AN ABSTRACT OF THE THESIS OF

RONALD NORMAN KNOSHAUG for the MASTER OF ARTS (Name) (Degree)

in MATHEMATICS presented on October 10, 1968 (Major) (Date)

Title: GENERALIZED DEFORMATION-RATES IN SECONDARY FLOWS OF VISCOELASTIC FLUIDS BETWEEN ROTATING SPHERES Redacted for Privacy kbstract approved: M.N.L. Narasimhan

By various experiments, it has been found that the re-

sponse of real materials to external forces is, in general,

nonlinear in character. In classical continuum mechanics, the use of ordinary measures of strain have forced the

constitutive equations to take complex forms and since the

orders of these measures are not fixed, many unknown re-

sponse coefficients have to be introduced into the consti- tutive equations. In general, there is no basis of choos-

ing these coefficients. Seth attempted to resolve this difficulty by introducing generalized measures in continuum mechanics and Narasimhan and Sra extended these measures in such a way as to adequately explain some rheological behav-

ior of materials. The constitutive equation of Narasimhan and Sra essentially contains two terms and four rheological constants and, unlike some previous theories, it does not contain any unknown functions of the invariants of kine- matic matrices while at the same time explains many visco- elastic phenomena. In the present investigation, a theorem has been proved establishing certain criteria for fixing the orders of generalized measures suitably so as to predict different types of viscoelastic phenomena, such as dilatancy. We have found during the course of this investigation that the constitutive equation of Narasimhan and Sra does not ade- quately explain such physical phenomena as pseudoplasticity.

However, in order to donstruct a constitutive equation so that it does explain such phenomena, we have found it nec- essary to construct combinations of sets of generalized measures. The resulting constitutive equation is found to be quite general and is able to explain a vast range of physical behavior of fluids. To illustrate the use of this constitutive equation based upon combined generalized measures, we have investi- gated the important problem of secondary flows for fluids in the presence of moving boundaries. This problem is very important, since the investigation of secondary flows al- lows us to obtain a clearer picture of the actual motion of the fluid. For the problem of flow of a fluid in thean- nulus of two rotating spheres, we have obtained the solu- tion for the velocity and pressure fields. In order to in- vestigate the secondary flow pattern more thoroughly, we have obtained the streamline function of the flow. The streamlines in meridian planes containing the axis of ro- tation are found to be closed loops and the nature of the closed loops is found to be strongly dependent upon the viscoelastic parameter S. For S less than critical value, the flow is found to be very much like that of a

Newtonian fluid, with the fluid advancing toward the inner sphere along the pole and outward along the equation. At the critical value the flow region is found to split into two subregions, each containing closed loops of stream- lines. As S is increased further, another critical value is reached whereby the streamlines again become one set of closed loops, with the sense of rotation reversed from that of Newtonian fluid. Generalized Deformation-Rates in Secondary Flows of Viscoelastic Fluids Between Rotating Spheres

by

RONALD NORMAN KNOSHAUG

A THESIS submitted to Oregon State University

in partial fulfillment of the requirements for the degree of

Master of Arts

June 1969 APPROVED:

Redacted for Privacy

Professor of Mathematics in charge of major

Redacted for Privacy

Acting Chairman of Department cif Mathematics Redacted for Privacy

Dean of Graduate School

Date thesis is presented October 10, 1968

Typed by Charlene Laski for Ronald Norman Knoshaug Acknowledgements

The author wishes to express his sincere appreciation to his major professor, Professor Narasimhan, for his sug- gestion of the problem, criticisms and suggestions during the course of the research, and especially his giving of unlimited time during the writing of this thesis.

The author wishes to express his sincere appreciation to his wife Judy for the moral support she gave him during the course of research.

The author thanks Charlene Laski for the excellent typing of the thesis The author is indebted to the Atomic Energy Commission for their financial support, (AT(45-1)-1947). Table of Contents

Chapter Page

I. Introduction

1.1 Classical Strain Measures 1 1.2 Present Constitutive Theory of Materials and Their Limitations 3 1.3 Purpose and Need for the Present Investigation 9 1.4 Plan of the Present Investigation 10

II. Measure Concept in Continuum Mechanics 2.1 Introduction 11 2.2 Generalized Measures of Deformation in Elasticity 12 2.3 Generalized Measures of Deformation-Rates in Fluid Dynamics 16 2.4 Generalized Measures of Deformation-Rates in Viscoelasticity 19

III. Extension of Generalized MeaSure Concept 3.1 Introduction 22 3.2 A Theorem on the Classification of Visco- elastic Fluid based on Orders of Measures 23 3.3 Combined Generalized Measures in Visco- elasticity 34 3.4 Cpnstitutive Equation based on Combined Generalized Measures 36 IV. Primary and Secondary Flows in the Presence of Moving Boundaries 4.1 Introduction 39 4.2 Secondary Flows in Classical Fluids 40 (Theoretical Results) 4.3 Secondary Flows in Non-Newtonian Fluids (Theoretical Results) 42 4.4 Experimental Results on Secondary Flow 50

V. Application of Generalized Measures of Deformation- Rates to Secondary Flows of Viscoelastic Fluids between Rotating Spheres 5.1 Formulation of the Problem 54 5.2 Method of Solution of the Equations of Motion 65 5.3 Primary Flows in Pseudoplastic Fluids 66 Chapter Page

5.4 Secondary Flows in Pseudoplastic Fluids 68 5.5 Discussion of the Results 78 VI. Summary, Discussion, and Scope of Further Work

6.1 Summary and Discussion 82 6.2 Scope of Further Work 85

Bibliography 86 Appendix 89 List of Figures

Figure Page

3.1 Simple shear model 25

3.2 Graph of apparent coefficient of viscosity for a pseudoplastic fluid 38

4.1 Typical Streamlines of steady radial-axial flow. One disk oscillating 41

4.2 Typical Streamlines of steady radial-axial flow. Two disks oscillating 41

4.3 Streamlines of secondary flow about a sphere rotating in a fixed spherical container 42

4.4 Secondary flow in a tube with elliptical cross- section 43

4.5 The secondary flow pattern in one quadrant of annulus of two spheres for a fluid whose para- meters have passed the critical values 46

4.6 The flow pattern of a viscoelastic fluid between two rotating cones 48

4.7 The projections of the particle paths on any plane containing the axis of rotation when m = 1/6. 49

4.8 The projections of the particle path on any plane containing the axis of rotation when m = 1/4. 50

4.9 Drawing showing flow characteristics 51

4.10 Vortices generated by a rotating cone immersed in 7.5% solution of Polyisobutylene in Cetane 52

5.1 Notation 56

5.2 Typical streamlines of the secondary flow for the case S = 0[01 = 0.2, Q2 = 0.1, and A = 2]. 80 List of Figures (con't)

Figure Page

5.3 Typical streamlines of the secondary flow for the caseS = 4.5[Q1 = 0.2, Q2 = 0.1, and A = 2]. 80

5.4 Typical streamlines of the secondary flow for the case S = 5.5[Q = 0.2, Q = 0.1, 1 2 and A = 2]. 81

Notation

We will be using the notation of tensor analysis and

hence shall assume some knowledge of tensors. We shall ex-

plain the meaning of the various symbols used as and when

they arise. However, we shall list the more frequently used symbols which may be referred to whenever needed:

Symbol Meaning

ai ithcomponent of acceleration.

a,b,c physical components of acceleration in spherical-polar coordinates along the r-, 8-, and cp- directions res- pectively.

A,B,C non-dimensional components of accel- eration in spherical-polar coordin- ates along the r-, 8-, and (1)- directions respectively.

dimension correcting constants. a2 b.. second deformation-rate tensor. 13 Symbol Meaning b*.. generalized second deformation-rate 1J tensor.

B second deformation-rate matrix.

B* generalized second deformation-rate matrix

B** combined generalized second deforma- tion-rate matrix

radius of inner sphere

d.. first deformation-rate tensor 13

d*.. generalized first deformation-rate 13 tensor

D first deformation-rate matrix

D* generalized first deformation-rate matrix

D** combined generalized first deforma- tion-rate matrix

a13.. kronecker delta

A Laplacian in spherical-polar coordin- ates

A* non-dimensional Laplacian in spheri- cal-polar coordinates

D material derivative Dt

a intrinsic derivative e strain tensor e.. strain tensor 1J Symbol Meaning e*.. generalized strain tensor 13

E strain matrix

E* generalized strain matrix

dimension correcting constant y dimension correcting constant h non-dimensional rheological parameter

H coefficient determinant

I unit matrix

I II III first, second, and third invariants B' B' B of the second deformation-rate matrix

II III first, second, and third invariants D ' D' D of the first deformation-rate matrix

II , III first, second, and third invariants e e e of the strain tensor

J, set of positive integers, positive j0' je odd integers, positive even integers respectively k, 1(1, k2, k' dimension correcting constants

K non-dimensional rheological parameter deformed and undeformed lengths res- 510 pectively

L length

A ratio of outer sphere to inner sphere m, m, m, m', m dimension correcting constant 1 2 M mass

1.1 dimension correcting constant Symbol Meaning

n, n1,, n2,n' measure indices v apparent viscosity a 0 0 angular velocities of the inner and 1' 2 out sphere respectively

hydrostatic pressure

P non-dimensional hydrostatic pressure

irreversibility indices q, q2, q' r, e, spherical-polar coordinate system density of fluid

S non-dimensional rheological parameter t time

stress tensor

deviatoric part of stress tensor

t( , t(yz) ,t , physical components of the stress xy) (xz) tensor in the rectangular coordinate etc. system t t physical components of the stress (rr)' (re)' (r41)) tensor in the spherical-polar coor- etc. dinate system

T stress matrix

T time u,v,w physical components of velocity in the spherical-polar coordinate system

U,V,W non-dimensional components of veloc- ity in the spherical-polar coordinate system Symbol Meaning

.th V. component of velocity

velocity component in the rectangular coordinate system in z-direction

spatial coordinates

cartesian coordinates GENERALIZED DEFORMATION-RATES IN SECONDARY FLOWS OF VISCOELASTIC FLUIDS BETWEEN ROTATING SPHERES

CHAPTER I

INTRODUCTION

1.1 Classical Strain Measures

The fundamental conservation laws of continuum mech- anics are applicable to all types of media. But different materials with the same mass and geometry when subjected to the same external forces in an identical manner are found to respond differently. This diversity in behavior is owing to the differences in their internal constitution. Therefore, to explain the response of a material when a force is applied, we need to establish a relationship be- tween the loading to which the material is subjected and its deformation or motion. For a viscous medium we must find a relationship between the stresses and the rate of deformation, while for an elastic medium we must find a re- lationship between the stresses and the resulting deforma- tion. This leads to the formulation of the constitutive equation of the material, which is a relation between the stress tensor and the deformation or the rate of deforma- tion tensor. It should be observed that the measure of stress is a definite quantity, that is, the force per unit area; however, the measure of strain of deformation is quite flexible, as it should be, to allow for different 2 types of deformations and flows.

In elasticity theory, various measures of strain have been used. In an uniaxial strain of a loaded wire, if ko

and it represent the undeformed and the deformed lengths of the wire respectively, then the various strain measures already in use in elasticity are (Seth, 1966h)

C 0 Cauchy measure: e ' 0

32-9, S 0 Swainger measure: e (1.2) 2,

H Hencky measure: e = log( R- (1.3) 0

A Almansi measure: e =1 [1-(I-9)21 (1.4)

2 G ( Green measure: e = jL) -1]. (1.5) 2 2, 0

In the classical theory of elasticity, the Cauchy measure is used when the strain is small and only the linear terms of displacement gradients referred to the unstrained state are retained; the Swainger measure uses linear displacement gradients, but referred to the strained state; the Hencky measure is useful in plasticity; and the Almansi and Green measures are used in the theory of finite deformations. 3

In fluid dynamics, the rate of strain must be used and the classical measure of the deformation-rate is

1 d1.. =(v. . + v3..), (1.6) 3 2 1;3 ;1 where

_ th v1 . = 1 component of velocity,

and ;j denote covariant partial differentation with respect to xi and xj, respectively.

1.2 Present Constitutive Theory of Materials and Their Limitations

We now briefly mention some of the commonly used con- stitutive equations in current literature and point out their limitations.

Newtonian Fluid

T = (-p + xle)I + 2pD (1.7) where

T E stress matrix,

1.1,X E coefficients of viscosity, 1 constant at given temperature, D E deformation-rate matrix,

oE dilatation, P E hydrostatic pressure. 4

Even though equation (1.7) describes very well the be- havior of some fluids, such as certain gases, water, al- cohol, and so on, many materials such as high polymer sol- utions, pastes, paints, colloidal solutions, paper pulp, etc. exhibit certain phenomena which cannot be described by the classical Newtonian stress-strain velocity relation

(1.7). Examples of such phenomena are: the swelling of a fluid at the exit of a tube, known as the Merrington effect

(Merrington, 1943), and the climbing of a fluid along the inner cylinder of a viscometer under rotation, known as the

Weissenberg effect (Weissenberg, 1947), both very common in technological and biophysical experiments; variable vis- cosity, visco-elasticity, plasticity, creep, fatigue, time dependency, pseudo-plasticity, dilatancy, relaxation, etc. which abound in everyday experiences, as well as in in- dustry. Secondary flows are worthy of special mention (for more details on secondary flows, see Chapter IV). In order to describe these phenomena, numerous investigators have posed various constitutive equations, of which the main ones are:

Reiner (1945) and Rivlin (1948) Fluids

2 T = -pI +a1lp+a2D , (1.8) where 5

T E stress matrix, a E coefficient of viscosity, 1 D E deformation-rate matrix, a E coefficient of cross-viscosity, 2 p E hydrostatic pressure.

This equation appears mathematically simpler than others, but the rheological coefficients a and a are unknown 1 2 functions of the invariants of the deformation-rate tensor and the theory offers no means of specifying them expli- citly. This theory always predicts the existence of two equal normal stresses in certain steady viscometric flows, which is contrary to experiments performed by Padder and

Dewitt (1954).

Oldroyd Fluids (1950)

(1 + --)t!. 2k(d. t! + d. !m) 1 Dt 13 1 im 3m

= 21.1(1 + A2 - 811k d. . , (1.9) 2 Dt ij 2 im 3 where

X E relaxation-time constant, 1

a _ 77-t_intrinsic derivative,

DA. . aA.. = _L-1+ A. vm + A vm+ A. vm. at at ij;m raj ;i im ;3 ' 6

A.. being any given tensor, 13 t'. E deviatoric part of the stress tensor, 13 k and k E arbitrary scalar constants, 1 2 d..E deformation-rate tensor, 13 p E coefficient of viscosity,

X E retardation-time constant. 2

Oldroyd's theory involves a number of material constants and the non-linearity has been introduced in a very arbi- trary manner.

Rivlin and Ericksen Fluids (1955)

N T aoI + V. ak(7rk + 71), (1.10) k=1 where

a(k = 0,1,...,N) are unknown functions of the k invariants of the kinematic matrices D(1) D(2)

(n) ,

Tr are certain matrix products formed from the k (1) (2) (n) kinematic matrices D , D ,..., D ;

Trk is the transpose of Trk;

the kinematic matrix D(r) EMd(r)iillis defined

(1) 1 by d = = + v. ) and ij 2 1;j 3;i 7

-1) ) (r) _ ij m(r-1) (r-1)m (r-1)m d = + vd . + d . v + d . v . ij 1j;m 1m ;j 3m 7.1

(r > 2) .

Rivlin and Ericksen's theory is successful in obtaining normal stresses which need not be equal, but the resulting

constitutive equation has been made very complicated by the

introduction of higher order kinematic matrices and unknown

functions of their invariants.

Green and Rivlin Fluids (1957)

R = E (t,T1,T2,...,TN) J N=0 -.0 3Plql PNqN

x (T)...g (T)dT...dT p q N 1 N Plql 1 N where

the kernels cpij...pNqN are continuous functions (0) of the indicated arguments and also of D , pa (1) D D") , defined in Rivlin and Ericksen's

theory,

E X (T)X (T), gPg(T) k,p k,q

t..E stress tensor. 13

Green and Rivlin's theory is a further generalization of 8

Rivlin and Ericksen's theory, hence the remarks on the limitations of Rivlin and Ericksen's theory apply here as well. Equations (1.11) become quite unwieldy except pos- sibly in a few simple cases.

Noll Fluids (1958)

CO T' = 0.4- (G(s)), (1.12) s=0 where

,cyl is the constitutive functional and.

G(s) is the history of the relative deformation

gradient.

Noll's theory, somewhat similar in idea to the theory of

Green and Rivlin, depends upon the experimental determina- tion of the viscosity function and two normal stress func- tions. The source of all the complications in the constitu- tive equations mentioned above is the use of ordinary mea- sures. Often the linear measures of strain or strain-rate are used even though we know from experiments that the strains and strain-rates are nonlinear in character. Since the order of these measures is not fixed, thereis no basis of choosing the response coefficients in these constitutive equations. 9

In order to overcome this difficulty, Seth (1966a,

1966b) introduced generalized measures of strain into elas- ticity. He also suggested the use of generalized measures of deformation-rate in viscoelasticity. But Narasimhan and

Sra (1968) showed that the mere use of generalized measures of deformation-rate always predicts two of the normal stresses to be equal in certain viscometric flows. Since the viscoelastic behavior depends not only on the deforma- tion-rate involving velocity gradients but also on a second deformation-rate involving acceleration gradients, they suggested the use of generalized measures of a second de- formation-rate along with the generalized measure of the first deformation-rate to overcome the above difficulty. They illustrated this point of view by constructing a con- stitutive equation for dilatant fluids.

1.3 Purpose and Need for the Present Investigation

In the present investigation, we find that the mere use of just one set of generalized measures of deformation- rates is inadequate to describe phenomena such as pseudo- plasticity, creep, relaxation, and so on. It will be shown in the text of this thesis that it is necessary to combine generalized measures of different orders to explain a var- iety of viscoelastic or viscoplastic phenomena. In order to illustrate the need for combining such generalized mea- 10

sures, suitable constitutive equations based on the mea-

sures are constructed for pseudoplastic fluids, which are characterized by the property that their apparent coeffi- cient of viscosity decreases with the increase in rate of

shear. The problem of secondary flows of such fluids be- tween two rotating concentric spheres is formulated and

solved to illustrate the pseudoplastic behavior.

1.4 Plan of the Present Investigation

The work is divided into six chapters. In Chapter I, we give the present constitutive theory of materials and their limitations. Chapter II, deals with the present state of the measure concept in continuum mechanics.

Chapter III is devoted to classifying viscoelastic fluid based on orders of measures and developing a constitutive equation based on combined generalized measures. In Chap- ter IV, we discuss primary and secondary flows in the pre- sence of moving boundaries. In Chapter V, we use the con- stitutive equation based on combined generalized measures to illustrate the secondary flows in the annulus of two spheres rotating about a common axis. Chapter VI contains the summary, discussion and scope of further work. 11

CHAPTER II

MEASURE CONCEPT IN CONTINUUM MECHANICS

2.1 Introduction

In recent years, many research workers have devoted themselves to the construction of suitable constitutive equations in order to explain the rheological behavior of materials. These constitutive equations involve many non- linear terms as well as unknown response coefficients.

There are two types of non-linearities which are commonly introduced. One may be termed as parametric non-linearity involving a variable rheological parameter in the constitu- tive equation and the other is the well known tensorial non-linearity involving a non-linear tensorial relationship between stress and strain or strain-rate. Both of these types of non-linearities have led to the introduction of many unknown functions and response coefficients in the constitutive equations. The source of all this trouble is the use of ordinary measures of strain or strain-rate in the constitutive equations. Any such restriction placed on the measure of strain will naturally result in compli- cating the constitutive equation since the order of the measure used is not fixed in relation to the actual phy- sical phenomena. In order to overcome this difficulty,

Seth (1966a, 1966b) introduced generalized measures of strain 12 and explained a number of irreversible phenomena in contin- uum mechanics.

2.2 Generalized Measures of Deformation in Elasticity

In section 1.1, the various classical strain measures used in elasticity theory were listed. Seth (1966b) served that these strain measures (1.1)-(1.5) could be given Riemann integral representations with suitable weight functions. These are

z

ec=Jr clZ , (2.1) k k 0 0

S rk (k0)2dk e = (2.2) -r- r- ' 2. 0 0

Z (2, eH Jr 0 dk (2.3) k \Z ZO 0

A 1 k CO) 3 dk e = (2.4) 9, k k 0 0

G irk (2'0)-1 dk e = (2.5)

'0 2'

Upon examination of these integral representations, Seth noted the following generalizations: If e* represents a general strain measure, then

irk(fton+1 * 1 e = 1 = (2.6) k k ' ZO ° 13 with

k n+1 ( , as the we'weight function.. (2.7) Q)

Note that, forn = -1,1,0,2,-2 in (2.6) we obtain (2.1)- C A (2.5) respectively. If eii , eii , eii , and eii (no sum

ali)are the principal Cauchy, Swainger, Almansi, and Green measures respectively, then the generalized principal mea-

sures et. (no sum on i) in Cartesian coordinates may be 11 written as C e.. 11 -n-1 et.=ie1..)(1 + de1.. (2.8) 11 1 1 0

1 C -n + 1) -1], = n[(e.

S e n-1 et =f 1i(1- de.1. (2.9) t 11 1 J

1 n H(1 (1 eii)

A e.. A f- 1 A et (1 ) de (2.10) .4 2e ii 0

n 1 7 = -(1 - 2e..) ] 11 ' 14

G -w -1 G z G et = ell(1 + ) de (2.11) ii ii 0

7 = + 2eG. ) 1], where n may be called the measure index.

A further generalization is (Seth, 1966b)

et. (1 , (2.12) 11 2me..)11 71c1 (mn) q[

where k and mare dimension correcting constants and n and q are the measure and irreversibility indices respectively.

If n is a positive even integer and q a positive inte- ger, then by the use of the Cayley-Hamilton theorem all powers of e.. higher than the second can be expressed in 11 2 terms of e1.. and e.., obtaining 1 11

e. + F e,. + Fe2. . (2.13) 11 1 11 2 11 '

where F.(i = 0,1,2) contain only a finite number of

terms in the invariants I , II , III of the e e e strain tensor e. By a rigid rotation of the frame of 15

reference, (2.13) becomes

e*. = F d. . + F e. . F e e . (2.14) 13 0 13 1 13 2 im m3

where F.1 (i = 0,1,2) remain unchanged since I ' II ' e e III are invariant under rigid body rotations. For gen- e eral values of n and q, Fi(i = 0,1,2) are infinite

series.

Likewise, the generalized measure (2.12) can also be

written as

e*, = - 2me..) (2.15) 13 .racinq 13 13 13

or in matrix form

E* [I-(1 -2mE)7] , (2.16) mqnq

where

= 116ii II

and E = II

The constitutive equation based on the generalized strain measure E* can be constructed from the classical

Hookean constitutive equation 16

T = X'6I + 211E, (2.17)

where

T E stress matrix

and

xt,n E Lame's constants,

by replacing E by E* to obtain

T = X6I + 211E*, (2.18)

or from (2.14) we obtain

2 T' = F'I + F'E + F'E (2.19) where

F!(i = 0,1,2)

are known functions of the invariants of E. This consti- tutive relation (2.19) has been successfully used by Seth in explaining a number of phenomena in non-linear elasti- city.

2.3 Generalized Measures of Deformation-rate in Fluid Dynamics

Using the concept of section 2.2, Seth (1966b) general- ized the measure of deformation-rate appropriate to fluid 17

dynamics to obtain

d11.. 2 "" d. = (1-2md..) d(d1..1 ), (2.20) 11 11

-41-(1-2md..)7), m1n 11

wheremis a dimension correcting constant and

dii 1 viii]. A further generalization (Seth, 2 1;3 1966b) would be

n- d = 1- (1-2mdii)2 (2.21) 11 q mn where k is again a dimension correcting constant and

n and q are the measure and irreversibility indices,

respectively. Following section 2.2, by use of the Cayley-

Hamilton theorem, (2.21) can be expressed as

2 d. = G+ G d.. + G d.. (2.22) 11 o 1 11 2 11 ' where G.(i = 0,1,2) contain only a finite number of terms

in the invariants I II III of the deformation- D ' D ' D rate tensor D if n is an even positive integer and q

a positive integer. By a rigid rotation of the frame of reference, (2.22) becomes

dt. = G S.. + G,d.. (2.23) 0 13 13 G22 didim m3m3, 18

where G.( i = 0,1,2) remain unchanged since I , II D D ' and III are invariant under rigid body rotations. D Likewise, the generalized measures can also be written as (Narasimhan and Sra, 1968)

n, 7 q D* = [I-(I - 2mD) (2.24) mqnq where

The constitutive equation based on the generalized deformation-rate measureD* can be constructed from the classical Newtonian constitutive equation by replacing

D by D* to obtain for incompressible fluids (Narasimhan and Sra, 1968)

T = -PI + 2pD* (2.25) or from (2.23)

T = (-P + + Gip +G2D2 (2.26) where

T E stress tensor,

G.(i = 0,1,2) are known functions of the invariants of D. 19

Hence the need for assuming unknown response functions in constitutive equations is eliminated and the non-linearity is condensed into just two terms.

2.4 Generalized Measures of Deformation-Rates in Viscoelasticity

In certain viscometric flows of viscoelastic fluids, equations (2.25) and (2.26) always predicted two normal stresses, contrary to experiments (Narasimhan and Sra,

1968). Since the viscoelastic behavior depends not only on the deformation-rate involving velocity gradients but also on a second deformation-rate involving acceleration gra- dients, Narasimhan and Sra (1968) suggested the use of generalized measures of a second deformation-rate along with that of the first deformation-rate to remedy the a- bove difficulty. It should be noted here that the deform- ation-rate matrix D of (1.6) shall henceforth be called the first deformation-rate matrix and B shall be called the second deformation-rate matrix. The generalization of the measure of the second deformation-rate, following the same procedure as for the first deformation-rate, led to the equations

k' 2 B* I (I 2m1B)11' (2.27) m c1 ,c1 20

where

B* E b E generalized second deformation-rate ij matrix,

1 B* , with b.. =(a. .+ a. + = jjbi3 13 2 1;3 3;1 m v v .) (2.28) mvi ;3

and

1. .th a1 component of acceleration, D

k' and m'E dimension correcting constants,

n' and q'E measure and irreversibility indices, respectively,

and

2 B* = H01 +H1B+H2B , (2.29) where H.1 (i = 0,1,2) are known functions of the invariants

I II III of the second-deformation-rate tensor B. B ' B ' B The constitutive equation based on the generalized

deformation-rate tensors D* and B* can be constructed from the classical Newtonian constitutive equation for

incompressible fluids T = -pI + 2pDby replacing D by

the generalized deformation-rate tensors D* and B* to obtain (Narasimhan and Sra, 1968)

T = -pI + 2pD* + 4nB* (2.30) or 21

2 2 T = (-p + Go + HI +G1D+G2D+H1B+H2B (2.31)

where, = 0,1,2) are known functions of the invariants

ID , IID , IIID , and Hi(i = 0,1,2) are known functions

of the invariants IB , IIB , IIIB.

Narasimhan and Sra (1968) observed that the use of generalized measures of deformation-rates in viscoelastic fluids helps to simplify the constitutive equations by prescribing a priori the order of the measures of deform- ation-rates, to eliminate the need for assuming additional response coefficients, and to explain adequately the rheo- logical phenomena usually observed in experiments. 22

CHAPTER III

EXTENSION OF GENERALIZED MEASURE CONCEPT

3.1 Introduction

The constitutive equation (2.25), using only the

generalized measure of the first deformation-rate tensor,

always predicts two of the normal stresses to be equal in

certain viscometric flows, contrary to experiments.

Narasimhan and Sra observed that viscoelastic behavior de-

pends not only on the deformation-rate involving velocity

gradients but also upon deformation-rate involving acceler-

ation gradients. Hence, to eliminate the deficiency of equation (2.25), they suggested the use of generalized measures of the second deformation-rate along with those of the first deformation-rate, resulting in equation (2.30).

By using the generalized measures of deformation-rates in viscoelasticity, a constitutive equation is constructed which contains known response coefficients and is adequate

in explaining most rheological phenomena. However, we ob- serve that the mere use of the generalized measure men- tioned above is inadequate to describe such well known phenomena as pseudoplasticity, creep, stress-relaxation, and so on. In order to explain the above mentioned pheno- mena, it is found necessary to construct combinations of sets of generalized measures of different orders instead 23

of just using one set of generalized measures. In this

chapter, we extend the generalized measure concept by com- bining two sets of generalized measures of different orders

and construct a constitutive equation which will adequately describe pseudoplasticity.

3.2 A Theorem on Classification of Viscoelastic Fluids Based on Orders of Measures

Using the constitutive equation (2.30) by Narasimhan

and Sra (1968), we shall classify viscoelastic fluids by

specifying the measure and irreversibility indices n,n'

and q,q' respectively in equations (2.24) and (2.27). But first, we must define precisely what is meant by a fluid exhibiting pseudoplastic, Newtonian, or dilatant behavior.

Definition 3.1. The apparent coefficient of viscosity of a fluid is defined to be the ratio of the shear va stress to the rate of shear.

The apparent coefficient of viscosity is defined thus in order to compare it with the classical viscosity coef- ficient and determine the behavior of a viscoelastic fluid

in relation to the classical viscous fluid. For a real

fluid, the apparent coefficient of viscosity should be positive. 24

Definition 3.2. Fluids for which the apparent coefficient of viscosity decreases with increase in rate of shear are

defined to be pseudoplastic fluids, while those for which

it increases with increase in rate of shear are defined to

be dilatant fluids.

Fluids which satisfy equation (1.7) are classified as

Newtonian fluids and it should be noted that the coeffi-

cients of viscosity for these fluids are constant for

given temperature. It is interesting to note that, theo- retically at least, it should be possible by choosing a

suitable combination of orders of measures to obtain a fluid for which the apparent coefficient of viscosity turns out to be a constant, quite analogous to the situation in

the classical viscous case but the fluid is not described by the Newtonian constitutive equation (1.7).

The behavior of apparent coefficient of viscosity can be studied by considering a shear model in which a flow is maintained between two parallel plates, one of which

(x = h) is moving relative to the other (x = 0) with velocityVparallel to the z-axis (see figure 3.1). Since the channel is taken to be of infinite extent in the y-direction, all quantities will be independent of y. As- sume the pressure gradient to be zero. Then from the equa- tion of continuity, it follows that if v=;(u,v,w), 25

awl 0 or w = f (x) (3.1) az 1 x

A x = h v = (0,0,v)

x = 0 z v = (0,0,0)

(Figure 3.1) Simple Shear Model

In this case the first and second deformation-rate tensors as given by equations (1.6) and (2.28) are

1 0 0 --w,2 1(x)

p ild..13il = 0 0 0 and 1 --w '(x) 0 0 21

2 [w'(x)] 0 0 1

B = = 0 0 0 13

0 0 0 where the prime indicates differentiation with respect to x 26

Then it followsif p is a positive integerthat

1 2p '-1 0 0 [-2 w11 (x) ]

2p' -1 D 0 0 0 (3.2)

2p'-1 [w 1(x)] 2 1 0 0

1 2p' fw '(x)] 0 0 2 1 2p' D 0 0 0 (3.3)

2p1 0 0 [7w1 (x)]

and

p [101 (x) ] 0 0

BP 0 0 (3.4)

0 0 0

Definition 3.3 Define J = {x:x is a positiveinteger}, J = {x:x is a positiveoddinte- 0 ger}, and J = {x:x is a positive even inte- E ger}.

TheoremSuppose the measure indices n,n' c JEand the

irreversibility indices q,q' c Jin the equations (2.24) and (2.27) and further

(i) if n = 2 or 4 and q = 1, then the fluid described by (2.30) is, in general, non-Newtonian 27

in nature but exhibits a constant apparent co-

efficient of viscosity.

(ii) If n = 2, q > 2with q c JO , or if n > 4,

q = 1, or if n > 2, q > 1with q e J, then the fluid described by (2.30) either exhibits

dilatant behavior or does not exist because the

apparent coefficient of viscosity is non positive

(iii) If n = 2, q e JE, then the fluid described by

(2.30) does not exist because the apparent co-

efficient of viscosity is non-positive.

Proof Since we have required n and n'e JE, let n = 2r andn' = 2s with r and s e J. Then the con- stitutive equation (2.30) becomes

T = -pI + 2pD* + 4flB *, (3.5) where D* I-(I 2mD) (3.6) (2rm)cl [- and k' B* = [I-(I 2m'BH (3.7)

(2sml)'

By use of the binomial expansion, (3.6) and (3.7) become

k r. D* = E(f)(-2mD) (3.8) (-2rm)n.-1 L=1 and 28

s k' 0 1-.q B* 1: ()(-21TI'B)- (3.9) (-2sm1)~ k=1

k Since B contributes only to the normal stress of- feats in the simple shear model (see 3.4), then B* con- tributes nothing to the shear stress and hence nothing to the apparent coefficient of viscosity. Thus the only term in the constitutive equation (3.5) which contributes to the shear stress is D* and hence D* is the only term we shall examine. Since m in equation (3.6) is an arbi- trary rheological parameter which can take on either posi- tive, negative, or zero values, then for convenience, we replacemby -m in (3.8) to obtain

r k D* (r)(2mD1) . (3.10) (2rm)q[k=1

Case (i): If n = 2(r = 1) and q = 1, then. (3.10) be-

D* = k D (3.11) and eauation (3.5) becomes

T = -pI + 2aD + 4nB* (3.12) where

a E pk. 29

For m' = 0 or k' = 0 in B*, equation (3.12) reduces

to the classical Newtonian equationT = -pI + 2aD for

incompressible fluids. For m' 0 and k' 0 in B*,

the shear stress components are

t = 0 = t and t = awe (x) (3.15) (xy) (yz) (xz)

and the apparent coefficient of viscosity becomes

shear stress txz v = a, a constant. a shear rate w'(x) (3.14)

Again, if n = 4(r=2) and q = 1, then (3.10) be-

comes 2 D* = kD+km D (3.15)

and equation (3.5) becomes

2 T = -pI + 2aD + 2amD + 4flB *. (3.16)

The shear stress components are

t(xy) = 0 = t(yz) and t(xz) = aw1(x) (3.17)

and the apparent coefficient of viscosity becomes

= a, a constant, (3.18) a which proves case (1). 30

Case (ii): If n = 2(r=1) and q > 2 withq e J0 , then

(3.10) becomes

2k+1 D* = k D (3.19)

where q = 29, + 1 and 2. e J. Equation (3.5) becomes

2k+1 T = -pI + 2aD + 4flB *, (3.20)

the shear stress components are

2k+1 a t = 0 = t and t w' (3.21) (xy) (yz) (xz) 2k xd 2

and the apparent coefficient of viscosity is

w' (x) 2k a, (3.22) a 2

an increasing function of the rate of shear W(x) since

a >0 for a real fluid in view of (3.22).

If n > 4(r > 2) and q = 1, then (3.10) becomes

r k k D* q) (2mD) (3.23) 2mr L,7 k=1

and (3.5) becomes

r 2, T = -pI +r_nT_ Eq)(2mD) + 4nB*. (3.24) k=1

Note that only the odd powers of D contribute to the 31

shear stress, with the even powers contributing to the nor- mal stress. The shear components are

a 1)(2f+1)111 .21 2i+1 t(xy)=0 = t(yz) and t (xz) T [W(x)] (3.25) where

r-1)/2 if r6 J0 ' R = r-2)/2 if rs JE '

and the apparent coefficient of viscosity is

R 21 v = ) [m w' (x) , (3.26) a r (2i +1) 1=0 a monotonically increasing function of shear rate since a > 0, for otherwise if a < 0, va <0 which is impos-

sible for a real fluid.

If n > 2(r > 1) and q > 1 withq e J, then using

(3.10), equation (3.5) becomes

2a T = -pI + (r)(2mD) + 4nB*. (3.27) k (2mr) [9,=1

Since only the odd powers of D contribute to the shear stress, the shear stress components become 32

R 2 a 7 1, 2i+l-q t = 0 = t and t 2i+1 xy yz xz a.L., 1j2i+112 [W (x)] (2r) 1=Q (3.28)

where h 2i+1 is the appropriate coefficient and is posi- tive for each i since it consists only of combination of fn, kt,

-1) /2 if q c Jo ,

E g/2 if a 6 J E

and

R = (rq -1)/2 if rq eJo,

[(rq -2) /2 if rq c JE.

The apparent coefficient of viscosity becomes

2a 2i NA (3.29) a 2i+1,221+1-a[P(x)] (2r)"1 i=c)

We must now examine the coefficient of each [W(x)]2i to see its nature, whether negative or positive, since it determines the nature of the apparent coefficient of vis- 2i+l-a cosity. That is, we must examine am since

2b 2i+1/ (2r)cl > 0 for all i. 33

(a)

cll2i+1-q_(.0(.0>0; For a>0, m>0, and geJwith 2i+l-q for a<0, m<0, and qcJE, am -(-)(-)>0; 2i-1-1.-1,(.0(4.)>0; for a>0, m<0, and geJo withq>1, am then v is a monotonically increasing function of shear- a rate.

(b)

2i+l-q_(_)(+)<0; For a<0, m<0, and q6.70, 2i+1-(4_(.0(-)<0; for a>0, m<0, and qcJE, am 2i+l-q_(_)(4.)<0; for a<0, m>0, and geJ, am then v is negative. a

Here the case (b) is impossible for a real fluid. Thus case (ii) is proved.

Case (iii): If n = 2(r=1) and q e JE, then (3.10) be- comes 22, D* = k D (3.30) where q = 22, and k 6 J. Then equation (3.5) becomes

T = -pI + 2aD29,+ 4nB* (3.31) and the shear stress components are

t = t = t = 0 (3.32) (xy) (yz) (xz) 34

which implies that the apparent coefficient of viscosity is

zero. Hence the proof of the theorem.

3.3 Combined Generalized Measures in Viscoelasticity

As we have seen in the theorem proved in the last

article, the study of the behavior of the apparent coef-

ficient of viscosity provides useful information regarding

non-Newtonian behavior of materials. We also find that the mere use of generalized measures of the deformation-rates

in the constitutive equation is not adequate to explain

non-Newtonian phenomena such as pseudoplasticity. In order to describe such phenomena, it is found necessary to com-

bine generalized deformation-rate measures of different or-

ders. In this article, a suitable constitutive equation is constructed based on the combined generalized measures in

order to explain pseudoplastic behavior of fluids.

Since the generalized measures of the rates of deform- ation are involved, one can combine these generalized mea- sures of different orders to obtain a suitable constitutive

equation based on the combined measures. In an actual physical situation involving creeping motion or pseudoplas-

tic flow, these orders of measures will be determined by

experiments. By choosing the orders of the generalized measures (mrxnrxkr xcTr) for the first deformation-

rate D and (m' x n'x k'x a') for the second deforma- - 35

tion-rate B, r = 1 and 2, and combining these result- ing sets of generalized measures, we obtain

D** = D* + D* -1 -2 (3.33)

where n k 1 q 1 D* I-(I 2mD) (3.34) ql (11 1 m n 1 1

and 2q, 2 2 D*= [I-(I 2mD) (3.35) 2 q q 2 m2n2 2 2

and

B** = B* + B* (3.36) 1 2

where

, n'lg, 1 B* = [I-(I- 2m'B)2 1- (3.37) qkl ql m' n1' 1

and

n' 2 (4., k' 2 2 B*= [I-(i 2m'B) 2 2- (3.38) c12 m' n' 2 2

In the next article we obtain constitutive equations based on the combined generalized measures. 36

3.4 Constitutive Equation Based on Generalized Measures

The constitutive equation based on the combined gen-

eralized measures D** and B** can be constructed from

the constitutive equation (2.30)(Narasimhan and Sra, 1968)

by replacing D* by D** and B* by B** to obtain for incompressible fluids

T = -pI + 2pD** + 471B** (3.39)

where D** and B** are given by (3.33) and (3.36) res-

pectively.

This constitutive equation (3.39) is very general and includes many of the previously constructed constitutive equations as special cases. By choosing the orders of the measures (mr xnr xk ,qr ) for D** and r (m'x n'x k'x q') for B**, with r = 1,2, r r r r we obtain the following constitutive equations:

If (m1 x2 x kl x 1),(m2 x n2 x0 x q2),

(n' x n' x0x q') with r = 1 and 2, then equation (3.39) becomes

T = -pI + 2aD,

where a = pk , the classical Newtonian equation for in- 1 compressible fluids. 37

If (m x 4 x kl x 1), (m x n2 x0 x ), and 1 2 -2 (m' x n' x0x a') with r = 1 and 2, then eauation

(3.39) becomes

2 T = -pI + 2aD + 2aD , 1 2

where a, = pk, and a2 = pmlki ,a special case of the

Reiner and Rivlin equation (1.8) for incompressible fluids.

If (r. ), (m x n2 x 0 x q ) and lxnlxklxa1 2 2 (ml x ni x ki x gi), xn2x 0 x r-), then. eauation

(3.39) becomes equation (2.30), due to 7'7arasimhan and. Sra

(1968).

If (m1 x2 x kl x 1),(m2 x 2 x x 3) and

(ml x2 x ki x 1), x n2 x 0 x ap, then eauation

(3.39) becomes

3 T = -pI + 2a 1D 2aD + y71, (3.4e) 2 where

a = pki , 1

a -pk2 2 and

= 4nk'. V

Using the simple shear model of section 3.2, the aprarent coefficient of viscosity va is 38

2 va = al 42 [w ' (x) I .

A graph of the apparent coefficient of viscosity vs. rate

of shear is shown in figure (3.2). Since the apparent viscosity is a decreasing function of the shear rate, the fluid is called pseudoplastic.

a 1

rate of shear

(Figure 3.2) Graph of apparent coefficientof viscosity for a pseudoplastic fluid. 39

CHAPTER IV

PRIMARY AND SECONDARY FLOWS IN THE PRESENCE OF MOVING BOUNDARIES

4.1 Introduction

In the study of non-Newtonian fluids, it is convenient to break the flow into two catagories, primary flow and secondary flow.

Definition 4.1 The primary flow is defined to be the mo- tion of the fluid which results by considering only the flow contributed by the inclusion of the Newtonian coeffi- cient of viscosity, and neglecting the contribution from the inertial and non-Newtonian terms.

Definition 4.2 The secondary flow is defined to be the motion due to the contribution from the inclusion of the inertial and non-Newtonian terms which is super-imposed upon the primary motion. Hence, the inertial effects and non-Newtonian effects, regarded small in comparison to the effect of the viscosity term, are considered only as secondary effects. By neg- lecting the secondary flow in fluid flow problems, espe- cially in problems dealing with moving boundaries, a valid picture of the actual flow of the fluid is not obtained.

Thus, the study of secondary flow allows a more thorough 40

understanding of the behavior of non-Newtonian fluids.

Also, by investigating the effects due to the non-Newtonian

terms, that is, secondary flows, one obtains a means of

evaluating the rheological parameters in the constitutive equation by relating the experimental results obtained by

using some non-Newtonian fluid with the corresponding

theoretical results.

4.2 Secondary Flows in Classical Fluids (Theoretical Results)

Rosenblat, S.(1960) investigated the flow of a clas- sical viscous fluid between two parallel infinite plates which oscillate torsionally about a common axis. Two cases were of specific interest: (i) only one disk is in motion,

and (ii) both disks oscillate with the same frequency and

amplitude, but 180° out of phase. Solutions for the equations were obtained for both small and large Reynolds numbers. For one disk oscillating, the fluid is drawn in- wards along the axis towards the moving disk and thrown radially outward near the oscillating disk (see figure

4.1). For both disks oscillating 180° out of phase with the same amplitude and frequency, the flow field consists of two regions symmetric about the midplane, withno flow across it. This plane acts as an interface separating the fluid into two self-contained portions, in each of which 41

there is axial flow towards the moving disk (see figure

4.2).

axis of rotation 1.0 1.0

y y

0 (c 0

(Figure 4.1) One disk oscil- (Figure 4.2) Two disks os- lating (y=0): typical stream- cillating: typical stream- lines of steady radial-axial lines of steady radial- flow. axial flow.

Haberman, W.L.(1962) considered the problem of sec-

ondary flow of a classical Newtonian fluid for a sphere rotating inside a coaxially rotating spherical eontainer. The equations of motion were made linear and the results showed that the secondary flow does not appear to contri- bute to the turning moment required to maintain rotation of

each sphere. The flow field consists of four regions each

of which contains streamlines that are closed loops (see

figure 4.3). 42

(Figure 4.3) Streamlines of secondary flow about a sphere rotating in a fixed spherical container.

4.3 Secondary Flows in Non-Newtonian Fluids (Theoretical Results)

Green, A.E. and R.S. Rivlin (1956) investigated the flow of the fluid described by the constitutive equation

T = -pI + ai(IID, IIID)D + a2(IID, IIID)D2

in a non-circular tube. They discovered that it is not possible to have a rectilinear flow in a non-circular tube without an appropriate body force distribution in addition to a uniform pressure gradient along the tube. They ob- served that for a tube with an elliptical cross-section, without the application of body forces, a steady-state flow 43

was produced consisting of a primary flow (rectilinear flow) plus a flow (secondary flow) super-imposed in planes perpendicular to the axis of the tube (see figure 4.4).

(Figure 4.4) Secondary flow in a tube with ,7 1irticF1.1 cross-section

Other investigators of the problem of flow through a non-circular tube are Ericksen (1956), Stone (1957), and Criminale, Ericksen, and Filbey (1958).

Jones, J.R.(1960) examined the flow of a fluid des- cribed by the constitutive equation

T = -pI + a1D + a2D2 in a curved pipe with circular cross-section. In the analysis, it was assumed that the curvature of the pipe was small, that is, the radius of the circle in which the line of the pipe is curved is large in comparison with the 44

radius of the cross-section of the pipe. The method of solution was by successive approximations, taking the

zeroth order approximation to be the Newtonian flow. It was found that to the order of the approximation consider-

ed, the relation between the axial pressure gradient and

the rate of out-flow is the same as if the pipe were

straight. However, a secondary flow was present which af-

fected the disposition of the streamlines.

Bhatnagar, P.L. and G.K. Rajeswari (1962) investigated

the flow of the non-Newtonian fluid described by the con-

stitutive equation

T = -pI + 2aD + 4a2D2+ 2ct B 1 3

between two parallel infinite plates which oscillate tor-

sionally about a common axis. Like Rosenblat (1960), they

investigated two specific cases: (i) only one disk in mo- tion, and (ii) both disks oscillate with the same frequency and amplitude, but 180° out of phase. Also, solutions for the equations were obtained for both small and large Rey- nolds numbers. For small values of the non-Newtonian para- meters, the results showed that the fluid behaves like

Newtonian fluid and possesses radial-axial flows similar to the one predicted by Rosenblat (1960). However, it was found that there exists critical values of the non-Newton- ian parameters, depending upon the Reynolds number above 45

which the flow corresponding to the steady part reverses in

sense. Also, for fixed values of the non-Newtonian para- meters, the authors observed that a value of the Reynolds

can be found for which the flow field divides into sub-

regions having distinct flow characteristics.

The same authors in a later paper (Bhatnagar, P.L. and

G.K Rajeswari, 1963) considered the problem of secondary

flow in the annulus between two concentric spheres, rotat- ing about a common axis. Again, they used the non-Newton- ian fluid described by the constitutive equation

2 T = -pI + 2aD + 4aD + 2aB. 1 2 3

The primary motion, obtained by just considering the vis- cosity term, consists of the streamlines which are concen- tric circles with centers on the axis of rotation. In the case of the viscosity term with inclusion of inertia terms, the secondary flow was observed to be the same as described by Haberman (1962). With the inclusion of the non-Newton- ian parameters, it was observed that there exist critical values, depending upon the Reynolds number, in which the flow field changes its nature. For values of these para- meters less than the critical values, the secondary flow resembles the secondary flow in the case of Newtonian fluid with the inclusion of inertia effects. However, when the 46 values of these parameters are chosen to be greater than the critical values, it was observed that the quadrants are

further subdivided into sub-regions by a streamline run- ning between vertical and horizontal radii. In each of

these sub-regions the streamlines form closed loops and the sense of the flow in these sub-regions is opposite (see figure 4.5). Also, it was found, after the critical values have been reached, the interface between the two regions moves towards the outer shpere as the non-Newtonian para- meters are increased and for a definite value of the para- meters, the outer part vanishes and each quandrant is fil- led up with closed loops as prior to the breaking, but of opposite sense.

axis of rotation

(Figure 4.5) The secondary flow pattern in one quadrant of an- nulus of two spheres for fluid whose parameters have passed the critical values. 47

Other investigators working on the problem of flow in

the annulus of two rotating spheres are Langlois, W.E.

(1962) and Waters, N.D.(1964).

Bhatnagar, P.L. and S.L. Rathna (1963) examined the

secondary flow between two rotating coaxial cones using the

non-Newtonian fluid described by the constitutive equation

2 T = -pI + 2aD + 4aD + 2aB. 1 2 3

Starting with the rectilinear motion of a Newtonian fluid

in which the streamlines are circles with centers on the

common axis of the cones, the equations were built up for studying the modification of the flow by taking into ac-

count the inertia terms according to Oseen's scheme and

regarding the effects of the non-Newtonian terms as small.

With the inclusion of the inertia terms in Oseen's approxi- mation, they observed that a secondary flow occurs in which the fluid particles move away from the vertex almost paral- lel to the surfaces of the inner cone, which is rotating

faster than the 0u-to:cone and the particles near the outer surface move parallel to it towards the vertex. When the non-Newtonian terms were included, it was found that the flow field divided into two domains. In the domain nearer the vertex, the streamlines of the secondary flow formed closed loops, and outside this domain, the fluid particles moved nearly parallel to the generators of the cones to 48

which they were closest, moving toward the vertex near one cone and away near the other (see figure 4.6)

(Figure 4.6) The flow pattern of a viscoelastic fluid between two rotating cones.

Thomas, R.H. and K. Walters (1964) considered the flow of an elastico-viscous liquid designated as liquid B', due

to a sphere immersed in it and rotating about a vertical diameter. The method of solution is by successive approx-

imations of a certain parameter. The primary flow is the

zeroth order approximation and is the same as for a New-

tonian fluid. The results show that to first order approx-

imation, the motion of fluid B' due to the rotating sphere is strongly dependent on a parameter, say m, des- cribed in the problem. It was found that there exist three different cases depending upon the values of m, which

1 1 1 are: (i) 0< m < 1, (ii) 17 < m < T ,and (iii) m > T.

For case (i), the flow pattern for fluid B' is similar to that for a purely viscous liquid (m=0); the liquid re- 49 cedes from the sphere at the equator and approaches it again at the poles. For case (ii), the flow field is di- vided by a streamline in the form of a sphere. Inside the sphere, the streamlines are closed curves and the liquid recedes from the sphere at the poles and approaches it again at the equator. Outside this sphere, the curves are open and the liquid recedes from the sphere at the equator and approaches it again at the poles (see figure 4.7).

(Figure 4.7) The projections of the particle paths on any plane containing the axis of ro- tation when m = 1/6. 50

For case (iii), the streamlines are closed curves and their

directions are such that the liquid approaches the sphere

at the equator and recedes from it at the poles (see figure

4.8) .

(Figure 4.8) The projections of the particle path on any plane containing the axis of ro- tation when m = 1/4.

4.4 Experimental Results on Secondary Flow

Hoppman, W.H. and C.H. Miller (1963) used the tech- nique of injecting a dye into the moving fluid bya hypo- dermic needle to observe the secondary flow of the Newton- ian fluid U.S.P. castor oil in a cone-plate viscometer.

The experiment was conducted four times, each time using a cone of different apex half angle, with the idea of see- ing how different shaped cones affected the fluid flow. 51

The results of the experiment showed that the fluid parti-

cles do not simply rotate in concentric circular paths a-

bout the axis of rotation of the cone, as expected, but

spiral around the central curve of a vortex as they move

around the cone axis (see figure 4.9).

(Figure 4.9) Drawing showing flow characteristics. (left) Schematic diagram of a streamline. (right) Schematic diagram of cross-section of toroidal-like surface of flow.

Hoppman, W.H. and Baronet (1964), using the same geo- matric configuration as above, investigated the secondary 52 flow using three Newtonian fluids, which were castor oil, water, and decalin, and two non-Newtonian fluids, which were sweetened condensed milk and solutions of polyisobuty- lene in cetane. It was observed that the flow phenomena for non-Newtonian fluids are much more complicated than for

Newtonian fluids. On examination of the median cross-sec- tions, vortices were found and the velocity field of the flow depended sharply upon the concentration of the fluid.

For example, for 10% solution of polyisobutylene in cetane, only one vortex center was observed, but the direction of the flow was opposite from that of a Newtonian fluid. For

7.5% solution of the fluid, two vortex centers were ob- served (see figure 4.10), and for 5% solution, the fluid behaved much like Newtonian fluid, with one vortex center and flow in the same direction as that of the Newtonian fluid.

(Figure 4.10) Vortices generated by a rotating cone immersed in 7.5% solution of Polyisobutylene in cetane. 53

In the preceding work, (see figure 4.10), the secondary

flow was obtained for fluids which were described by using

classical deformation-rate measures and not generalized measures. The use of classical measures leads to unknown response coefficients and in order to use such equations,

the response coefficients are arbitrarly assigned. In chapter 5, we illustrate the use of generalized measures

in describing the secondary flow of a fluid. By the use of generalized measures, the response coefficients are known exactly. 54

CHAPTER V

APPLICATION OF GENERALIZED MEASURES OF DEFORMATION- RATES TO SECONDARY FLOWS OF VISCOELASTIC FLUIDS BETWEEN ROTATING SPHERES

5.1 Formulation of the Problem

The Fluid

In this chapter we shall illustrate the secondary

flows using the constitutive equation (3.39) based on

combined generalized measures by fixing the orders of the measures as

x2 x ki x 1),(m2 x2x k2 x 3) and (5.1)

(m' x2x k'x 1), (m2x n' x0 x a') 1 1 2 -1 and obtaining the constitutive equation

T = -pI + 2aD - 2aD3+ yB, (5.2) 1 2 or by the Cayley-Hamilton theorem

T = -pI + 2a1D - 2a2(IIIDI-IIDD) + yB, (5.3) where

T = stress matrix, p E hydrostatic pressure,

IE 6k.fiE unit matrix, 11 55

al E ukl , a rheological parameter,

D =II d1, 11-=first deformation-rate matrix,

a2 E -uk2 , a rheological parameter,

II Dand III second and third invariants of D, res- pectively,

y ELink' , a rheological parameter,

and B EIlbijIIE secondary deformation-rate matrix.

For more discussion on the fluid, see (3.40).

The Geometric Configuration

The fluid behavior is investigated in the annulus of

two concentric spheres which rotate slowly with respect to

each other about a common axis. The spherical-polar coor-

dinate system (r1 , 0, 0 is used with the origin at the

common center of the spheres and the polar angle 0 and

the azimuthal angle (1) being measured from the common axis of rotation and some convenient meridian plane, respective-

ly. Let the spheres be of radii 0. and X3(A > 1), and

rotating with angular velocities 01 and Q2 , respect- ively (see figure 5.1). 56

(Figure 5.1) Notation

The Equations of Motion

Since we are assuming an incompressible fluid in

steady-state motion, the density p of the fluid is con-

stant and all quantities are independent of time. By sym- metry with respect to the axis of rotation, all quantities

are independent of cl). If u,v,w are the physical compon- ents of velocity; a,b,c are the physical components of acceleration; and t etc. are the (rr)' t(re)' t(r(p)/ physical components of the stress tensor, then the physical components of stress are: 57

2 = -p-F2alur -2a2 [IIID-IIDur ] +y[ari+uriFvr2 il-wr2i, t i (rr) 1 1 (.5.4a) 2a II 1 D t(ee) = -p + (u+ve) 2a[III ] 2 D r (u+v e) 1 1 5.4b)

, 2 2 2 + - [rbe + r + (u-v) + (u+v ) + we] , 1 la 0 6 r 1

2a II D = -p (u+vctne ) -2a[III (u+vctne) ] t r 2 D r (44) 1 1 5.4c) 2+ 2 2e] + I [1. + r bctn0+ (u+vctne ) wcsc , la r 1

al aII , 2 t = tu+rv -v)+ [ue+r (re ) r 0 1r1 r 1v 1 1D r 1-v] 5 . 4d)

/ [1(a+r b -b)+u (u e- v)+vr (u+ve) +wrwel +r 2 1 r r 1 1 1

al a , 2 D t = --kr w -w)+ [rwr- w] (r(1)) r1 1 r 1 1 1 )5.4e)

+-1--[1(rc -c) -wu -wv ctne+w (u+vctn0) ] , 2 1r r r r 1 1 1 1 1

a a II 1 2 D tee = --(w wctn6 )+ [we -wctn0 ] r 0 r 1 1 >5.4f) y 1 + - [-(rc-rc ctne-w (u-v) -ctnew (u+v)+w (u+vctrie ) ] 2 2 10 1 0 0 0 r 1 58

the equation of continuity is

2u ve vctne u + (5.5) r1 r1 1 1 and the momentum equations are:

2v 2-7u 0 2ctne pa = -pr + al Au - ---7 2 v. r1 r1 r I 1 (5.6a)

+ ySi(ri,e) aK(r,e) 21 1

2u Po 0 pb = + a[LW+ 1 2 2v. 2 1 r r sin e 1 (5.6b) + yS2(ri,e) a2K2(ri,e),

w I pc = a[Aw + yS (r 0) aK(r 0),(5.6c) 1 2 2 3 1, 23 1, r sin e where 2 9 2 9 1 a2 ctn0 A (5.7) E ---7 r Dr 2 2 ' 2 90 ' ar 1 1 r ae r 1 1 1

2 2 aEuu - -u --1-(v + w ) , (5.8a) r r 0 r 1 1 1

uv w2ctne bEuv + v + (5.8b) r1 TI 0 r r 1 1 59

vw C E U W + e uw vwctne (5.8c) r r r r 1 1 1 1

2a r 1 1 1 ctne S (r e)E a + b + ---- a 1 1, r r + 2a®0Oe 2r1 rle r1 2 0 1 1 2r 2r 1 1

3b 2a ctn8 0 3ctnOb 2 + u +v2+w2 2r br 2 2 r1 r1r rjr ri 1 1 2r 2r 1 1 1

12[1. 2 2 2 1 )+ v (u+v)+w wI+-- u +v + w r 6 r e r2 r r r 1 l l l l rl 1 1

1 2 2 2 te [-(ela -v)+(v+u) + w + (u -v)+v (u+v )+w w 3 e 8 r e r 8 r r r 1 1 1 1 1

1 + vctne w2 (5.9a) 3 2 r r sin 1 1

2a 1 1 1 0 1 S (r 0) = a + b + b + + b 2 1, 2r r0 2rr ee 2 1 1 11 r2 r 1 rl l 1

ctne 1 b b + T ur v)+vr (u+v )+ww 2 e 2 2 (u0 rl r r r sin 0 1 1 1 l 1 1

1 2 , 2 2 I 2 (u-v) +tu+v ) +w + [-u (u-v)+v (u+v )+w w 3 0 e 6 2 r 0 rl 0 r .] r 0 r 1 1.0 1 1

pe_v)2. . 2 2] ctnill2 2 +t 0+u) +w w csc 0+(u+vctn0)2 1, (5.9b) e 3 r 1 1 60

1 1 1 ctn0 S (r 0)- -C +----c +--c ----c 3 11 2 rr 2 00 r r 2e 2 . 2 11 2r 1 1 2r 2r sin e 1 1 1

1 2(wu + wv ctn0 - w (u + vctne)) 3 1 r r 1 1 1 1 1

1 [ 2 sine(w(u-v)+ wctne(u+v)w(u+vctne)) ,(5.9c) 3 2 e e e r sin e

1 K(r 8) E2[III IIDur11 IID(ue+r,v - v) 1 D- r1 e rl 1

11, + 4u - 4ru uectn6 + 3vctn0 - r-vctn0 + 1r r1 ri 1 (5.10a)

II 1 2 a K(r e) E (u + rv v) +T-1III (114-v D 1r1 D e 2 r1

211 D[ + --F vcsc 28 - vctne - r1vr - u (5.10b) e e 1 1 and

1 K(r e) - T w) -17 IID(we-wctne) 3 11 1 L1 ri r1

211 (rw - w)+ ctn0 (w wctne) (5.10c) 2 1r e 1 1 with 61

u u , r 1 , l II "-L(u+v ) + --7(u+v )(u+vctne) + D ri (u+vctne) r 1 1

2 2 (u+r v v)+(rw w)+(w -wctne)2 I, (5.11a) e rl 1rl 71; -17-

u III (u+v)(u+vctne) - T(w wctne) D r22 e e 1

1 (u +r v v)(rw w) (w -wctne) 3 e 1 rl 1r- 4r l 1

1 2 2 (u +r v v) (u+vctne)+(rw- w) (u+v 3 e 1 rl 1r 4r 1 1 (5.11b)

and w and we denote partial differentiation ofw r with respect to r and 0 respectively.

The above equations are subjected to the following boundary conditions:

u = 0 = v, w = 6Q1sine at r = 6, 1 (5.12) u = 0 = v, w = X6Q2sin0 at r = X6. 1

Let A,B,Cbe the non-dimensional components of ac- celeration andU,V,W be the non-dimensional components of velocity. If the following substitions:

2 2 v vS2 A, b = B and c = C; (5.13) 3 ' 13 R313 62

v u = 7 u, v =7v, and w = 8QW; R

2 py n P --T 8

and r = 8r, (5.16) 1

where a 1 v = and Q = Q 1 2 '

are made in equations (5.6), then we obtain the

Non-Dimensional Equations of Motion:

equation of continuity

ve vcten U + 2T + o (5.17) r

and momentum equations

2V 2ctnel A = -P+ *U - + SS*(r8)+ KK*(r 8) r 2 2 1 ' r TT r " (5.18a)

P8 2U 8 V B = - +[*V+ + SS* (r 8) +KK*(r8) 2 2 2 2 " r r sin (5.18b) W C =[*W 11+SS1(r,9) + KK1(r,0), (5.18c) r 2sin. 2e where 63

2 2 2 D 1 a ctne D A*= - (5.19) 2 r Dr 2 - 2 2 aa Dr r de r

V V2 hW2 A =UUr + r U0 (5.20a)

V UV hW2ctn0 B =UV + - V + (5.20b) r r 0 r

V UW vw = UV 4. w + ctne, (5.20c) r r 0

1 1 2 ctne sl(r,e) = A + ---A + B + Ar + A rr 2 00 2r ru r 2r 2r

2A ctne 3 3ctne u2 v2 hw2 p - B 2r r 2 r2 2r 2r2

2 {-12 2 2 2 + 1 [1J (U - V)+V(U + Ve)+hW + + V+ hW r r r r r

WM, c[utne - -r] U0-V)2+(Ve+U)2+hW v) +vr (u+ve ) +hwrIA10._ r [ r2

hW2 ---k+Vct41 - (5.21a) r r 3sin 20 '

B 1 1 1 2 S (r,e) = --A + -B + --B + A + r 2 2r re 2 rr 2 ee 2 0 r r r

ctne B + u(U -V)+V(U+V )+hW 2 2 2 r r rWA] r r sin e r

1 2 2 2 2 (U. -V)+(U+Vd+hWI+ (U (U +V0))+hW +hWrWO 3 2 r r r e r 64

ctn3 6 2+ 2 1 ct3n 0[ 2 2 [EJ- V) (ve+u)+hW hWcsc6+ (U+Vctne)21 e 6 r r

1 c tn S3 *(r 6) = -C + ---C+ -C + --C 2 rr r r 0 2 2 2r2ee 2r2 2r sin

1[I2 r (WUr + WV rctn0 Wr (U + Vctn6) ) r3

2 sine(w(u v)+wctne (Li+ (U±Vctne) ,(5.21c) 3 2 v w r sine

1 Kt (r, 0) = 2 III** + rVr V) r

II* D 4U-4rU-Uctn6 + 3VctnO - rVctne + 2V (5.22a) 2 r 0 r r

II* 1 2 K2(r, 6) = (U + rVr- V)1+-[II* (U + r e r e r Dr 2IIIS[ Vcsc 2e V ectne - rVr - U (5.22b) 2 r

1 1 (r,0) =-kIVrWr-W)1 EIle) (We Wctne) r r 211*

(rW r-W) + ctn0 (We Wctn6)] , (5.22c) 2 r with U U 1 r II* = (U+V ) +(u +ye) (U +VctnB) + r(U+Vctn8) D r e r2

1 , , kU + rVr -V)2 +h (rwr-W.)+h (We wctne ) (5.23a) 2 e 4r 65

Ur IIID= (U+Ve)(U+Vctne)- 111-(W Wctne)2 4 0 r

+ (U + rV-V) (rW-W)(W Wctne) 3 e r r e 4r

2 2 1 (U+rV-V ) (U +Vctne) +h (rVr -W) (U+V )J, (5.23b) 3 e r 4r e

2 -1 -32 mOLOTO, S E y/03 with [S] = ML /ML L (5.24a)

v 1T.T 11,2 a2 = -0 0 0 K E --- with [K] mL T , (5.24b) -3 PI3 ML LL4

and (2)2 T-2 L4 000 h with [h] = MLT , (5.24c) T-2 L4

where [ ] denotes the dimensions of the quantity in gues-

tion in terms of mass M, length L, and time T.

The boundary conditions in non dimensional form are

1 U = 0 = V, W = sine at r = 1 (5.25) U = 0 = V, W = 2Asine at r = A 2

5.2 Method of Solution of the Equations of Motion

If the spheres are rotating in the same direction with nearly the same velocities, that is, if Q = Qi Q2 is

'small', if v is "large" meaning al is large, then 66

e82)2 11 - (5.26)

2 is very small so that h << h. The method of solution will be to expand the expression for pressure and the velo- city components in ascending powers of the parameter h.

That is, assume

2 P = P + hP + hP + (5.27) 0 1 2

2 U = U + hU+ hU + (5.28a) o 1 2

2 V = V+ hV + hV + (5.28b) o 1 2

2 W = W0 + hW + hW + (5.28c) 1 2

5.3 Primary Flows in Pseudoplastic Fluids

We shall call the rectilinear motion with the neglect of the inertia terms and the non-Newtonian S- and X- terms in the equations of motion the primary motion. We shall specify the primary motion by the zero subscript. The equations of motion for the primary flow are: continuity equation

2U V Vctne 08 U + o Or 0; (5.29) and 67

momentum equations

2U 2V Oe 2ctne -P + U0 - V = 0, (5.30a) Or 2 2 0 r

2U I Poe +[Alw + 08 = 0, (5.30b) o 2 2V02 r r sin 0 w o 4 *W O. (5.30c) 0 r 2sin. 20

The boundary conditions for the primary flow are

1 0 = V Wo = T sine at r = 1, U0 = 0' (5.31) 2 0 = V W = Xsine at r = X. U0 = 0' 0 Q

In the primary motion, the streamlines in any plane perpendicular to the common axis of rotation are concentric circles with centers on the axis. Thus

U0 = 0 = V0 (5.32) 0 and P = 0 = P which implies P= constant (5.33) Or oe

LetWO = f(r)sine, then (5.28c) becomes

2 df 2 df 2f = 0 (5.34) 2 T dr 2 dr r and 68

f(r) = +er) , (5.35) 2 r

where 3 3 0X X Q1 d = and e 2 (5.36) 3 3 X-1 NX-1)

Thus W = +er)sine (5.37) 0 (-2

5.4 Secondary Flows in Pseudoplastic Fluids

In this investigation, we shall consider only the con-

tribution due to the first order terms of h. By including

the first order terms of h, we include the effects due to

the inertia terms and the non-Newtonian S- and K- terms

The equations of motion for the first-order approximation

are

the equation of continuity

2U V Vctne 1 10 1 U + 0 lr (5.38) and the momentum equations

2 W 2U 2V 2ctnEW 1 le Plr+ 6,*U +ss**(r e) 1 2 2 1 r r

+ KK**(r"A) (5.39a) 69

2 2U V WO 0 ctne Pre+[A*V 1 2 27 2 .1 r r sin 0

+ SS*2 *(re)+KK**(r8) (5.39b) ' 2 "

C*(r,e) = 0 *W SS** (r 8) + KK**(r 8) (5.39c) 1 r 2sin. 2e 3 3 where

2 -36d 2 S**(r 8) sin 0, (5.40a) ' 7 r

S*2 *(re) = 0, (5.40b) '

1 1 1 ctne C* S**(r e) = -C* + + -c* __--C* 3 ' 2 rr 2 ee r r 2 8 2 2 2r 2r 2r sin

1 [2 rtwouir+woVir,ctne WOr(Ul+Victn0) 1

1 [-sin 2e{w(u v )-1-wctne(u+v )w (u+vctne)} r 3sin. 20 0 10 1 0 1 10 00 1 1 (5.40c)

K**(r8) = 0 (5.41a) 1 '

K2* (r, 8) = 0 (5.41b)

81d3 3 K**(r e) = sin 38 (5.41c) 3 ' 10 2r

V, o, U, Wo ViWo C*(r,e) = UW + ° + ----ctne, (5.42) 1Or

+ er sine, as given in (5.37). WO0 2 r 70

Boundary Conditions

By the no-slip condition of a viscous fluid on the

boundary, the boundary conditions for the first approxima-

tion of the secondary flow are

U =V1 =W1 =0 at r= 1 1 (5.43) U =V1 =W1 =0 1 atr=

Solution for the Velocity Field

Solutions of the coupled equations (5.39a-b) are ob-

tained by eliminating the pressure term P in the equa-

tions and introducing the streamline function tpl(r,e)

where 4 1 94)1 -1 U and VV1 (5.44) 1 2 e rsine Dr ' rsine

which satisfy the continuity equation

2 (r sinelyr + (rsineV1)0 = 0 (5.45)

exactly. Thus, equations (5.39) become

cl;\ 2 4 72dS 2 D11)1 = -6 + 7 sinecose + sin°cos° (5.46) r r where 71

2 1 3 ctne 3 D (5.47) 2 2 2 r2 De 11)1 3r r DA ) and d,e, and S are given by (5.36) and (5.24a) respect- ively.

From (5.43), the streamlines 11(r,8) must satisfy the boundary conditions

at r = 1, 1P18(rie) 4)1r(r'el) (5.48) q)16(r,8) = 0 lr(r,0) at r =

In order to solve equation (5.46) for the streamline

1p (r 0) assume a solution of the form 1

Yr,e) =g(r)sin28cose (5.49) and note that

_2 41 2 6g(r)] T) sin20cose (5.50) 2 dr r

Then equation (5.44) becomes

2 2 2 dz(r) 6z(r) d de) 72Sd 6( + + (5.51) 2 2 2 2 7 dr r r r r where

2 dg(r) 6g(r) z(r) (5.52) 2 2 dr r 72

Solving (5.51) for z(r), then (5.52) for g(r), we ob-

tain

(r,e) = E a rJ sin20cose, (5.53)

and by (5.44)

U(r0) = a.ri-6)(2-3sin 20), (5.54a) 1 ' 3

V(r,0) = aj(j -4) r3 -6 sinecose, (5.54b) 1 where

2 a = Sd/2, 1

4 2 4-X2) 5 3 a = -E2X 7-5A+3A)C2+15(X (C-C)+(3X-5X+2)C 2 1 3 41/H'

2 a = d/4, 3

7 2 7 2 4 2 3 a = (4X+21A-25)C+(2A-7A+5)C+(-25A+21X+4/X)C 4 1 2 3

5 3 2 + (5X-7X+2/X)C1/H (5.55)

a = 0, 5 a = -de/4, 6 a =I(- 5x4 +7x2- 2/X3)C2+10(1/x3- X4)(Cl-C3) 7

+ (-2X5+7-5/X2 )C41/1-I a8 = 0, 73

2 3 2 3 3 2 a = (3A-5+2/A)C+6(A-1/A)(C-C)+(2X-5+3/A)C /H 9 2 1 3 4, and

Q3 = 0 for all j < 1 and j > 9;

2 2 C = (de-d-2Sd)/4, 1

2 2 C = (2de + d + 6Sd)/4, 2

2 2 C = (deX d/X 2Sd2/X3)/4, 3

2 2 2 4 C = (2deX + d/X + 6Sd/A)/4, 4 with

10 7 3 3 H = (4X - 25A +42X5 25A + 4)/X (5.56a) or 4 (X-1) 6 5 4 3 2 H = (4X+16A+40X+55A+40A+16A + 4) (5.56b) 3 A

Note that H > 0 for X > 1 and <0 for 0

Substituting U1 (5.54a) and V1 (5.54b) into equa- tion (5.39c), we obtain

W 1 j-11 )sine j-11 3 A*W =( Ew. r + ETr. r sin0 1 r 2sin. 20 j 3 j 3 \ ( 5 . 57) where

w =-[2(j-5)a +S(3j2 -27D+78)aid-2(j-ll)a.. e (5.58a) j-2 J-5

42j-.9) 0j2-15j+118)a./;Sd+2 (j-12) a e J-5

(5.58b) 74 with

1 d. = 1 if j = 1 and 0 if j 1, and a. and d,e given by (5.55) and (5.36) respectively.

By assuming a solution of the form

3 W (re) = N(r)sin0 N(r)sin6, (5.59) 1 ' 1 2 equation (5.57) can be written as the system of equations

2 dN(r) dN (r) 2N (r) 8N (r) 1 2 1 1 2 -11 Ew3. r 2 r dr 2 2 dr r r (5.60a)

2 dN(r) dN (r) 12n2 2 2 2 TT r (5.60b) 2 r dr 2 dr r j with

N(1)= N(1)=0 1 2

N (X)= N(X)=0. 1 2

Solving the system of equations (5.60), we have

w.(j-5)(j-12)-87. -2 N1 (r) = dr + r + 5 6 j5,7,10,12(j-75)(j-7)(j-10)(j-12)rj-9

47 1 1 -2 1 - ..1-(w + - )r knr + -(w - )rknr 7 3 10 5 10

(Continued on page 75) 75

w 2-86)r3 -4, 4 3f 7 knr)+ 35",5r '10 2411) 35712 '10 16

(w 867) 5 7 -4, 10 r (5.61a)

7 3 -9 N 2(r) =67r-4+ Sr + ) r 8 j5,12(j-5) (j-12)

Tr5 71 -4 2 3 7F knr + knr, (5.61b) r 7 r where w. (j-5) (j-12)-87ri X2 5 (j-5) (j-7) (j-10) (j -12) 225(11.5"12) X3-1 [j5,7,10,12

w(j-5)(j-12)-£37r1. + (w -86)+ 16 12 8 10(w5-867d+ [j5,7,10,12(j-5)(j-7)(j-10)(j-12)XJ- 47r7 1 -2 1 4 -5-(w7 -3)X 9JIA 7(w10 r10)AkriX

4 -4 4 3 7 3 + 3575A (-170+56nX)+T5-712X (TT) knA)+ (w12 868)16

(wc 86-) / 4 + '10 (5.62a)

1 f w.(j-5) (j-12)-87, 66 2 _L, j5,7,10,12(j-5) (j-7) (j-10) (j-12)+25'5"12'

wi(j-5)(j-12)-8Tri j5,7,10,12(i-5)(j-7)(i-2°)(7-1/

4Tr 1 7 -2 1 4 (w7 --)X knA 3(w10 5T10)"nA 76

3 -4 7 4 3 7 X 31T5X (DTI thX)/- -5r12X (IU ftiA)+(w12 868)I6

( w - 8 6 ) _4] 10 (5.62b)

A4 jTr.X -9 _x J (j -5) (j -12) / , -1 j5,12 j5,12k7-D)tJ-12)

knAtff A3 5,-4) (5.62c) 12 71.

j-9 TT. knA 6 = r 3 8 4(IA4 (J-5) (j-12) 7T12X ff5X-4] -1 j5,12

7j j5,12 (j-5)(j-12)) (5.62d)

withw.and TT. given by (5.58).

Solution for the Pressure Field

The equations for the pressure field are obtained by substituting (5.37) and (5.54) into equations (5.39) and

(5.39b) as:

2 2de 2 \ 36d2S3sin2e P =[( d5-+ + er)-3Tcy(j-2)(j-7)r7-8 lr r r ii r

+ 2).a.(j-2)(j-7)ri-8, (5.63a) i 3 77

2 d 2de 22 _4( + er (j-2) (j-6) (j-7)(5.7ri-71sinecose, le T.74 (5.63b) where ap DP 1 1 and P Firlr Dr le De

Solving these systems of equations, we have

2 F/d2 2de 2 2\ Pi e r )- (j-2)(j-6)(j-7)ari-71sin =Lk T + T + 3 J 2

+ ).2a.(j-2)ri-7+ P (5.64) j7 10 where the constant of integration P10 can be determined as the pressure corresponding to any given r and 0, say r = r with 1 < r < A and 0 = 0; d and e are given 0 0 by (5.36); a, given by (5.55).

Summary of Solutions

Primary Flow (zeroth approximation):

U0 = 0 = V0

P = constant, o

WO =( +er) sine,

where d and e are given by (5.36). 78

Secondary Flow (first order approximation):

j-4 ) 2 Streamlines -- qr,0) = ( a.r sin °cos°,

4 )(2-3sin 2 Velocity Components t1(r,0) a.rj 0),

V(r e) a.(j-4)rj-6)sinOccs0, 1

3 W(r 0) = N (r) sine + N(r)sine, 1 2 where ay, N1 and N2 are given by (5.55), (5.62a) and

(5.62b) respectively.

Pressure Field -- P is given by (5.64).

5.5 Discussion of the Results

In order to make a complete investigation of the flow pattern, we first obtain the streamline function tP1from equation (5.53) by using the expressions for the velocity components given by (5.54a and b). We find that the stream- 7 37 line function IP' vanishes at = 0,7 , 727 , and 27 for all values of r and is zero at r = 1 and r = A for all values of 0. Thus the secondary flow in a meri- dian plane is divided into four quandrants by the axis of rotation and the equator (0 = 721). For a Newtonian fluid

(S = 0, K = 0), the secondary flow consists of closed loops, 79 with the direction of flow being toward the common center of the spheres at the poles and away from this common center at the equator (see figure 5.2). The secondary flow pattern for a non-Newtonian fluid is, to a first ap- proximation in h, strongly dependent upon the rheological parameter S characterizing viscoelasticity. There exist parametric values, say S0 and Si, which are dependent upon the rate of rotations of the spheres, for which the flow pattern changes. For S < S0, the secondary flow pattern is very similar to that of a Newtonian fluid with the inclusion of inertial terms. At the critical value

S0, the secondary flow in each quadrant is further broken into two subregions, each containing closed loops (see figure 5.3). As the parameter S is increased from S0 to S1, the interface between the two regions moves to- wards the outer sphere, and at Si, the outer subregion van- ishes and once again each quadrant is filled up with the same type of loops. The direction of these loops is, how- ever, in the opposite direction from that of Newtonian flow (see figure 5.4). This is due to the viscoelastic nature of the fluid.

To illustrate the above and give precise breaking values S and S1 of the streamlines, we let the inner 0 sphere rotate with angular velocityQi = 0.2 (non-dimen- sional form) and the outer sphere rotates withQ2 = 0.1 80

2 so thath (5.26)- .01 and h<< h. Letting A = 2, we obtain, by use of the computer program VELOCITY found in the appendix, S0 = 0.38 and S1 = 0.56.

(Figure 5.2) Typical streamlines of the secondary flow for the case S' = 0[Q1 = 0.2,S-22 = 0.1, and A = 2].

(Figure 5.3) Typical streamlines of the secondary flow for the case S = 4.5.[2i = 0.2, Q2 = 0.1, and A = 2]. 81

r = 1 2

(Figure 5.4) Typical streamlines of the secondary flow for the case S' = 5.5[Q1 = 0.2, Q = 0.1, and A = 2]. 82

CHAPTER VI SUMMARY, DISCUSSION, AND SCOPE OF FURTHER WORK

6.1 Summary and Discussion

The response of real materials to external loading is, in general, nonlinear in character. The classical theories which were designed to explain the behavior of materials subjected to deforming forces use classical measures of de- formation and assume complicated constitutive equations in- volving many unknown response coefficients. Hence a wide spread interest in the search for more general theories has arisen. In our present work, we have given a brief outline of the various theories governing the behavior of real mat- erials. The pioneering work done by Reiner, Rivlin, Erick- sen, Green, Oldroyd, Noll, etc. have used ordinary measures of deformation or deformation-rate. The use of linear or classical measures of deformation have strained the consti- tutive equations into complex forms involving many unknown response coefficients and powers and products of ordinary measures. The main source of the difficulty is the use of classical measures instead of generalized measures. An- other pioneer, Seth, however, attempted to resolve this difficulty by the introduction of generalized measures of deformation or deformation-rate in continuum mechanics.

Using this idea, Narasimhan and Sra extended these general- 83

ized measures of Seth in such a way as to explain adequate-

ly some rheological behavior of materials. The constitu- tive equations they set up, based on generalized measures, contain essentially two terms and at most four rheological constants. Unlike some previous theories, these constitu- tive equations do not contain any unknown functions of the

invariants of kinematic matrices, etc. In the present investigation, we have proved an im- portant theorem concerning the generalized measures of de-

formation rates which enables one to predict precisely, by fixing the orders of these measures, a variety of visco- elastic phenomena such as dilatancy, pseudoplasticity, and so on. We also found that the constitutive equation of

Narasimhan and Sra does not adequately describe the impor- tant physical phenomenon pseudoplasticity, which abounds in every day life. To remedy this, we found it necessary to construct combinations of generalized measures. We found that the constitutive equation using the combined measures still does not contain any unknown functions of the invariants of kinematic matrices, etc.; but is much more general and hence is able to describe a wider range of physical phenomena.

To illustrate the use of combined generalized measures in constitutive equations, we have considered the very in- teresting and important problem of secondary flows in 84

fluids in the presence of moving boundaries. For this pur- pose we arbitrarily fixed certain orders of the measures of the rates of deformation D** and B**. In actual prac- tice however, the choice of the orders of measures will have to be determined by experiments. In examining the fluid flow in the presence of moving boundaries, we found it necessary to investigate secondary flows in order to ob- tain a true picture of the actual flow. For the problem of flow of a fluid in the annulus of two rotating spheres, we have, for the first time, obtained a satisfactory-- scienti- fic basis for explaining the secondary flow phenomena based on the combined generalized measure concept. Expressions for the velocity and pressure fields have been obtained by solving the basic field equations. In order to study the secondary flow pattern, the streamline function has been obtained. The streamlines of the flow in meridian planes containing the axis of rotation are found to be closed loops and the nature of these loops is found to be strongly dependent upon the viscoelastic parameter S. There exist parametric values, say So and S1, which are dependent upon the rate of rotations of the spheres, for which the flow pattern changes. For S less than the critical value S0, the flow is very much like 85

that of a Newtonian fluid, with the fluid advancing toward the inner sphere along the pole and outward along the equa-

tor. At S , the flow region splits into two subregions, 0 each containing closed loops of streamlines. As S in-

creases toward S , the interface between the two sub- 1 regions advances toward the outer sphere, and at S1 , the streamlines again become one set of closed loops, with the sense reversed from that of Newtonian fluid. With the aid of the CDC 3300 computer, we were able to find precise values of S0 and S1. With Qi = 0.2, Qi = 0.1, and

A = 2, we found that S = .38 and S = .56. 0 1

6.2 Scope of Further Work

It appears at this time that nonlinear theory based on generalized measures has a clear advantage over the non- linear theories based on ordinary measures. Therefore, there is a natural desire to exploit the idea of general- ized measures in the investigation of nonlinear phenomena. We saw by combining sets of generalized measures, we were able to adequately describe pseudoplastic behavior. An- other extension of the generalized measure concept would be to describe materials which have memory, a very impor- tant phenomenon in mechanics of deformable bodies. 86

BIBLIOGRAPHY

Bhatnagar, P.L. and G.K. Rajeswari. 1962. The secondary flows induced in a non-Newtonian fluid between two parallel infinite oscillating planes. Journal of the Indian Institute of Science 44:219-238.

1963. Secondary flow of non-Newtonian fluid between two concentric spheres rotatingabout an axis. Indian Journal of Mathematics15:93-112. Bhatnagar, P.L. and S.L. Rathna. 1963. Flow of a fluid be- tween two rotating coaxial cones having the same ver- tex. Quarterly Journal of Mechanics and Applied Math- ematics 16:329-346. Criminale, W.O., J.L. Ericksen and G.L. Filbey, Jr. 1958. Steady shear flow of non-Newtonian fluids. Archive for Rational Mechanics and Analysis 1:410-417. Ericksen, J.L. 1956. Overdetermination of the speed in rectilinear motion of non-Newtonian fluids. Quarterly of Applied Mathematics 14:318-321. Eringen, A.C. 1962. Nonlinear theory of continuous media. New York, McGraw-Hill. 477 p.

Green, A.E. and R.S. Rivlin. 1956. Steady flow of non- Newtonian fluids through tubes. Quarterly of Applied Mathematics 14:299-308.

1957. The mechanics of non-linear mater- ials with memory. Part I. Archive for Rational Mech- anics and Analysis 1:1-21. Haberman, W.L. 1962. Secondary flow about a sphere rotating in a viscous liquid inside a coaxially rotating spher- ical container. The Physics of Fluids 5:625-626. Hoppmann, W.H., II and C.N. Baronet. 1964. Flow generated by a cone rotating in a liquid. Nature 201 (Part 2): 1205-1207.

Hoppmann, W.H., II and C.H. Miller. 1963. Rotational fluid flow generator for studies in rheology. 87

Jones, J.R. 1960. Flow of a non-Newtonian liquid in a curv- ed pipe. Quarterly Journal of Mechanics and Applied Mathematics 13:428-443. Langlois, W.E. 1963. Steady flow of a slightly viscoelastic fluid between rotating spheres. Quarterly of Applied Mathematics 21:61-71.

Merrington, A.C. 1943. Flow of viscoelastic materials in capillaries. Nature 159:310-311.

Narasimhan, M.N.L. and K.S. Sra. 1968. Generalized measures of deformation-rates in viscoelasticity and their ap- plications to rectilinear and simple shearing flows. International Journal of Nonlinear Mechanics.(In press).

Noll, W. 1958. A mathematical theory of the mechanical be- havior of continuous media. Archive for Rational Mech- anics and Analysis 2:197-226.

Oldroyd, J.G. 1951. The motion of elastico-viscous liquid contained between coaxial cylinders, I. Quarterly Journal of Mechanics and Applied Mathematics 4:271-282

Reiner, M. 1945. A mathematical theory of dilatancy. Ameri- can Journal of Mathematics 67:350-362.

Rivlin, R.S. 1948. The hydrodynamics of non-Newtonian fluids, I. Proceedings of the Royal Society (London), Ser. A, 193:260-281. Rivlin, R.S. and J.L. Ericksen. 1955. Stress-deformation relations for isotropic materials. Journal of Rational Mechanics and Analysis 4:323-425.

Rosenblat, S. 1960. Flow between torsionally oscillating disks. Journal of Fluid Mechanics 8:388-399.

Seth, B.R. 1966a. Generalized strain and transition concept for elastic-plastic deformation-creep and relaxation. In: Proceedings of the Eleventh International Congress of Applied Mechanics, Munich, 1964. Berlin, Springer, p. 383-389.

1966b. Measure-concept in mechanics. International Journal of Non-linear Mechanics. 1:35-40. 88

Stone, D.E. 1957. On non-existence of rectilinear motion in plastic solids and non-Newtonian fluids. Quarterly of Applied Mathematics 15:257-262. Thomas, R.H. and K. Walter. 1964. The motion of an elasti- co-viscous liquid due to a sphere rotating about a diameter. Quarterly Journal of Mechanics and Applied Mathematics. 17:39-53.

Waters, N.D. 1964. A note on the secondary flow about a sphere rotating in an elastico-viscous liquid inside a rotating concentric spherical container. British Journal of Applied Physics 15:600-602.

Weissenberg, K. 1947. A continuum theory of rheological phenomena. Nature 159:310-311. 89

Appendix

To aid us in finding numerically the velocity compon-

ents U andV ((5.54a) and (5.54b) respectively), for 1 1 each r and 0, we used the following Fortran program.

First, however, we shall explain the symbols used in the

program.

Notation: Problem:

A X

AD increment change in 0

Bl, B2 end points of 0 range Cl, C2, C3, C4 C1,C2,C3,C4 respectively D d E e H H 0M1, 0M2 Q1, Q respectively 2 OM RR radius of inner sphere S S, rheological parameter SI(J) a.

U ( ), V( ) U and V respectively 1 1, W increment change in 0

Fortran Program

PROGRAM VELOCITY

DIMENSION SI(9), U( , ), V( , ) READ (60,1) A,OM1,0M2,S,RR,W,B2,B1,AD

1 FORMAT ( ) 90

D=A**3/(A**3-1) OM=0M1-0M2 E=(0M2*A**3-0M1)/(0M*(A**3-1.0)) C1=0.25*(D*E-D*D-2*S*D*D) C2=0.25*(2*D*E+D*D+6*S*D*D) C3=0.25*(D*E*A**2-D*D/A-2*S*D*D/A**3 C4=0.25*(2*D*E*A+D*D/A**2+6*S*D*D/A**4) H=(4*A**10-25*A**7+42*A**5-25*A**3+4)/A**3 SI(1)=0.5*S*D*D SI(2)=-((2*A**7-5*A**4+3*A**2)*C2+15*(A**4-A**2)* (C1-C3)+(3*A**5-5*A**3+2)*C4)/H SI(3)=0.25*D*D SI(4)=((4*A**7+21*A**2-25)*C1+(2*A**7-7*A**2+5)*C2+ (-25*A**4+21*A**2+4/A**3)*C3+(5*A**5-7*A**3+2/A**2)* C4)/H SI (5) =0.0 SI(6)=-0.25*D*E SI(7)=((-5*A**4+7*A**2-2/A**3)*C2+10*(1/A**3-A**4)* (C1-C3)+(-2*A**5+7-5/A**2)*C4)/H SI (8) =0.0 SI(9)=((3*A**2-5+2/A**3)*C2+6*(A**2-1/A**3)*(C1-C3 +(2*A**3-5+3/A**2)*C4)/H MAX R=(A-1.0)*RR/W+1.0 MAX B=(B2-B1)/AD+1.0 DO 3 K=1, MAX R DO 2 L=1, MAX B 2 U(K,L)=0.0 3 V(K,L)=0.0 DO 7 K=1, MAX R

R =RR+ (K -1) *W DO 6 L=1, MAX B B =B1+ (L -1) *AD DO 4 J=1, 9 91

4 U(K,L)=U(K,L)+SI(J)*R**(J-6) U(K,L)=U(K,L)*(2-3*sin(B)*sin(B)) DO 5 J=1, 9 5 V(K,L)=V(K,L)+SI(J)*(J-4)*R**(J-6) 6 V(K,L)=-V(K,L)*sin(B)*cos(B) 7 CONTINUE PRINT 8

8 FORMAT ( ) STOP END