) that

ω = C2α +C3 + 0(β2). (2.125)

We can obtain the constants of integration, C2 and C3 from the boundary conditions (2.103), sothat,

ω = Ωα

β+ O(β2). (2.126)

We can determine an approximation for γ , denoted by γA by evaluating (2.124) using (2.126),

γA =Ωβ

+O(β2). (2.127)

Combining these results, we see that the approximate expression for η, which we’ll denote by ηAis:

ηA(γ2A) =

3mz

2πR3o

β

Ω(γA)

Ω =β

γA

assuming

1. Velocity field of the form v = rω(φ)eθ2. Negligible body forces3. Negligible Inertial Effects4. Free Surface of Fluid at r = R

5. Small Cone Angle (β << 1)

(2.128)

54

If mz and Ω are measured for a given geometry (R0, β), the functional form ηA(γA2) can bedetermined for the range of measured values. Note that for the small angle approximation (ignoringinertial effects) both τφθ and ˙γA are constant through out the fluid domain.

2.5.2 Finite Cone Angle

The relationship, (2.128), relating ηA to of the applied torque, Ω, β and Ro was derived for smallcone angle and negligible inertial effects. It is natural to wonder how much the solutions changewhen these assumptions are not imposed. While it is more complicated to consider the effectsof inertia, we can obtain some closed form solutions even when we do not make the small angleapproximation. To obtain the corresponding relationship to (2.128) for a finite cone angle, we mustconsider particular constitutive equations. In the following section, we obtain solutions for coneand plate flow for finite angle for Bingham fluids and some Power-Law fluids. We still impose thenegligible inertia assumption.

Power-Law fluid in cone and plate rheometer at finite angle

For a Power-Law fluid undergoing steady flow of the form, (2.106), the only non-zero componentof the extra stress tensor is,

τφθ = −η(γ2)γ = −Kγn. (2.129)

Combining this result with the form for τφθ obtained from the equation of linear momentum for ageneralized Newtonian fluid τφθ = C1/sin

2φ, we have that

γ = − sinφd

dφ

(ω

sinφ

)=

(C1

K

)1/n

sin−2/n φ (2.130)

and therefore

vθ = r sinφ(C1

K

)1/n ∫ π/2

φsin−1−2/n η dη (2.131)

We can obtain an expression for Ω by evaluating (2.131) at the cone surface and making use of theboundary condition (2.103). The integral in (2.131) can be easily evaluated when (1 + 2/n) is aninteger. If we define f(β) as,

f(β) =∫ π/2

π/2−βsin−1−2/n η dη (2.132)

then,

Ω =(C1

K

)1/n

f(β), (2.133)

where the function f(β) depends on the particular Power-Law fluid, in particular, on the choice ofn. From (2.130) and (2.133), we have that

γ =Ω

f(β) sin2/n φ, γw =

Ω

f(β) cos2/n β. (2.134)

55

If we now evaluate (2.129) at the surface of the cone (denoting the value of γ at φ equal to (π/2−β)by γw), we obtain

η(γ2w) = −τφθ(π/2− β)

γw=

C1

Ωf(β)(cosβ)2(1/n−1). (2.135)

We now consider specific Power-Law fluids, and calculate the values of f(β) for these fluids.

n = 1/2We first consider a shear thinning fluid with n = 1/2, In this case,

f(β) =∫ π/2π/2−β sin

−5 η dη. (2.136)

From (2.135)

η(γ2w) =

C1

Ωf(β) cos2 β. (2.137)

and therefore, at the wall,

η(γ2w) =

3mz

2 πR3oΩ

[−3 sin3 β + 5 sin β8 cos2 β

− 316

cos2 β ln(1− sin β1 + sin β

)]. (2.138)

In the limit of β much less than one, the expression in brackets in (2.138) is equal to β plus higherorder terms. As expected, the expression for η(γ) can be approximated by ηA, where

ηA =3mz β

2 πR3oΩ

, (2.139)

as deduced in the preceeding section on small cone angle.

n = 2For a shear thickening fluid with n = 2,

f (β) =∫ π/2

π/2−βsin−2 η dη = tanβ. (2.140)

Therefore at the wall of the cone, η is,

η(γ2w) =

3mz

2 π R3o

sin βΩ cos2 β

. (2.141)

In the limit of β much less than one, we can approximate η by ηA, (2.139).

We see from Fig. 2.5.2 that the error for both shear thickening and shear thinning fluids is onthe order of a few percent for typical values of cone angles of 5-6 degrees.

56

Figure 2.5: Effect of small angle approximation: η/ηA for Power-Law fluids with n=1/2 and n=2.

Bingham Fluid in a cone and plate rheometer with finite cone angle

In this subsection, we consider the flow of a Bingham fluid in a cone and plate rheometer. Recallthat the only non-zero components of the extra stress tensor are τφθ = τθφ as given in Eq. 2.112.It then follows that the square root of the magnitude of the second invariant of the extra stresstensor is, √

|IIτ | = |τφθ| =3mz

2πR3o sin

2 φfor φ ∈ [π/2− β, π/2], (2.142)

which is a maximum at φ = π/2 − β. We therefore expect that as the torque is increased fromzero, the yield criterion

√|IIτ | = τY will first be met at the surface of the cone. The critical torquenecessary for this condition to be met is,

mY =τY 2πR

3o cos

2 β

3. (2.143)

For mz < mY , the cone will not move. For mz greater than (2.143), the fluid will flow in someregions of the fluid domain. The yield surface is defined by the angle where

√|IIτ | = τY ,

φY = sin−1

[√3mz

2πR3o

]. (2.144)

57

Therefore, for a Bingham fluid,

Dφθ =

12η

1− τY√

|IIτ |

τφθ, for φ ∈ [π/2− β, φY ]

0 for φ ∈ [φY , π/2]

(2.145)

From (2.112) and (2.119) we see that τφθ is negative and therefore, we can rewrite (2.145) as

Dφθ =

12η

[− C1

sin2 φ+ τY

]for φ ∈ [π/2− β, φY ]

0 for φ ∈ [φY , π/2].

(2.146)

From (2.107), (2.146) and (2.108), we can now solve for ω(φ).

sinφd

dφ

(ω

sinφ

)=

1η

(− C1

sin2 φ+ τ

Y

)for φ ∈ [π/2− β, φY ]

sinφ∂

∂φ(

ω

sinφ) = 0 for φ ∈ [φY , π/2].

(2.147)

Integrating (2.147) we find,

ω =sinφη

[C1

2

(cosφsin2 φ

− ln(tan(φ

2)))+ τ

Yln(tan(

φ

2)) +C2

]for φ ∈ [π/2− β, φY ]

ω = C3 sinφ for φ ∈ [φY , π/2].

(2.148)The constants of integrationC2 and C3 must be chosen such that the no-slip condition is met at thelower plate (ω(π/2) = 0) and so that the velocity is continuous throughout the flow field. From thecondition at the lower plate, we find that C3 is equal to zero and therefore vθ is zero at all pointsin the fluid where φ is greater than or equal to φ

Y, in particular it is zero at the yield surface. We

can obtain an expression for C2 from the condition that the velocity field be continuous at the yieldsurface. Hence,

vθ =r sinφη

C1

2

(cot φsinφ

− cotφY

sinφY

)+ ln(

tan(φ

2)

tan(φY

2))(−C1

2+ τY

) for φ ∈ [π/2− β, φY ]

vθ = 0 for φ ∈ [φY , π/2].

(2.149)

58

The boundary condition on the upper plate, (2.103), then provides a relationship between η, Ω andmz :

η =1Ω

C1

2

(sin βcos2 β

− cotφY

sinφY

)+ ln(

tan(π/2− β

2)

tan(φY

2)

)(−C1

2+ τY

) . (2.150)

(recall that C1 = 3mz/(2πR3o). Equation (2.150) holds when φ

Yis greater than or equal to π/2−β.

If not, the value of Ω is zero. Namely, the applied torque must be sufficiently large that the fluidbegins to yield at the surface of the cone.

Once the fluid is yielding throughout the domain, (φY = π/2),

η =1Ω

[C1

2

(cotφsinφ

)+ ln(tan(

φ

2))

(−C1

2+ τY

)]. (2.151)

For small β, (2.151) becomes,

η ≈ β

Ω

[3mz

2πR3o

− τY

]where mz ≥ mY = 2πR3

o τY cos2 β/3.

(2.152)

59

60

Appendix A

The Relationship between CoordinateFree Notation and Indicial Notation

It is sometimes convenient to write expressions in terms of coordinate free notation rather thanwith respect to a particular coordinate system. For example, in coordinate free notation, we wouldrefer to a vector by the symbol v where as in indicial notation, we would write the vector as thesum of components relative to a particular basis, ei,

v = viei (A.1)

where in (A.1) it is understood that repeated indices are summed from one to three. The followingis a table containing some commonly used relationships between coordinate free and index notation.Consider vectors v and w and second order tenors A and B,

Coordinate Free Notation Indicial Notationv = vi ei

v · w = vi wi

v × w = εijk viwj ek

A = Aij ei ⊗ ej

A · v = Aij vj ei

AT · v = v · A = Aji vj ei

A ·B = Aij Bjk ei ⊗ ek

AT · B = AjiBjk ei ⊗ ek

A ·BT = Aij Bkj ei ⊗ ek

trA = Aii

A : B = AijBij

(A.2)

61

The Definition of Some Frequently Used OperatorsIn this section, we recall the definitions with respect to a fixed basis ei of some commonly usedoperators. We will consider the result of operations on scalar fields φ(x), vector fields v(x) andw(x) as well as second order tensor fields A(x). Unfortunately, the definition of these operatorsis not universal in the literature. In this class, we will use the definitions given in the followingtable. Significantly, in many fluid mechanics books these operators are defined as the transpose of

Operator Symbol Definition

grad φ ∇φ ∂φ

∂xjej

grad v ∇v ∂vi∂xj

ei ⊗ ej

div v ∇ · v ∂vi∂xi

div A ∇ ·A ∂Aij

∂xjei

curl v ∇× v εijk∂vk∂xj

ei

div grad v ∇ · (∇ v)or ∇2 v∂2vi

∂xj∂ xjei

(∇w) · v vj∂wi

∂xjei

(A.3)

those operations given here. It is important to take care when reading and writing publicationswhich use these operators. For example, in fluid mechanics literature, the operator (∇w) ·v is oftenwritten as v · ∇w. In addition, sometimes the curl of a vector is defined as the negative of thatgiven above.

62

Appendix B

Isotropic Tensors

The discussion in this section follows from [21]. Consider a rectangular Cartesion coordinate systemwith basis, (ei), and a second Cartesion coordinate system which is rotated relative to the first,with basis (e′i). The relationship between (ei), and (e′i) can be represented as

ei = Aije′j (C.1)

where Aij are components of a proper orthogonal tensor,

AijAkj = δik, detA = 1. (C.2)

Therefore, we also havee′j = Aijei (C.3)

Consider an arbitrary vector v. We refer to the components of this vector with respect to basis(ei), and (e′i) as vi and v′i, respectively. Namely,

v = viei = v′ie′i. (C.4)

Using this notation,v = viei = v′je

′j

= v′jAkjek(C.5)

Therefore,vl = v′jAkjek · el

= Aljv′j

(C.6)

Similarly, it can be shown that, the components Mijk...t of a tensor M will transform as

Mijk...t = AiaAjbAkc...AthM′abc...h. (C.7)

63

A tensor M is called isotropic if its components retain the same value regardless of the coordinateaxis rotation,

Mijk...t = M ′ijk...t (C.8)

for all proper orthogonal Aij. Alternatively,

Mijk...t = AiaAjbAkc...AthM′abc...h. (C.9)

Theorem 1.: All scalar invariants are isotropic tensors of order 0.

Theorem 2.: The only isotropic tensor v of order 1 is the zero tensor.Proof: If v is an isotorpic tensor of order 1, then the components of v are unchanged under anproper orthogonal transformations:

vi = Aijvj (C.10)

for all proper orthogonal A. Consider the case where the components of A are defined as

[A] =

−1 0 0

0 −1 00 0 1

. (C.11)

Upon consideration of equation (C.10), we see that this choice of A requires that

(v1, v2, v3) = (−v1,−v2, v3). (C.12)

This implies that v1 = 0, v2 = 0. Similarly, if we now consider the case where the components of Aare equal to

[A] =

1 0 0

0 −1 00 0 −1

. (C.13)

then necessarily (v1, v2, v3) = (v1,−v2,−v3). so that v3 must also be zero. Therefore, if v is anisotropic tensor of order 1, then necessarily v must be the zero vector. It is clear that this is also asufficient condition.

Theorem 3.: Any isotropic tensor of order 2, can be written as a scaler multiple of the iden-tity tensor. Namely, if B is a second order isotropic tensor, then the most general form of B canbe written as

B = λI, Bij = λδij (C.14)

where λ is a scalar.Theorem 4.: The components of the most general isotropic tensor B of order 3 can be written as

Bpqr = λ εijk (C.15)

64

where λ is a scalar.Proof: If B is an isotropic tensor of order 3, then the components of B are unchanged under anproper orthogonal transformations:

Bijk = AiaAjbAkcBabc (C.16)

for all proper orthogonal A. Consider the following three choices for the components of A,

[A] =

0 0 −1

−1 0 00 1 0

, [A] =

0 0 1

1 0 00 1 0

, [A] =

0 0 1

0 1 0−1 0 0

. (C.17)

If we consider (C.10) for these three proper orthogonal choices for A, we fined that There are onlysix non-zero components of B, and they are related as

B123 = B231 = B312 = −B321 = B213 = B132, (C.18)

and therefore necessarily Bijk must be of the form

Bijk = λεijk (C.19)

where lambda is a scaler. The form (C.19) is also sufficient:

AiaAjbAkcBabc = AiaAjbAkcλεabc= λ det(Aab)εijk= λεijk= Bijk

(C.20)

Theorem 5.: The components of the most general isotropic tensor, B of order 4 can be writtenas

Bijkl = λδijδkl + µδikδjl + γδilδjk. (C.21)

Exercise B.0.1 Prove Theorem 2, making sure to show both necessary and sufficient conditions.

65

66

Appendix C

Isotropic Tensor Functions

C.1 Scalar valued isotropic tensor functions

A scalar valued function of a second order tensor φ = φ(B) is said to be an isotropic tensor functionif

φ(Bij) = φ(B′ij) = φ(AikBklAjl) (D.1)

for all proper orthogonal A. Such a scalar valued function is also called simply an invariant. Thefollowing isotropic scalar functions IB , IIB IIIB are called the Principal Invariants of a second ordertensor B,

IB = trB = Bii

IIB =12(tr(B)2 − tr(B ·B)

)=

12(BiiBjj − BijBji)

IIIB = detB

(D.2)

The set of invariants (D.2) arise frequently as the coefficients of the characteristic equation for asecond order tensor. Another set of invariants are,

IB = trB = Bii

IIB = trB2 = BijBji

¯IIIB = trB3 = BijBjkBki

(D.3)

Representation of Invariant Scalar Valued functions of a Symmetric TensorA scalar valued function of a symmetric tensor φ(B) is an invariant if and only if it can be writtenas a function of the principal invariants of the symmetric tensor B.

φ(B) = φ(IB , IIB , IIIB) (D.4)

67

C.2 Symmetric Isotropic Tensor Functions

Consider a second order tensor valued function F = F (B1, B2, ...BN ) where Bi are second ordertensors. The second order tensor function F is said to be an isotropic tensor function if

A · F (B1, B2, ...BN ) · AT = F (A · B1 · AT , A · B2 · AT , ...A · BN · AT ) (D.5)

for all proper orthogonal A and all Bi in the domain of definition of the function F .

Representation of Symmetric Isotropic Tensor FunctionsA tensor function F = F (B) for which both F and B are symmetric second order tensors, isisotropic if and only if it has a representation of the form,

F = F (B) = φoI + φ1B + φ2B2 (D.6)

where φ1, φ2, φ3 are functions of the principal invariants of B.

This theorem can be generalized for tensors which are not of second order (see, for example,Truesdell, Non-Linear Field Theories of Mechanics, Handbuch Der Physik, Volume III/3, 1965, p32), [31].

68

Appendix D

Material Line Elements, AreaElements, Volume Elements

Some useful results for material line elements, area elements and volume elements are summarizedbelow:

ds2 = FiAFiBdXAdXB. (E.1)

dai = J XB,idAB. (E.2)

dv = J dV (E.3)

Dv

Dt=

DJ

DtdV = J vi,i dV (E.4)

Proofs will be added to this section at a later time.

69

70

Appendix E

Rectangular Coordinates

In this appendix we consider rectangular components (x, y, z) with corresponding orthonormalbasis, (ex, ey, ez).

An arbitrary vector may be written with respect to rectangular coordinates as,

v = vi ei = vxex + vyey + vzez. (F.1)

The following are some commonly used operators written with respect to rectangular components.Gradient of a scalar function f

∇f =∂ f

∂ xiei =

∂f

∂xex +

∂f

∂yey +

∂f

∂zez. (F.2)

Laplacian of a scalar function f

∂2f

∂x2+

∂2f

∂y2+

∂2f

∂z2. (F.3)

Gradient of a Vector, vThe gradient of a vector v, often denoted as L, is defined as

[∇v] =[∂vi∂xj

ei ⊗ ej

]=

∂vx∂x

∂vx∂y

∂vx∂z

∂vy∂x

∂vy∂y

∂vy∂z

∂vz∂x

∂vz∂y

∂vz∂z

(F.4)

Divergence of a Vector, v

∇ · v =∂vi∂xi

=∂vx∂x

+∂vy∂y

+∂vz∂z

(F.5)

71

Curl of a Vector, v

∇× v = εijk∂vj∂xi

ek =(∂vz∂y

− ∂vy∂z

)ex +

(∂vx∂z

− ∂vz∂x

)ey +

(∂vy∂x

− ∂vx∂y

)ez. (F.6)

Divergence of a Second Order Tensor, A

∇ ·A =∂Aij

∂xjei =

(∂Axx

∂x+

∂Axy

∂y+

∂Axz

∂z

)ex

+(∂Ayx

∂x+

∂Ayy

∂y+

∂Ayz

∂z

)ey

+(∂Azx

∂x+

∂Azy

∂y+

∂Azz

∂z

)ez.

(F.7)

Balance of Linear MomentumThe balance of linear momentum can be written with respect to rectangular coordinates, as

ρ

(∂vx∂t

+ vx∂vx∂x

+ vy∂vx∂y

+ vz∂vx∂z

)=

∂Txx∂x

+∂Txy∂y

+∂Txz∂z

+ ρgx,

ρ

(∂vy∂t

+ vx∂vy∂x

+ vy∂vy∂y

+ vz∂vy∂z

)=

∂Tyx∂x

+∂Tyy∂y

+∂Tyz∂z

+ ρgy,

ρ

(∂vz∂t

+ vx∂vz∂x

+ vy∂vz∂y

+ vz∂vz∂z

)=

∂Tzx∂x

+∂Tzy∂y

+∂Tzz∂z

+ ρgz.

(F.8)

If the stress tensor , T is decomposed into the sum of the spherical and deviatoric parts,

T = −pI + τ , (F.9)

then we can rewrite (F.8) as,

ρ

(∂vx∂t

+ vx∂vx∂x

+ vy∂vx∂y

+ vz∂vx∂z

)= −∂p

∂x+

∂τxx∂x

+∂τxy∂y

+∂τxz∂z

+ ρgx,

ρ

(∂vy∂t

+ vx∂vy∂x

+ vy∂vy∂y

+ vz∂vy∂z

)= −∂p

∂y+

∂τxy∂x

+∂τyy∂y

+∂τyz∂z

+ ρgy,

ρ

(∂vz∂t

+ vx∂vz∂x

+ vy∂vz∂y

+ vz∂vz∂z

)= −∂p

∂z+

∂τxz∂x

+∂τyz∂y

+∂τzz∂z

+ ρgz.

(F.10)

72

For the special case of a Navier-Stokes (incompressible Newtonian) fluid, the balance of Linearmomentum can be written as,

ρ

(∂vx∂t

+ vx∂vx∂x

+ vy∂vx∂y

+ vz∂vx∂z

)= −∂p

∂x+ µ

[∂2vx

∂x2 +∂2vx

∂y2 +∂2vx

∂z2

]+ ρgx,

ρ

(∂vy∂t

+ vx∂vy∂x

+ vy∂vy∂y

+ vz∂vy∂z

)= −∂p

∂y+ µ

[∂2vy

∂x2+

∂2vy

∂y2+

∂2vy

∂z2

]+ ρgy,

ρ

(∂vz∂t

+ vx∂vz∂x

+ vy∂vz∂y

+ vz∂vz∂z

)= −∂p

∂z+ µ

[∂2vz

∂x2 +∂2vz

∂y2 +∂2vz

∂z2

]+ ρgz.

(F.11)

73

74

Appendix F

Cylindrical Coordinates

In this appendix we consider cylindrical coordinates (r, θ, z), defined relative to rectangular com-ponents (x, y, z) through,

Figure G.1: Cylindrical Coordinate System

x = r cos θ, y = r sin θ, z = z. (G.1)

The corresponding orthonormal base vectors (er, eθ, ez) are related to the rectangular basis (ex, ey, ez),through

ex = cos θer − sin θeθ, ey = sin θer + cos θeθ, (G.2)

and ez is identical for both coordinate systems. An arbitrary vector may be written with respectto cylindrical coordinates as,

v = vr er + vθ eθ + vz ez. (G.3)

Gradient of a scalar function f

∇f =∂f

∂rer +

1r

∂f

∂θeθ +

∂f

∂zez. (G.4)

75

Laplacian of a scalar function f

∇f =1r

∂

∂r(r∂f

∂r) +

1r2

∂2f

∂θ2 +∂2f

∂z2 . (G.5)

Gradient of a Vector, v

[L] = [∇v] =

∂vr∂r

1r

∂vr∂θ

− vθr

∂vr∂z

∂vθ∂r

1r

∂vθ∂θ

+vrr

∂vθ∂z

∂vz∂r

1r

∂vz∂θ

∂vz∂z

(G.6)

Divergence of a Vector, v

∇ · v =1r

∂(rvr)∂r

+1r

∂vθ∂θ

+∂vz∂z

(G.7)

Curl of a Vector, v

∇× v =(1r

∂vz∂θ

− ∂vθ∂z

)er +

(∂vr∂z

− ∂vz∂r

)eθ +

1r

(∂(rvθ)∂r

− ∂vr∂θ

)ez. (G.8)

Divergence of a Second Order Tensor, A

∇ · A =(∂Arr

∂r+

1r

∂Arθ

∂θ+

∂Arz

∂z+

Arr − Aθθ

r

)er

+(∂Aθr

∂r+

1r

∂Aθθ

∂θ+

∂Aθz

∂z+Aθr + Arθ

r

)eθ

+(∂Azr

∂r+

1r

∂Azθ

∂θ+

∂Azz

∂z+Azr

r

)ez.

(G.9)

76

The operator ∇A · v for a second order tensor A

[∇A · v]rr =(vr∂Arr

∂r+

vθr

∂Arr

∂θ+ vz

∂Arr

∂z− vθ

r(Arθ + Aθr)

)

[∇A · v]rθ =(vr∂Arθ

∂r+

vθr

∂Arθ

∂θ+ vz

∂Arθ

∂z+

vθr(Arr − Aθθ)

)

[∇A · v]rz =(vr∂Arz

∂r+

vθr

∂Arz

∂θ+ vz

∂Arz

∂z− vθ

rAθz

)

[∇A · v]θr =(vr∂Aθr

∂r+

vθr

∂Aθr

∂θ+ vz

∂Aθr

∂z+

vθr(Arr − Aθθ)

)

[∇A · v]θθ =(vr∂Aθθ

∂r+

vθr

∂Aθθ

∂θ+ vz

∂Aθθ

∂z+

vθr(Arθ + Aθr)

)

[∇A · v]θz =(vr∂Aθz

∂r+

vθr

∂Aθz

∂θ+ vz

∂Aθz

∂z+vθrArz

)

[∇A · v]zr =(vr∂Azr

∂r+

vθr

∂Azr

∂θ+ vz

∂Azr

∂z− vθ

rAzθ

)

[∇A · v]zθ =(vr∂Azθ

∂r+

vθr

∂Azθ

∂θ+ vz

∂Azθ

∂z+vθrAzr

)

[∇A · v]zz =(vr∂Azz

∂r+

vθr

∂Azz

∂θ+ vz

∂Azz

∂z

)

(G.10)

Balance of Linear MomentumThe balance of linear momentum can be written with respect to cylindrical coordinates, as

ρ

(∂vr∂t

+ vr∂vr∂r

+vθr

∂vr∂θ

+ vz∂vr∂z

− v2θ

r

)=

∂Trr∂r

+1r

∂Trθ∂θ

+∂Trz∂z

+Trr − Tθθ

r+ ρgr,

ρ

(∂vθ∂t

+ vr∂vθ∂r

+vθr

∂vθ∂θ

+ vz∂vθ∂z

+vrvθr

)=

∂Trθ∂r

+1r

∂Tθθ∂θ

+∂Tθz∂z

+ 2Tθrr

+ ρgθ,

ρ

(∂vz∂t

+ vr∂vz∂r

+vθr

∂vz∂θ

+ vz∂vz∂z

)=

∂Trz∂r

+1r

∂Tθz∂θ

+∂Tzz∂z

+Tzrr

+ ρgz.

(G.11)

77

Using the decomposition of the tensor T into its spherical and deviatoric parts, we can rewrite(H.10) as

ρ

(∂vr∂t

+ vr∂vr∂r

+vθr

∂vr∂θ

+ vz∂vr∂z

− v2θ

r

)=

−∂p

∂r+

∂τrr∂r

+1r

∂τrθ∂θ

+∂τrz∂z

+τrr − τθθ

r+ ρgr,

ρ

(∂vθ∂t

+ vr∂vθ∂r

+vθr

∂vθ∂θ

+ vz∂vθ∂z

+vrvθr

)=

−1r

∂p

∂θ+

∂τrθ∂r

+1r

∂τθθ∂θ

+∂τθz∂z

+ 2τθrr

+ ρgθ,

ρ

(∂vz∂t

+ vr∂vz∂r

+vθr

∂vz∂θ

+ vz∂vz∂z

)=

−∂p

∂z+

∂τrz∂r

+1r

∂τθz∂θ

+∂τzz∂z

+τzrr

+ ρgz.

(G.12)

For the special case of a Navier-Stokes (incompressible Newtonian) fluid, the balance of linearmomentum can be written as,

ρ

(∂vr∂t

+ vr∂vr∂r

+vθr

∂vr∂θ

+ vz∂vr∂z

− v2θ

r

)=

−∂p

∂r+ µ

[∂

∂r

(1r

∂(rvr)∂r

)+

1r2

∂2vr

∂θ2 +∂2vr

∂z2 − 2r2

∂vθ∂θ

]+ ρgr,

ρ

(∂vθ∂t

+ vr∂vθ∂r

+vθr

∂vθ∂θ

+ vz∂vθ∂z

+vrvθr

)=

−1r

∂p

∂θ+ µ

[∂

∂r

(1r

∂(rvθ)∂r

)+

1r2

∂2vθ

∂θ2+

∂2vθ

∂z2+

2r2

∂vr∂θ

]+ ρgθ,

ρ

(∂vz∂t

+ vr∂vz∂r

+vθr

∂vz∂θ

+ vz∂vz∂z

)=

−∂p

∂z+ µ

[1r

∂

∂r

(r∂vz∂r

)+

1r2

∂2vz

∂θ2 +∂2vz

∂z2

]+ ρgz.

(G.13)

78

Appendix G

Spherical Coordinates

In this appendix we consider spherical coordinates (r, φ, θ), defined relative to rectangular compo-nents (x, y, z) through,

Figure H.1: Spherical Coordinate System

x = r sinφ cos θ, y = r sinφ sin θ, z = r cosφ. (H.1)

The corresponding orthonormal base vectors (er, eφ, eθ) are related to the rectangular basis (ex, ey, ez),through

ex = cos θ(sinφer + cosφeφ

) − sin θeθ,

ey = sin θ(sinφer + cosφeφ

)+ cos θeθ,

ez = cosφer − sinφeφ.

(H.2)

An arbitrary vector may be written with respect to rectangular coordinates as,

v = vr er + vφ eφ + vθ eθ. (H.3)

79

Gradient of a scalar function f

∇f =∂f

∂rer +

1r

∂f

∂φeφ +

1r sinφ

∂f

∂θeθ. (H.4)

Laplacian of a scalar function f

∇2f =1r2

∂

∂r(r2 ∂f

∂r) +

1r2 sinφ

∂

∂φ(sinφ

∂f

∂φ) +

1r2 sin2 φ

∂2f

∂θ2 . (H.5)

Gradient of a Vector, v

[L] = [∇v] =

∂vr∂r

1r

∂vr∂φ

− vφr

1r sinφ

∂vr∂θ

− vθr

∂vφ∂r

1r

∂vφ∂φ

+vrr

1r sinφ

∂vφ∂θ

− vθrcotφ

∂vθ∂r

1r

∂vθ∂φ

1r sinφ

∂vθ∂θ

+vrr

+vφr

cotφ

(H.6)

Divergence of a Vector, v

∇ · v =1r2

∂(r2vr)∂r

+1

r sinφ∂(vφ sinφ)

∂φ+

1r sinφ

∂vθ∂θ

(H.7)

Curl of a Vector, v

∇×v =1

r sinφ

(∂(vθ sinφ)

∂φ− ∂vφ

∂θ

)er +

(1

r sinφ∂vr∂θ

− 1r

∂(rvθ)∂r

)eφ +

1r

(∂(rvφ)∂r

− ∂vr∂φ

)eθ.

(H.8)Divergence of a Second Order Tensor, A

∇ · A =(∂Arr

∂r+

1r

∂Arφ

∂φ+

1r sinφ

∂Arθ

∂θ+

2Arr − Aφφ − Aθθ +Arφ cotφr

)er

+(∂Aφr

∂r+

1r

∂Aφφ

∂φ+

1r sinφ

∂Aφθ

∂θ+

2Aφr + Arφ + (Aφφ − Aθθ) cotφr

)eφ

+(∂Aθr

∂r+

1r

∂Aθφ

∂φ+

1r sinφ

∂Aθθ

∂θ+

2Aθr + Arθ + (Aφθ + Aθφ) cotφr

)eθ.

(H.9)

80

Balance of Linear MomentumThe balance of linear momentum can be written with respect to cylindrical coordinates, as

ρ

(∂vr∂t

+ vr∂vr∂r

+vφr

∂vr∂φ

+vθ

r sinφ∂vr∂θ

− (v2φ + v2

θ)r

)=

∂Trr∂r

+1r

∂Trφ∂φ

+1

r sinφ∂Trθ∂θ

+2Trr − Tφφ − Tθθ + Trφ cotφ

r+ ρgr,

ρ

(∂vφ∂t

+ vr∂vφ∂r

+vφr

∂vφ∂φ

+vθ

r sinφ∂vφ∂θ

+(vrvφ − v2

θ cotφ)r

)=

∂Trφ∂r

+1r

∂Tφφ∂φ

+1

r sinφ∂Tφθ∂θ

+3Trφ + (Tφφ − Tθθ) cotφ)

r+ ρgφ,

ρ

(∂vθ∂t

+ vr∂vθ∂r

+vφr

∂vθ∂φ

+vθ

r sinφ∂vθ∂θ

+(vrvθ + vφvθ cot φ)

r

)=

∂Trθ∂r

+1r

∂Tφθ∂φ

+1

r sinφ∂Tθθ∂θ

+3Tθr + 2Tφθ cotφ

r+ ρgθ.

(H.10)Using the decomposition of the tensor T into its spherical and deviatoric parts, we can rewrite(H.10) as

ρ

(∂vr∂t

+ vr∂vr∂r

+vφr

∂vr∂φ

+vθ

r sinφ∂vr∂θ

− (v2φ + v2

θ)r

)=

−∂p

∂r+

∂τrr∂r

+1r

∂τrφ∂φ

+1

r sinφ∂τrθ∂θ

+2τrr − τφφ − τθθ + τrφ cotφ

r+ ρgr,

ρ

(∂vφ∂t

+ vr∂vφ∂r

+vφr

∂vφ∂φ

+vθ

r sinφ∂vφ∂θ

+(vrvθ − v2

θ cotφ)r

)=

−1r

∂p

∂φ+

∂τrφ∂r

+1r

∂τφφ∂φ

+1

r sinφ∂τφθ∂θ

+3τrφ + (τφφ − τθθ) cotφ

r+ ρgφ,

ρ

(∂vθ∂t

+ vr∂vθ∂r

+vφr

∂vθ∂φ

+vθ

r sinφ∂vθ∂θ

+(vrvθ + vφvθ cot φ)

r

)=

− 1r sinφ

∂p

∂θ+

∂τrθ∂r

+1r

∂τφθ∂φ

+1

r sinφ∂τθθ∂θ

+3τrθ + 2τφθ cot φ

r+ ρgθ.

(H.11)

81

For the special case of a Navier-Stokes (incompressible Newtonian) fluid, the balance of linearmomentum can be written as,

ρ

(∂vr∂t

+ vr∂vr∂r

+vφr

∂vr∂φ

+vθ

r sinφ∂vr∂θ

− (v2φ + v2

θ )r

)= −∂p

∂r+

µ

[∂

∂r

(1r2

∂(r2vr)∂r

)+

1r2 sinφ

∂

∂φ(sinφ

∂vr∂φ

) +1

r2 sin2 φ

∂2vr

∂θ2 − 2r2 sinφ

(∂

∂φ(vφ sinφ) +

∂vθ∂θ

)]+ ρgr,

ρ

(∂vφ∂t

+ vr∂vφ∂r

+vφr

∂vφ∂φ

+vθ

r sinφ∂vφ∂θ

+(vrvφ − v2

θ cotφ)r

)= −1

r

∂p

∂φ+

µ

[1r2

∂

∂r

(r2∂vφ

∂r

)+

1r2

∂

∂φ

(1

sinφ∂

∂φ(vφ sinφ)

)+

1r2 sin2 φ

∂2vφ

∂θ2+

2r2

∂vr∂φ

− 2 cotφr2 sinφ

∂vθ∂θ

]+ ρgφ,

ρ

(∂vθ∂t

+ vr∂vθ∂r

+vφr

∂vθ∂φ

+vθ

r sinφ∂vθ∂θ

+(vrvθ + vφvθ cot φ)

r

)= − 1

r sinφ∂p

∂θ+

µ

[1r2

∂

∂r

(r2∂vθ

∂r

)+

1r2

∂

∂φ

(1

sinφ∂

∂φ(vθ sinφ)

)+

1r2 sin2 φ

∂2vθ

∂θ2+

2r2 sinφ

∂vr∂θ

+2 cotφr2 sinφ

∂vφ∂θ

]+ ρgθ.

(H.12)

82

Bibliography

[1] G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill Book Company (UK) Limited, 1974.

[2] H. A. Barnes, The yield stress- a review of π, αντα - everything flows?, J. Non-NewtonianFluid Mech., 81 (1999), pp. 133–178.

[3] R. B. Bird, Useful non-Newtonian models, Ann. Rev. Fluid Mech., 8 (1976), pp. 13–34.

[4] R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, VolumeI, John Wiley & Sons, second ed., 1987.

[5] R. B. Bird, G. C. Dai, and B. J. Yarusso, The rheology and flow of viscoplastic materials,Reviews in Chemical Engineering, 1 (1983), pp. 1–70.

[6] B. D. Coleman, Kinematical concepts with applications in the mechanics and thermodynam-ics of incompressible viscoelastic fluids, Arch. Rational Mech. Anal., 9 (1962), pp. 273–300.

[7] B. D. Coleman, R. J. Duffin, and V. J. Mizel, Instability, uniqueness, and nonexistencetheorems for the equation ut = uxx − uxtx on a strip, Arch. Rational Mech. Anal., 19 (1965),pp. 100–116.

[8] B. D. Coleman, H. Markovitz, and W. Noll, Viscometric Flows of Non-NewtonianFluids, Springer-Verlag, 1966.

[9] M. M. Cross, Rheology of non-Newtonian fluids- A new flow equation for pseudoplastic sys-tems, J. Colloid. Sci., 20(5) (1965), pp. 417–437.

[10] J. E. Dunn and R. L. Fosdick, Thermodynamics, stability and boundedness of fluids ofcomplexity 2 and fluids of second grade, Arch. Rational Mech. Anal., 56 (1974), pp. 191–252.

[11] G. Duvant and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, 1982.

[12] G. P. Galdi, M. Grobbelaar-Van Dalsen, and N. Sauer, Existence and uniqueness ofclassical solutions of the equations of a second-grade fluid, Arch. Rat. Mech. Anal., 124 (1993),pp. 221–235.

[13] G. P. Galdi and B. D. Reddy, Well-posedness of the problem of fiber suspension flows, J.Non-Newtonian Fluid Mech., 83(3) (1999), pp. 205–230.

83

[14] A. E. Green and P. M. Naghdi, A note on invariance under superposed rigid body motions,J. Elasticity, 9 (1979), pp. 1–8.

[15] C. Guillope and J. C. Saut, Existence results for the flow of viscoelastic fluids with adifferential constitutive law, Nonlinear Analysis, Theory, Methods and Applications, 15 (1990),pp. 849–869.

[16] C. Guillope and J.-C. Saut, Existence and stability of steady flows of weakly viscoelasticfluids, Proc. Royal Soc. Edinburgh, Section A-Mathematics, 119 (1991), pp. 137–158.

[17] D. D. Joseph, Instability of the rest state of fluids of arbitrary grade greater than one, Arch.Rational Mech. Anal., 75 (1981), pp. 251–256.

[18] D. C. Leigh, Non-Newtonian fluids and the second law of thermodynamics, Physics of Fluids,5 (1962), pp. 501–502.

[19] C. W. Macosko, Rheology: Principles, Measurements and Applications, VCH Publishers,Inc., 1994.

[20] P. M. Naghdi, Mechanics of solids, in Handbuch der Physik, C. Truesdell, ed., vol. VIa/2,Springer-Verlag, 1972, ch. The Theory of Shells and Plates, pp. 425–640.

[21] , Lecture notes for Continuum Mechanics, ME 185, Department of Mechanical Engineer-ing, University of California at Berkeley, 1987.

[22] J. G. Oldroyd, A rational formulation of the equations of plastic flow for a Bingham solid,Proc. Camb. Philos. Soc., 43 (1947), pp. 100–105.

[23] , Two-dimensional plastic flow of a Bingham solid, Proc. Camb. Philos. Soc., 43 (1947),pp. 383–395.

[24] T. C. Papanastasiou, Flows of materials with yield, J. of Rheology, 31 (1987), pp. 384–404.

[25] W. Prager, Introduction to Mechanics of Continua, Ginn, 1961.

[26] E. L. Reiner, Handbuch der Physik, vol. 6, Springer-Verlag, 1958.

[27] Z. Tadmor and C. G. Gogos, Principles of Polymer Processing, John Wiley and Sons,1979.

[28] C. Truesdell, A new definition of a fluid. II. The Maxwellian fluid, J. Math. Pure Appl.,30 (1951), pp. 111–55.

[29] , The mechanical foundations of elasticity and fluid dynamics, J. Rational Mech. Anal.,1 (1952), pp. 125–300.

[30] C. Truesdell and W. Noll, Non-linear field theories of mechanics, in Handbuch der Physik,S.Flugge, ed., vol. III/3, Springer-Verlag, 1965, pp. 41–47,.

84

[31] , Non-linear field theories of mechanics, in Handbuch der Physik, S.Flugge, ed., vol. III/3,Springer-Verlag, 1965.

[32] K. Yasuda, R. C. Armstrong, and R. E. Cohen, Shear flow properties of concentratedsolutions of linear and star branched polystyrenes, Rheol. Acta, 20(2) (1981), pp. 163–178.

85