COUPLING OF LINEAR AND ANGULAR MOMENTUM

IN CONCENTRATED SUSPENSIONS OF SPHERES

by

SHIHAI FENG, B.S., M.S.

A DISSERTATION

IN

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty

of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Chairperson of the Committee

Accepted

Dean of the Graduate School

August, 2003

ACKNOWLEDGEMENTS

I would like to thank my family for their support and understanding.

I would like to thank my advisor and mentor. Professor Alan L. Graham, for

his guidance, help, patience and understanding during my study. I am indebted to

Professor James R. Abbott and Professor Howard Brenner for their guidance and

help throughout my study. I would also like to thank Professor David Chaffin and

Dr. Samuel R. Subia for their help in numerical modeling presented in this work.

I would like to thank the students, faculty, and staff at the Department of Chemical

Engineering, Texas Tech University, for the enriching years they provided me. I would

also like to thank the High Performance Computing Center at Texas Tech University

for its support in numerical modeling in this work.

I would like to thank Professor Marc S. Ingber, Dr. Lisa A. Mondy, Professor

Richard Christian, Professor Jeremy W. Leggoe and Professor Philip W. Smith for

their help and thoughtful discussions. I would also like to thank Dr. Pat T. Rear-

don and Dr. Wenxian Lin for their assistance with the manuscript preparation. I

gratefully acknowledge support for this work from the U.S Department of Energy,

Division of Chemical Sciences, Geosciences and Biosciences, Office of Basic Energy

Science, U.S. Department of Energy, Environmental Projects Division, National En­

ergy Technology Laboratory and the Advanced Research Program from Texas Higher

Education Coordinating Board.

CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT v

LIST OF FIGURES vii

I INTRODUCTION 1

II NUMERICAL METHOD OVERVIEW 5

HI VORTEX VISCOSITY IN SUSPENSIONS 14

3.1 Introduction 14

3.2 General Theory 16

3.2.1 Internal Angular Momentum Equation 16

3.2.2 Linear Flow 19

3.2.3 Parabolic Flow 24

3.3 Numerical Experiments and Results 27

3.4 Conclusions 36

IV SPIN VISCOSITY IN SUSPENSIONS 38

4.1 Introduction 38

4.2 General Theory 39

4.3 Numerical Experiments and Results 43

4.4 Conclusions 50

V ENERGY DISSIPATION IN SUSPENSIONS 52

5.1 Introduction 52

5.2 General Theory 54

5.2.1 Equation of Energy Dissipation Rate 55

5.2.2 Velocity Flow Fields Examined 59

5.2.3 Linear Velocity Flow 59

5.2.4 Parabolic Velocity Flow 61

5.2.5 Cubic Velocity Flow 65

5.3 Numerical Experiments and Results 68

iii

VI CONCLUSIONS 76

BIBLIOGRAPHY 78

IV

ABSTRACT

In Newtonian fluids, the proportionality constant that relates the flux of linear

momentum to gradients in the velocity is the shear viscosity. Similarly, the flux

of angular momentum is related to gradients in the spin field by the spin viscosity

for simple fluids. In fluids in which external torque per unit volume is applied, the

equations of linear and angular momentum are coupled. In simple fluids, the equations

are coupled through the vortex viscosity, which relates the transfer of angular to linear

momentum to the difference in the local spin rate and one-half of the vorticity of the

fluid. Apparent viscosity is normally measured, for example, in a pressure driven flow

experiment, as a proportionality constant that relates the force acted on system and

the volume flow rate of the system. The purpose of this study is to evaluate shear

viscosity, vortex viscosity, and spin viscosity in suspensions by solving the coupled

momentum equations and study their relationships to the apparent viscosity and

energy dissipation.

In this study, we investigated the influence of particle spin imposed by external

torque on the apparent viscosity of non-colloidal suspension by using a simulation

based on the boundary element method (BEM). The numerical results reveal that

particles spin has a pronounced effect on apparent viscosity of the suspension. For

example, in suspension subjected to a pressure gradient in a tube, vary the spin

gradient from positive to negative by exerting external torque on the particles, the

apparent viscosity changes from close to zero to infinite with asymptotic change to

apparent negative viscosity. Apparent "negative" viscosity is due to torque driven flow

through vortex viscosity that is the proportionality constant between the transfer of

the Hnear to angular momentum flux and the difference between the spin rate and

the one-half local vorticity. Measurements of vortex viscosities from the numerical

experiments agreed the theoretical prediction of vortex viscosity of dilute suspensions

and extended the results to concentrated suspensions. The energy efficiency of torque

driven flow is compared to the energy efficiency of force driven flow.

External torques are applied to neutrally buoyant particles in boundary element

simulations to generate a cubic velocity profile in a suspension confined in a rectan­

gular channel. The quantitative values of spin viscosity are determined as a function

of solids fraction by calculating the volume averaged stress and kinematics of these

flows and combining these results with a theoretical analysis of the coupled equations

describing the conservation of linear and angular momentum. Many configurations

of particles at each concentration are generated and analyzed until reproducible av­

erages of the spin viscosity at each solids fraction are determined. The scaling of

the spin viscosity with the square of the particle size predicted in earHer theoretical

studies is verified in this investigation.

The conservation equation describing the rate of energy dissipation in suspensions

is derived, and the contributions due to spin viscosity are included. The energy

dissipation rate is shown to consist of the linear combination of terms that include the

shear viscosity, vortex viscosity and spin viscosity. Predictions from these equations

using the coefficients determined in the this work are compared with the energy

dissipation rates obtained from a macroscopic balance using the force and velocity on

the boundaries of the suspension in numerical simulations of a number of different

flow fields. Over the range of our data, these independent predictions are shown to

be in excellent agreement for both dilute and concentrated suspensions.

VI

LIST OF FIGURES

2.1 Mesh for cubic box 9

2.2 Mesh for tube 10

2.3 Error in T for cubic box 11

2.4 Error in T for tube 12

2.5 Error for two spheres 13

3.1 Torque driven linear flow 29

3.2 Torque driven parabolic flow 30

3.3 Pressure and torque driven flow 31

3.4 Apparent viscosity 33

3.5 Vortex viscosity 34

3.6 Shear viscosity 35

3.7 Efficiency of torque driven flow 36

4.1 Torque driven cubic flow 45

4.2 Vortex viscosity in cubic flow 46

4.3 Antisymmetric stress tensor in cubic flow 47

4.4 Spin viscosity is proportional to a^ 48

4.5 The relative contribution from ^g in angular momentum equation . . 50

4.6 Spin viscosity increases with (f) 51

5.1 Linear velocity flow can be induced by constant spin field 60

5.2 Parabolic velocity fiow can be induced by linearly spin field 63

5.3 Cubic velocity flow was induced by particles whose 0^ = a{y'^ — H^). 66

5.4 Viscosities are functions of cj) 69

5.5 The ratio of E^^ to El is a function of R/a 72

5.6 The relative contribution of energy dissipation rate from JJLS-, jJ-v and // 73

5.7 JE^ is equal to El at steady state 75

vi i

CHAPTER I

INTRODUCTION

The equations for the conservation of linear and angular momentum are decoupled

in fluids and suspensions in which the molecules and suspended particles are free

from external body couples. Small, neutrally buoyant spheres in dilute suspensions

rotate with one-half the local vorticity, w = V x v, of the average macroscopic flow

field [1]. Here v is the velocity of the fluid. One consequence of decoupling the

equations of the linear and angular momentum is that the resulting stress tensor, TT,

is symmetric [2, 3]. In Newtonian fluids, gradients in the velocity are proportional to

the flux of linear momentum where the proportionality constant is the shear viscosity,

/i. Dahler proposed a similar constitutive relationship [5, 6] for simple liquids in

the angular momentum conservation equation. Here, the proportionality constant

between gradients in the spin fields and the flux of angular momentum is spin viscosity,

The equations for conservation of linear and angular momentum are coupled and

the stress tensor is asymmetric in fiuids and suspensions in which the molecules and

suspended particles can be acted upon by external body couples. In these dipolar

fluids in which external torques act on the fluid, there is an inter-conversion of bulk

and internal angular momentum and these forms of angular momentum are not con­

served separately [8]. In these simple fluids, linear and angular momentum equations

are coupled through the vortex viscosity, /jLy, that is the proportionality constant

between the transfer of the linear to angular momentum flux and the difference be­

tween the spin rate and the one-half w. The resulting stress tensor is asymmetric

[9, 10, 11, 13, 14, 15, 16, 17, 18]. By balancing the hydrodynamic and external cou­

ples on a single particle [10] in dilute suspensions, Brenner predicted ^y = l.5(f)^o.

Here /XQ is the pure fluid shear viscosity, and (f) volume fraction of the particles.

Free rotation is possible only when no external couples act on the individual

particles. However, this condition is not always met. For example, if the center of

mass of each particle is displaced from its geometric center owing to inhomogeneities

in the internal mass distribution, couples will arise from the action of gravity whenever

the embedded gravity dipole does not lie parallel to the direction of gravitational field

[11, 10].

An external torque may promote the particles to rotate faster or slower than | w ,

or even rotate in opposite |u; . Particle spin can be induced by external torque. A

spatially inhomogeneous spin field can induce fluid flow in fluids that would otherwise

be quiescent [6, 26]. For example, magnetic or electric dipoles embedded within the

suspended particles will give rise to couple in the presence of external magnetic or

electric fields [19, 20, 21]. Magnetic fluids (MF) are colloidal solutions of magnetic

nano-particles in a fluid carrier [21]. Since the properties and location of this fluids

can be easily influenced by an external magnetic field, they have recently attracted

much scientific and technological interest [22, 23, 24, 25, 26, 27, 28, 29]. Some of

the studies [22, 23, 24, 26, 30] observed a reduction of apparent viscosity, r], of MF

in alternating magnetic field. This is so called apparent "negative" viscosity. One

possible explanation [24] is: The alternating magnetic field forces the particles in the

MF to rotate faster than |u; . The magnetic energy is partially transformed into the

angular momentum of the particles, which in turn is converted into a hydrodynamics

motion of the fiuid.

The first objective of this study is to use BEM simulations to study the interre­

lationship of r), /J, and fiy in dipolar suspensions. We intend to verify the theoretical

predictions for fj,y in dilute suspensions [10] and extend the results to the higher

concentration suspensions. Furthermore, we are interested in comparing the energy

efficiency of torque driven flows to that of force driven flows such as pressure drive

flow and flow generated by moving boundary.

The second objective of this study is to evaluate ^g in suspensions in which the

suspended particles are subjected to external torque. There are no quantitative values

reported for /i^ in dipolar fluids or in dilute or concentrated suspensions. As will be

discussed in greater detail later, this analysis requires a flow field in which there is a

gradient in the spin gradient on the particles. The simplest such rectilinear field is

a cubic velocity field generated by a quadratic spin field. By measuring the velocity

profile resulting from the quadratic spin field one can determine u and //„ from the

coupled set of equations describing the conservation of linear and angular momentum.

The last objective of this study was to investigate the role of fiy and yu in the

conservation of energy and test the consistency of dissipation rates derived by us­

ing the results of viscosities obtained by using volume averaged stress with those

obtained using energy dissipation rates determined by using forces and velocities on

the boundaries. Most text books do not include the fj.s in the energy equations. We

find that gradients in spin contribute to the overall energy dissipation rate in flow

fields in which the spin rate is not constant. The energy dissipation rate equations

derived here include these effects. The intent is to parallel the derivations using en­

ergy dissipation rate method [38] and by volume average stress tensor, TT, [7] that

led independently to identical predictions of the apparent suspension viscosity as a

function of solid fraction, (j). Einstein initially predicted the suspension viscosity, /x,

as a function of cj) by equating the energy dissipation rates. El, in suspensions to that

of a hypothetical one phase fluid in the same average flow field.

In Chapter II, the BEM codes used for the particle-level simulations are described.

The domain meshes and error analysis associated with the simulations performed in

this study are described. In Chapter III and Chapter IV the theoretical basis to

obtained fXy and fig for the data reduction is developed respectively. Then a series of

numerical simulations are conducted and the simulation results are analyzed by using

the equations obtained from the theoretical analysis to obtain jj,y and fig respectively.

In Chapter V, equations that describe the rate of dissipation of energy in suspensions

are derived. These equations include the contributions due to fiy and (Xg- A series

of numerical simulations were conducted on different flow fields in suspensions where

external torques were applied to the particles. The simulation results were analyzed

by using the energy dissipation rate equations obtained from the theoretical analysis

and Hy, and fig obtained in Chapter III and Chapter IV. The final chapter discusses

the major findings and presents the conclusions.

CHAPTER II

NUMERICAL METHOD OVERVIEW

Boundary element method (BEM) simulations are used in this study to model the

flow of neutrally buoyant suspensions of uniform spheres in Newtonian fluids. A basic

feature of the method is that the object needs to be discretized into elements only

along the boundaries. These are line contours for a two-dimensional flows and the

containing surfaces for three-dimensional flows. The basic governing equations of the

problem are solved for the whole domain but written in a manner that the unknowns

involved are the values of the parameters (such as velocity, traction) at nodes located

on the boundary only. Because no nodes exist in the interior of the object, the

unknowns of the problem are reduced significantly. This reduction decreases the

number of equations that must be solved. However, the resulting matrix is fully

dense. Having determined the nodal parameters on the boundary, the governing

equations can be used again to derive simple algebraic relations to obtain values at

the interior points with reference to parameters along the boundary nodes [32].

The governing equations for quasi-static creeping flow of particles are the Stokes

equations:

V - V = 0 , X G K (2.1)

V-7r = 0 , x G y , (2.2)

where V is the region exterior to the particles containing the fluid whose boundary

can be decomposed as N

S = Sj + Y^S„ (2.3)

where Sj represents the fixed portion of the boundary and Si represents the surface

of the zth particle and A' is the total number of particles.

In Newtonian fluids, the total stress field n is given by

7r^p5- ^( Vv + Vv^) - - £ . TT,, X G V, (2.4)

here (J is a unit tensor.

Equations (2.1) and (2.4) can be recast into integral form by considering a weighted

residual reformulation of the governing equations with weighting functions given by

the fundamental solutions of the Stokes equations [31, 35]. The fundamental solution

for the velocity field, V* ., and the associated fundamental solution for the stress field,

n-jfc, are given by

K^(x ,y ) = - ^ i 5 . , + S,S,), (2.5)

n^-.(-'y) = - i f ^ - (2-6)

Here 5ij is the Dirac delta function. Si is a unit vector in the ith direction, r is the

distance between the field point x and the source point y, Vfi^{x,y) represents the

fundamental solution of the ith velocity component at x point due to a force in the

A;th direction, F^, applied at y point, n*j^(x,y) represents the fundamental solution

of the components of TT at x point due to a force in the kth. direction applied at y

point. The relationship between fundamental solutions and velocity and traction can

be written as:

V * - F = v, (2.7)

n* • F = TT. (2.8)

The resulting boundary integral equation (BIE) can be written as:

cik{x)v,{x)+ / Illkix,y)vJ{y)S^{y)dS

^-jvUx,y)7r.{x)dS, (2.9)

where S represents the boundary of the exterior domain V, TT,- are the components

of the traction along the surface S and 5^ are the components of the unit outward

normal vector to the boundary S.

The velocity on the surface of each sphere can be related to the six components

of the sphere centroid velocity (three Hnear and three angular) through

v = v ' + fi' X ( r - r ' ) , (2.10)

where v ' and ft' are, respectively, the centroidal translational velocity and the angular

velocity of the ith sphere, r is the position vector of a point on the surface, and r' is

the position vector of the ith. sphere centroid. In this way, the number of unknowns

to be determined is dramatically reduced.

The surface of each sphere is discretized into A ; triangular and quadrilateral

boundary elements. For the super-parametric treatment used in this study, the

stresses and velocities are assumed constant within each element, but quadratic shape

functions are used to define the surface geometry. By evaluating the surface tractions

or velocities at the center of an element, an equation is generated that relates the

unknown surface velocities or tractions for the elements. The discretized equation for

the mth element takes the following form

Cik{x)v^{x)

+ J ] / Ul,{x,y)vJ^iy)N^{y)5.{y)dS^ m=l "^^"^

NB .

= - E / ^Ux,y)K{y)N-{y)dSrn, (2.11) m = l •^^"'

where Cik = ll25ij, v" represents the value of ith components of velocity at the

nth element, u]" and TT™ represent the values of the j t h components of velocity and

ith components of tractions respectively at the mth element, Nm{y) are the shape

functions, and NE is the total number of elements on all the surfaces of the spheres.

The algebraic system of equations is closed in the quasi-static analysis by enforc­

ing the force and torque balance. That is, for rigid spheres suspended in a Newtonian

fluid, the resultant forces and moments on the spheres generated by the surface trac-

tions, body forces and torques are zero. Or,

n-ndS + g' = O, (2.12) L I

'Si

(v - r') X {n • 7r)dS + T' = O. (2.13) I Si

Here g' is the body force acting on the ith sphere, which in the present study is zero

as the spheres are neutrally buoyant. Here T ' is the external torque acting on the ith

sphere.

In this method, the fundamental singular solutions of the governing differential

equations are continuously distributed over the boundaries of the problem, and the

boundary conditions then lead to integral equations for the densities of the fundamen­

tal solution. Thus, the solution of a differential equation in n dimensions is reduced

to the solution of an integral equation in n — 1 dimensions. In addition to reducing

the dimensions of the problem, this method is attractive for Stokes' problems since

it is a very general approach independent of the body geometry and the form of the

external flow field. The analytical solution of the integral equations is, in general,

not possible, but they can be solved numerically. A more detailed description of the

BEM can be found elsewhere [33, 32].

A cylindrical tube and a cubic box are used to represent the domains studied in

this study. The meshes of the two domains and the error analysis for these domains

are shown below in Fig. 2.1.

In a Newtonian fluid, if a rigid spherical particle with angular velocity Q, is im­

mersed in an unbounded fluid which velocity at is zero, the torque, Ttheory, on the

sphere can be described as [36]:

Ttheory = STTfXa^n. (2 .14)

Here /j, is the fluid viscosity, a is the radius of the sphere.

We use this theory as a benchmark to check the BEM accuracy by comparing

the torque on the particle obtained from simulation to the theoretical result. As

8

Figure 2.1: A three-dimensional BEM mesh of spherical particles in a parallel channel. Note the front wall mesh has been cut in order to view the particles clearly. The length of the meshed box is 2H x 2H x 2H. In this example, particle radius is 0.15i/. There are 60 particles in this simulation and <;/> = 11 %. There are 720 elements on the cubic box boundary. The number of elements on each particle is 48. Note the figure is generated by Tecplot^^ that uses lines to connect the nodes. In our simulation, we used super-parametric element with quadratic curves to connect the nodes. In order to smooth the geometry of the ball in this Tecplot^^ figure which is more representative of the actual geometry in the our simulation, we used a 200 element ball mesh in this figure.

shown in Fig. 2.3 and Fig. 2.4, comparison with analytical results of a Newtonian fluid

containing one small particle shows that the results converge monotonically with mesh

T - T theory T , heory

, in the densification on the particle. There is less than 1 % error, error =

T with 50 mesh elements on the particle and 1520 elements on the container in our

simulation. The simulations in this study were performed with 50 mesh elements on

each particle, 1520 elements on the tube container and elements on the box container.

The hydrodynamic force experienced by either of two identical spheres approach­

ing each other, at low Reynolds number, with the same velocity along their lines of

Figure 2.2: A three-dimensional BEM mesh of spherical particles in a tube. Note the wall mesh has been cut in order to view the particles clearly. The length of the tube is 4H and the radius of the tube is IH. There are 160 particles in this simulation and (f) = 10 %. There are 1600 elements on the cylinder boundaries. The number of elements on the particle is 48.

center can be calculated as [36]:

Sn/xayP (2.15)

here /? is the required correction due to the presence of the opposing sphere. It has

an exact solution[36] as:

10