Non-Newtonian flow in incompressible fluids

CoA Note N o . 134

2 O dec. 6?

THE COLLEGE OF A E R O N A U T I C S

C R A N F I E L D

NON-NEWTONIAN FLOW IN INCOMPRESSIBLE FLUIDS

P a r t I A general rheological equation of state

P a r t II Some problems in steady flow

by

A. Kaye.

TECHNiSCHE HOGESCHOOL DELFT

VUECTUIGBOUWKUNDB

BIBLIOTHEEK

THE C O L L E G E OF AERONAUTICS

CRANFIELD

Non-Newtonian Flow in Incompressible Fluids Part I A general rheological equation of state P a r t II Some problems in steady flow

b y

-A. Kaye, M.-A.

SUMMARY

A rheological equation of state of the form, t

p..

-

p6..

= 2 f f

I T

s..

-

1^, s:.^

)

dt'

- 0 0

is proposed for an incompressible m a t e r i a l . fi is a function of J and J^, the invariants of Cauchy-Green deformation tensor S.. which relates the deformation at the present time t with that at some past time t'. Q is also a function of t and t ' . Some steady state flow problems are solved for a material obeying this equation. It is anticipated that this equation will be of some use in investigating the flow properties of concentrated polymer solutions and polymer melts,

CONTENTS

Page Summary

List of Symbols

P a r t I, A General Rheological Equation of State 1

P a r t II. Some problems in Steady Flow 4

(a) Simple Shearing Flow 4 (b) Flow through straight pipes of a r b i t r a r y cross-section 5

(c) Troughton or elongational flow 7 (d) Flow between a rotating cone and a stationary plane 9

(e) Flow between parallel planes 11 (f) Flow between rotating coaxial cylinders 13

Discussion 16 References 18 Appendix I - Proof that equation (1, 8) can be expressed

in the form (1.11) 19 Appendix II- The equation of state in t e r m s of convected

co-ordinates 20 F i g u r e s .

isotropic p r e s s u r e s current time

past time

current position of a general particle in rectangular cartesian co-ordinates

past position of a general particle in rectangular cartesian co-ordinates

position of a particle of an elastic body in the unrestrained state

stored energy per unit volume of a strained elastic body the deformation tensor for an elastic body

the invariants of C

13

the deformation tensor relating the deformation between the past time t' and the current time t of a flowing body the invariants of S..

13

a function of J^, J^,, J^ and t - t'

coefficients in the expansion of W in t e r m s of I, and I^ coefficients in the expansion of fi in t e r m s of J^ and J^ the shear rate

defined by equation (2.4) defined by equation (2. 5) defined by equation (2. 7) defined by equation (2. 8)

constants in the expansion of Ü in the form given by equation (2. 20)

List of Symbols (Continued)

6. convected co-ordinates

P the density of the liquid

Z the normal force per unit area T the tangential force per unit area g the acceleration due to gravity

P a r t I. A G e n e r a l Rheological Equation of State

In the p h e n o m e n o l o g i c a l t h e o r y of l a r g e e l a s t i c d e f o r m a t i o n s in i s o t r o p i c m a t e r i a l s a s developed by Rivlin and o t h e r s the s t r e s s - s t r a i n r e l a t i o n s have the g e n e r a l f o r m ,

p.. = _!,_ ; c . . r | ^ . i , | ? •)- c. c. l^ + i,!^ 6.. I (1.1)

13 I ^ t 13 V 9 I, ' 91^ y la a] 9 1^ 3 3 1^ ij J

3

in which the suffices t a k e the v a l u e s 1, 2 and 3 with the usual s u m m a t i o n convention; Pij a r e the p h y s i c a l components of t h e s t r e s s t e n s o r and Cj^^ a r e the c o m p o n e n t s of the C a u c h y - G r e e n d e f o r m a t i o n t e n s o r defined by ,

9x. 9 x .

S j = a x ; 3X„

w h e r e x. and X. a r e the r e c t a n g u l a r C a r t e s i a n c o - o r d i n a t e s of a t y p i c a l p a r t i c l e in the d e f o r m e d s t a t e and the undeformed s t a t e r e s p e c t i v e l y . I , I ^and I^ a r e

i n v a r i a n t s of the t e n s o r C^^ defined by the r e l a t i o n s

Cai3 V < ^ - 2 )

W i s the e l a s t i c a l l y s t o r e d f r e e e n e r g y p e r unit volume e x p r e s s e d a s a function of the i n v a r i a n t s I , I and 1 ; 6.. i s the unit t e n s o r .

1 2 ' 1 3

In t h i s p a p e r we s h a l l c o n s i d e r i n c o m p r e s s i b l e m a t e r i a l s , in which c a s e I j = 1 for all d e f o r m a t i o n s and W i s a function of I^ and I^ only. The s t r e s s -s t r a i n r e l a t i o n -s m a y then be w r i t t e n ,

p . . P6.. = 2 r 1 ^ C . . + ^ ( C . I, - C , C. .) ] (1.3) 13 13 L 9 1 , 13 a i ^ 1] ' ik k] J

in which p i s an a r b i t r a r y h y d r o s t a t i c p r e s s u r e following from the a s s u m p t i o n of i n c o m p r e s s i b i l i t y .

Rivlin (1956) h a s c o n s i d e r e d the p o s s i b l e f o r m s of W(I,, I^) and shown t h a t , for i n c o m p r e s s i b l e i s o t r o p i c m a t e r i a l s , the s t o r e d e n e r g y function m a y a l w a y s be w r i t t e n in the f o r m , I 1 I 2 I3 at S :x C«a

^Ka

det C . 13 W » W(I , I ) X ) ) C ' (I - 3 ) P ( I - 3 ) ' p-^1 q^l P^ ( 1 . 4 )

with Cóo= 0. C ' i s a s e t of c o n s t a n t s which defines the e l a s t i c p r o p e r t i e s of the m a t e r i a l .

In the k i n e t i c t h e o r y of r u b b e r - l i k e e l a s t i c i t y T r e l o a r (1958) d e r i v e d a specific f o r m for W for an ideal r u b b e r by i n v e s t i g a t i n g the p r o p e r t i e s of a n e t w o r k of i d e a l i s e d m o l e c u l a r c h a i n s . A s t a t i s t i c a l m e c h a n i c a l a n a l y s i s of t h i s n e t w o r k , involving c e r t a i n simplifying a s s u m p t i o n s , including that of i n c o m p r e s s i b i l i t y , l e a d s

to a stored e n e r g y function of the f o r m ,

W =. C ' ( 1 - 3 ) ( 1 . 5 )

10 1

in which C' = 2 NkT, w h e r e N i s the n u m b e r of n e t w o r k chains p e r unit voluntie, k i s B o l t z m a n n ' s constant and T i s the a b s o l u t e t e m p e r a t u r e .

F o r a m a t e r i a l obeying equation (1.5) the s t r e s s - s t r a i n r e l a t i o n s of equation ( 1 . 3) b e c o m e , p . . - P6.. = 2 f ^ C . 13 13 9 1 , 13 (1.6) « 2 C' C . 10 1]

Lodge (1956) h a s extended the s t a t i s t i c a l m e c h a n i c a l m o d e l of the i d e a l r u b b e r in such a way a s to p r o d u c e a model of a liquid which will exhibit v i s c o - e l a s t i c e f f e c t s . T h i s e x t e n s i o n i s achieved by a s s u m i n g that the network junction points of the ideal r u b b e r a r e no l o n g e r p e r m a n e n t but have a finite l i f e t i m e . The n e t w o r k can then u n d e r g o continuous s t e a d y d e f o r m a t i o n and, with v a r i o u s simplifying a s s u m p t i o n s , a r h e o l o g i c a l equation of s t a t e for the m o d e l i s d e d u c e d . T h i s h a s the f o r m , /• ax. ax. p . . - p 6 . . = / 2 S ' ( t - t ' ) — i - r - f d t ' ( 1 . 7 ) ^11 ^11 I 10 ax' ax' : / - « « in which x. a r e the r e c t a n g u l a r C a r t e s i a n c o - o r d i n a t e s of a p a r t i c l e at c u r r e n t t i m e t , and x^ a r e the c o - o r d i n a t e s of the p a r t i c l e at s o m e p a s t t i m e t ' .

S',o(t - t') i s a function of the e l a p s e d t i m e (t - t ' ) , which t e n d s to z e r o a s (t - t') t e n d s to infinity. Lodge i n t e r p r e t s t h i s function a s a lifetime d i s t r i b u t i o n function for the n e t w o r k c r o s s l i n k s and w r i t e s it ^^T j^^^ - t ' ) . "^^^ notation S' (t - t')

h a s b e e n adopted h e r e b e c a u s e equation ( 1 . 7 ) will be r e g a r d e d a s a s i m p l e m a t h e m a t i c a l g e n e r a l i s a t i o n of equation (1.6) in which the constant C,o i s r e p l a c e d by the function SJQ (t - t Odt' and the c o - o r d i n a t e s X^ by x'i ; a s u m m a t i o n o v e r all t i m e i s t a k e n up t o the p r e s e n t t i m e t .

It i s the p u r p o s e of t h i s p a p e r to w r i t e down a g e n e r a l i s a t i o n of e q u a t i o n s ( 1 . 3 ) , the s t r e s s - s t r a i n r e l a t i o n s for an i s o t r o p i c , i n c o m p r e s s i b l e e l a s t i c solid obtained by a m a t h e m a t i c a l g e n e r a l i s a t i o n f o r m a l l y analogous to that involved in going from equation (1.6) to ( 1 . 7 ) . It i s expected that t h i s new equation will define the p r o p e r t i e s of a c l a s s of v i s c o - e l a s t i c l i q u i d s . T h e p r o p e r t i e s of s u c h l i q u i d s , when subjected to a v a r i e t y of known flow h i s t o r i e s , will be i n v e s t i g a t e d .

T h e g e n e r a l i s a t i o n of equations (1.3) l e a d s to t

p.. - p5.. = 2 \ I f l S.. + 1^ (S..J -S. S .) Idt' (1.8)

1] 13 I X. 9J, 13 9J^ 1] 1 la a] J t ' = -00 w h e r e now, ax. ax S.. = — i - . - ^ (1.9) 13 a x ; a x ;

and J and J a r e i n v a r i a n t s of the d e f o r m a t i o n given in t e r m s of S.. by equations s i m i l a r to e q u a t i o n s ( l . 2).

A l s o ,

= (J,, J j )

S' (J, - 3)P (J, - 3)*^ and S' = 0 (1.10) p q 1 2 QQ

q=U

in which S a r e functions of (t - t ' ) , which m u s t tend to z e r o a s (t - t') t e n d s to infinity sufficiently quickly to e n s u r e c o n v e r g e n c e of the i n t e g r a l s in ( 1 . 8).

n i s w r i t t e n a s a function of J and J^^, the i n v a r i a n t s of the d e f o r m a t i o n t e n s o r Sj^-j. It i s the function which c h a r a c t e r i s e s the r h e o l o g i c a l b e h a v i o u r of the s y s t e m and it i s , in equation ( 1 . 1 0 ) , r e p r e s e n t e d a s a s u m m a t i o n of a s e r i e s of functions SpQ(t - t 0 by analogy with the c o r r e s p o n d i n g e x p r e s s i o n s in the e l a s t i c i t y t h e o r y . The p h y s i c a l significance of n i s not u n d e r s t o o d . In the p u r e l y e l a s t i c c a s e W i s known in the s e n s e that it i s the phenomenological v a r i a b l e of s t o r e d e n e r g y p e r unit v o l u m e . In s i m p l e c a s e s t h e r e i s a m o l e c u l a r i n t e r p r e t a t i o n of W, for e x a m p l e

the k i n e t i c t h e o r y of r u b b e r like e l a s t i c i t y p r e d i c t s W from m o l e c u l a r c o n s i d e r a t i o n s . H o w e v e r we do not even u n d e r s t a n d the p h e n o m e n o l o g i c a l n a t u r e of Q. It i s not

p r o p o s e d , t h e r e f o r e , to a t t e m p t a d i s c u s s i o n of t h e s e q u a n t i t i e s h e r e ; r a t h e r will it be a s s u m e d that such a g e n e r a l r e l a t i o n a s ( 1 . 8 ) i s p h y s i c a l l y p e r m i s s i b l e and i t s p r o p e r t i e s will be i n v e s t i g a t e d .

T h e r h e o l o g i c a l equation of s t a t e ( 1 . 8 ) can be r e w r i t t e n in the f o r m ,

t'= - . w h e r e , and,

^l

p'

ax.

= p + 2

ax'

a

ax.

3 t r ^

an

_9J, (1.12) dt' (1.13) T h e f o r m (1.11) i s m o r e convenient for a l g e b r a i c m a n i p u l a t i o n than the f o r m

( 1 . 8 ) . The proof of t h i s r e s u l t will be found in Appendix 1. T h i s equation of s t a t e will now be i n v e s t i g a t e d for v a r i o u s s t e a d y s t a t e flow s y s t e m s .

4

-P a r t II. Some -P r o b l e m s in Steady Flow (a) Simple S h e a r i n g Flow

C o n s i d e r the c a s e of s t e a d y r e c t i l i n e a r l a m i n a r s h e a r flow in which a x e s a r e chosen so that the s t r e a m l i n e s a r e p a r a l l e l to the x , a x i s and the s h e a r i n g p l a n e s a r e p e r p e n d i c u l a r to the x^ a x i s . T h e k i n e m a t i c e q u a t i o n s a r e t h e n ,

X, » x^ + G(t - t') x^

X » X 3 3

w h e r e G i s a c o n s t a n t , the s h e a r r a t e . By evaluating S.. and S.. we find, J "J ' 3 + GHt - t ' ) "

1 2

J , « 1

and using equation (1.11) we find, p - p = N + G* N - M w h e r e , a n d , ' 1 1 P22 P 3 3 N, 1 M. 1 - P - P P21 P 2 3 = 2 = 2 0 2 = N M -0 -0 = N - M 0 0 0 G^M i = (N + M ) G 1 1 *31 t

ƒ (t-t')^

~ 00 t r

j it-t')'

- 0 0 9n ^+ a j , '^^ 9 ; d t ' (2.1) ( 2 , 2 , ) ( 2 , 3 ) ( 2 , 4 ) ( 2 . 5 )

It i s obvious that the equations of motion and continuity a r e s a t i s f i e d . It i s to be o b s e r v e d that n ( J , J^) will be a function of G so that the v i s c o s i t y ,

P

- ^ » N + M , i s not n e c e s s a r i l y a c o n s t a n t . However we notice that p^^- P j j , P

P22" P11' r ^ ^^® ^^^ functions of G , in a g r e e m e n t with the g e n e r a l p r e d i c t i o n s of C o l e m a n and Noll (1961).

If.

t h e n N. and M. b e c o m e c o n s t a n t s independent of G. T h e s e c o n s t a n t s will b e called N ! and M', w h e r e , t N ! = 2 f S;„ (t - t ' ) ^ d t ' ( 2 . 7 ) J -eo t M'. = 2 f S' (t - t ' ) ^ d t ' ( 2 . 8 ) / 01 - oo

In t h i s c a s e p - p and p - p^j b e c o m e p r o p o r t i o n a l to the s q u a r e of the s h e a r r a t e and the v i s c o s i t y b e c o m e s c o n s t a n t , in a g r e e m e n t with r e s u l t s of Markowitz (1962) for low r a t e s of s h e a r .

(b) Flow t h r o u g h s t r a i g h t p i p e s of a r b i t r a r y c r o s s - s e c t i o n

Following the w o r k of E r i c k s e n , (1956), and O l d r o y d , (1958), it i s of i n t e r e s t t o e x a m i n e the flow of liquid down a pipe of a r b i t r a r y c r o s s - s e c t i o n . C o n s i d e r an infinitely long pipe whose w a l l s a r e p a r a l l e l t o the x., a x i s and whose s e c t i o n i s given by F(x , x ) =«0. Writing the k i n e m a t i c equations d e s c r i b i n g the flow a s ,

X, =• X,' + (t - t') f (x; , x'3)

X , ' x ; ( 2 . 9 )

X " x'

3 3

we s h a l l e x a m i n e w h e t h e r a flow of t h i s t y p e , in which the s t r e a m l i n e s a r e p a r a l l e l to the x, d i r e c t i o n , i s p o s s i b l e for a liquid obeying the r h e o l o g i c a l equation of s t a t e ( 1 . 1 1 ) .

E v a l u a t i n g S.. and S.' f r o m ( 1 . 9) and (1.12) we find, 13 13 1 + (t -t'fai + fi) S.. = [ (t - t')f 1 0 1 (2.10) (t - t'H^ S.'.^ - -(t - t')f 1 + f ( t - t ' ) ^ (t - t'fti \ (2.11) 13 z -(t - t')f w h e r e , ^ ^ af(x^, X3) ax 2 2 (t - t')f^ 1 0 -(t - t')f^ 1 + i^(t - t ' ) ^

(t -

t')ïf

2 3 , . 9f(x,. x ;

' ax

3 (t - t')f^ 0 1 - ( t - t ' ) f , (t - t')'f/3 1 + f^t - t ' ) 3 and h e n c e , J , » J , = 3 + (t - t'f(f^+ tp J ,

6

-Substituting S.. and S.. in (1.11) we get, p - p = N + N (f*+ f^) - M 11 o 2 2 3 O p - p = N - M - M f ^ ^ 2 2 "^ O O 2 2 p - p = N - M - M f ^ 3 3 O o 2 3 ( 2 . 1 2 ) P2, = <N, + M,)f^ P31 = <N + M , ) f , p = - M f f • ^ 3 2 2 2 3

T h e equation of continuity i s obviously s a t i s f i e d . Since, in t h i s c a s e , t h e r e a r e to be no body f o r c e s and the flow i s s t e a d y and r e c t i l i n e a r , the e q u a t i o n s of m o t i o n which have to be satisfied a r e ,

a p . .

^ = 0 (2.13)

ax. 3

If we f i r s t c o n s i d e r the s p e c i a l c a s e of ( 1 . 1 1 ) in which n t a k e s the f o r m ( 2 . 6 ) , we find, on i n s e r t i n g (2.12) into (1.11) and noting that N' and M'. a r e independent of X. 1' a£ (N' + M' ) (f + f ) = 1 1 22 3 3 O X 1 ^ ^ 2 2 + f 3 3 ) ^ ^ 2 f22+ f23 f3 - ^ ^^ ( 2 . 1 4 )

^P^.^^J•'i,^,,^^..'.

_i_ ap M ' 9X T h e s e e q u a t i o n s a r e c o n s i s t e n t if, ^22 "*" ^33 ^ constant = - P (say) (2.15) T h i s equation and the b o u n d a r y condition of no s l i p , o r , m a t h e m a t i c a l l y ,

fg = f, = 0 on F(x2, Xj) = 0, enable ik^^, x ^ to be d e t e r m i n e d . P(N,' + M*, ) i s in fact the p r e s s u r e g r a d i e n t down the t u b e , and l(x^, x,) i s of c o u r s e the velocity d i s t r i b u t i o n for a Newtonian liquid of v i s c o s i t y (N' + M* ). In o t h e r w o r d s , we have shown that a liquid for which n t a k e s the f o r m given by (2. 6) will flow down a tube of a r b i t r a r y c r o s s s e c t i o n with the p a r t i c l e s of fluid moving in r e c t i l i n e a r p a t h s .

In g e n e r a l , h o w e v e r , on i n s e r t i n g ( 2 . 1 2 ) into ( 1 . 1 1 ) , and r e m e m b e r i n g that now N. and M. a r e functions of x and x , we get,

9p' = A

TM

f n + -^

TM

f f 1

ax^ 9x, L 2 ^ J 9x3 L == ^ ' J (2.16)

K" 9 ^ J ^ ^ 0 ^ 9T [^3 ^2 f 3] where p ' = p + N^ - M^

This set of equations cannot normally be satisfied by some function f(x , xj. That i s , the liquid will not necessarily flow down the tube of arbitrary cross section with rectilinear streamlines. However, for the special case of tubes of circular cross section, it can be shown that the liquid will always flow down the tube with streamlines parallel to the axis of the tube.

(c) Troughton or elongational flow

Consider a state of flow in which a liquid filament is elongated at a constant rate of strain; that i s ,

^ = k, V ^ = k X ^ = k3X3 ( 2 . 1 7 )

dt ' ' dt ^ ' dt ' '

where k. are constants describing the flow. If now we take k, = a and k^= k^= -^a we have simple elongational flow. We have taken k + k + k = 0 since this is

required by the constant volume condition. The kinematic equations become,

.' „a(t-t') , _ , _-èa(t-t') ,_ _ ..' _-ea(t-t')

giving,

x^ = x^ e ^2 ^ ^2 ® S ^ ^3 ^ ( 2 , 1 8 )

^ 2a(t-t') ^ - -a(t-t') ^ -2a(t-t') ^ „ a(t-t') J = e + 2 e , J = e + 2 e

1 2 J , = 1

The equation of state gives,

p - p = p = p = p = 0

^2 2 ''^ 3 3 "^12 ^ 2 3 ^31

/ „ f a n / 2a(t-t') -a(t-f)\ an / -2a(t-t') a(t-t')VK./

" (2.19) We observe that since n is a function of (t - t') the transformation

T = t - t' proves that p - p is not a function of t. Let us consider the special case where n takes the form,

n = C e-^i<^-^') [ j , - 3 ] + C^ e-^^<^-*') [ J^ - 3] (2,20) where C^ and C^ are constants independent of (t - t ' ) . The reason for this is

8

-£ a

e-Wt

S,; = Z-/ ^ m ^ '"^^ ''^ <2.21)

m = l

and i s able to give a p h y s i c a l m e a n i n g to ^ ^ - A n a t u r a l e x t e n s i o n of (2. 21) t o the f o r m of n given in (1.10) i s to take m

\ ' -\ ( t - t ' )

S ' = /_, h e P^'^ (2.22) pq ^ 1 p q r

w h e r e b and X a r e independent of (t - t ' ) . Equation (2.20) r e p r e s e n t s a f o r m p q r p q r

of n in which b = Q , b „ = C „ X , = K and X = K . All the o t h e r b ' s and X's

101 ' # 1 ^ ^ \ ^ 1 ^ oil / 2

a r e z e r o . If a < min(2 K,, K^) we find, by evaluating ( 2 , 1 9 ) ,

rv r> - ^^' ^ ^^ + ^ a - 2C2 . ,

Pii P22 " (K, + a)(K^ - 2a) (K^ + 2a)(K^ - a) ^'^•'^'^' If we c a l l rj = " " "^^ , the T r o u g h t o n v i s c o s i t y , then

3 ! S 3 ! ^

ir-

_^^

' (:n-^) e^K^X'-tj

(2.24) and we note t h a t , a s a t e n d s to z e r o , =: '1' ( 3 ' 2C — 1 - +

^K

2C . 2

K

J

= 3 . ( s h e a r v i s c o s i t y )

The f o r m of the two functions involved in ??„ i s shown in F i g s . I and II. We s h a l l now d i s c u s s the p h y s i c a l significance of t h e s e r e s u l t s . It i s well known that e l a s t i c liquids r e a d i l y f o r m f i l a m e n t s of liquid: t h u s , if a rod i s dipped into a p o l y m e r solution and then w i t h d r a w n , a filament of liquid will be w i t h d r a w n with it. Most Newtonian liquids do not have t h i s p r o p e r t y , and it

s e e m s r e a s o n a b l e t h e r e f o r e that the e l a s t i c n a t u r e of t h e s e liquids m a y s o m e t i m e s explain t h i s effect.

Lodge (1960), h a s investigated the s t a b i l i t y of such elongational flow: for a filament which v a r i e s in t h i c k n e s s along i t s length, he h a s shown that if the e l o n g a t i o n a l v i s c o s i t y i n c r e a s e s sufficiently quickly with r a t e of s t r a i n then a thin s e c t i o n of the filament will d e c r e a s e in a r e a l e s s r a p i d l y than a thick s e c t i o n . T h e flow will then obviously be s t a b l e .

The above m a t h e m a t i c a l a n a l y s i s i n v e s t i g a t e s a s i m p l e m o d e l for the elongation of a liquid filament: we find that the elongational v i s c o s i t y r i s e s r a p i d l y with r a t e of s t r a i n , ( F i g s . 1 and II). It can be shown that c e r t a i n l y s o m e w h e r e in the r a n g e of s t r a i n r a t e s 0 < a < m i n (è K,, K,) t h e r e i s a s t r a i n r a t e a„ above which the elongational flow i s s t a b l e , in the Lodge s e n s e , and below which it i s u n s t a b l e .

b e the s p h e r e of r a d i u s a, with its c e n t r e at the apex of the c o n e . The s h e a r i n g l a m i n a e a r e a s s u m e d t o be c o n e s c o - a x i a l with, and with the s a m e v e r t e x a s , the r o t a t i n g c o n e . The fluid p a r t i c l e s a r e a s s u m e d to r e m a i n at fixed d i s t a n c e s f r o m the apex d u r i n g the m o t i o n . We use the equation of s t a t e in the f o r m A9 ( s e e Appendix II).

We take the s p h e r i c a l p o l a r c o - o r d i n a t e s ( r , 6, ^ ) a s our c u r v i l i n e a r o r t h o g o n a l s e t ( 6 ^ , 6^, 63) at t i m e t , and we i n v e s t i g a t e w h e t h e r the k i n e m a t i c equation (2.25) s a t i s f i e s the e q u a t i o n s of motion and continuity.

c^ = r c o s ^ s i n e c = r sin é sin 6 2 ^ c, = r cos 6 x^ = r cos ( ^ - r 6 ) s i n 6 x ' = r sin (é - TQ) s i n e 2 "^ x ' = r cos 6 3 (2.25) w h e r e ,

_{r - f (t-t')}

\ ) = ' _{( 2 ) ' ( 3 )} = r s i n e w h e r e h . . . i s defined by equation A 5 , Appendix II.

I n s e r t i n g e q u a t i o n s ( 2 . 25) into AlO we find,

w h e r e , S.. = 13 s.-.i = 13 J = / \ / \ J 1 0 0 ' . 0 0 3 0 1 T sin e 0 1 +r2 sirf e - r sin e + r^ s i n ^ e 0 T s i n e 1 +r^sin2 6 0 - r sin e 1 (2.26) (2.27) (2.28)

and we get f r o m A 9 , ( 2 . 26) and ( 2 . 27), with the u s u a l notation for the p h y s i c a l c o m p o n e n t s of the s t r e s s t e n s o r in p o l a r c o - o r d i n a t e s ,

10 -( r r ) - p

(ée) - p

(S<t>) - p

(e;^)

= = = = N - M o 0 N - M - - ^ sin'' e . M _{o o «2} N + — sin^ e . N - M o a'= 2 o (N, + M ) - s i n e ( P e ) = {r<f>) = O ( 2 . 2 9 ) T h e equation of continuity for i n c o m p r e s s i b l e n a a t e r i a l s i s

div V = O

w h e r e v i s the v e l o c i t y of a p a r t i e l e . U s i n g ( 2 . 25) we s e e that t h i s i s obviously s a t i s f i e d . T h e e q u a t i o n s of motion a r e ,

^ (rr) + - ^ (ft) + - 4 - ^ ^ (P^S) + - r2(r''r) - (66) - {fi>) + (A cot 6)1

9 r r 36 r s i n ö 9^ r L J y2 = - p g c o s e - pT s i r f e —2 e* a ,^, aF <^ _a_ ar (.2 = P g s i n e - pr sine c o s e — e^ a ( r > . i A (e"^) ^ - i _ . ^ ^ ^ (^-^) , i [3 (r'V) + 2 (é>) cot 6 ) ] (2.30)

w h e r e p i s the d e n s i t y of the liquid.

It can be shown that e q u a t i o n s ( 2 . 2 9 ) , when i n s e r t e d into ( 2 . 3 0 ) , a r e c o m p a t i b l e when 6 = 2ir , and when i n e r t i a l f o r c e s and body f o r c e s a r e i g n o r e d . That i s , an a p p r o x i m a t e solution to equation ( 2 . 30), when a i s s m a l l , m a y be obtained by putting 6 = ITT and i g n o r i n g i n e r t i a l and body f o r c e s . We then find that (2.30) b e c o m e s ,

|H + i

[M,

- N 1 4 = 0

a r r [_ 2 2J Q.2

F i = ° <2.3i)

9 £ 3^ and h e n c e , „ 2 p = - log r (M - N ) V + constant (2.32) 2 2 a"

U s i n g the b o u n d a r y condition that ( r r ) " 0 when r = a, we find,

(eè)=(N - M ) - ï ^ l o g - - M, 4 <2,33)

2 2 g[2 a a

* /\ Noting t h a t the n o r m a l f o r c e Z p e r unit a r e a on the plane i s -(66) when

e = i w , we obtain

Z * = ( N ^ - M ^ ) ^ l o g | + M ^ - ^ ( 2 . 3 4 ) a l s o ,

(^0) = (N + M ) ^

'^ 1 1 a

If the s h e a r f o r c e p e r unit a r e a on the plane is T , then T = (e?l) at e= ir/2

and

T = (N + M ) - at 6 = Tr/2 ( 2 , 3 5 ) 1 1 a

(e) Flow b e t w e e n p a r a l l e l p l a n e s

L e t the liquid be s h e a r e d between two h o r i z o n t a l p a r a l l e l p l a n e s , with one p l a n e

fixed and the o t h e r r o t a t i n g about i t s n o r m a l with a constant a n g u l a r v e l o c i t y u .

L e t the b o u n d a r y of the liquid be a c y l i n d e r , of r a d i u s a, c e n t r e d on the a x i s of r o t a t i o n . If the s h e a r i n g l a m i n a e a r e a s s u m e d to be p l a n e s p a r a l l e l to the fixed p l a n e , and the d i s t a n c e of any p a r t i c l e f r o m the a x i s of r o t a t i o n i s c o n s t a n t , we m a y i n v e s t i g a t e w h e t h e r the k i n e m a t i c e q u a t i o n s ,

(2.36)

s a t i s f y the e q u a t i o n s of motion and continuity. We u s e a s c u r v i l i n e a r c o - o r d i n a t e s ( 6 , 6 , 6^ ) the c y l i n d r i c a l p o l a r c o - o r d i n a t e s ( r , 6 , Z ) . K = J;! (t - t') w h e r e L

L i s the d i s t a n c e b e t w e e n the two p l a n e s . We note t h a t ,

h, , = 1 h, . = r h, . = 1

(i) (2) (3)

where h/.v i s defined in equation A5 (Appendix II), and, f r o m e q u a t i o n s ( 2 . 36) and

A l O . ^^' • ' ^ l ^ 2 ^^3 = r cos 6 = r sin 6 = Z x ' 1 x ' 2 t ^^3 = r COS ( 6 - ICZ) = r sin (6 - KZ) = Z S.. = 1] 1.1 ' 0

V

V

/

r

\ o

1 0 + r^K* r K 0 1 - r K 1 0 r K 1 0 - K r + r'=K' (2.37) (2.38)

12

-and,

J = J, = 3 + r* K* , J

1 2 ' 3

Equations (2.37), (2.38) and A9 give, using the usual notation for the physical components of the s t r e s s tensor in cylindrical polar co-ordinates,

( r r ) - p = N - M o o /> 2 2 (ZZ) - p = N - M - ^ M, o o L "

(Pe) = (rZ) = 0

(ez) = ^ (N + M ) <2.39)

L 1 1 (66) - p = N - M + ^ N, _{o o L^ 2} We note that N. and M. are functions of r .

1 1

The equation of continuity is obviously satisfied. The equations of motion a r e ,

±.^r%) + ^ ^ - ( ^ 6 ) + ^ ^ (r'z) + i ((r-r) - (6^6)) - - '-^ p

dr r ae a z r \ / L

fi^r) ^ 1 9 ( 6 ^ . ^ ( 6 %

^

^ = 0 (2.40)

a r r a e a z r

f(rz) .i-^(e"z).A(z'^z).i^ = -gp

3r r 36 a Z r . ( r ^ )

Inserting (2.39) into (2.40) we obtain,

3 r r \

_L'

r ^ N , ) ^ 3p' 36 3p'

az

r u j ^ Z ^ 0 - pg (2.41) where p ' = p + N - M ''^ ^ o o

We observe that these equations are inconsistent unless we ignore the inertia r u' 7?

term - 5— p . Ignoring the inertia t e r m and integrating these equations, we get, ^

Using the boundary condition that there is zero surface traction at the surface r = a for all Z , 6, we find that this condition can only be satisfied if we ignore the t e r m -PgZ in (2,42). Physically, this is equivalent to considering only very small gaps between the planes. Hence, ignoring the -pgZ in (2.42) and using ( r r ) = 0 at r = a, N + M + o o u —T L r Ngdr (2.43)

Using (2.43) and (2.39) we find,

(ZZ) =

a

_{r N dr}

-2

M .

(ze) = ^ (M, + N,)

Again the normal force Z per unit a r e a on the plane i s , a , 2 T ' 2 I" - ' 2 ,* _{M +} 2

n

r Njjdr (2.44) (2.45) (2.46)

The tangential force T per unit a r e a on the plane i s ,

T = Y^ (M + N ) (2.47)

(f) Flow between rotating coaxial cylinders

Let the liquid be contained between vertical coaxial cylinders of radii r^ and r^ rotating with angular velocities w^ and u^ . Let us assume that the shearing laminae a r e cylinders coaxial with the moving cylinders, and that any particle r e m a i n s at the same height between the cylinders. Taking cylindrical polar co-ordinates (r, 6, Z) for (6,, 6^, 63). with the Z axis along the axis of the cylinders, we may investigate whether the kinematic equations,

X = r cos e r sin 6 Z x | = r cos [ e - (t - t') q ( r ) ] x ' = r sin [p -(t - t') q(r) ] 3 3

satisfy the equations of motion and continuity. F r o m (2.48) and AlO we obtain,

(2.48) S.. 13 1 r(t - t ' ) q ' 0 r ( t - t ' ) q ' 1 + r^(t-t')==q'* 0 0 0 1 (2.49)

- 14 w h e r e q

/ I + rS(t - t ' ) V

: / - ( - H . - f>q'

V 0

d r - r ( t - f ) q ' 1 0 0 0 1 (2,50) a l s o , J^ = J^ = 3 + r2 (t - t ) q'^and J3 = 1. U s i n g t h e e q u a t i o n of s t a t e A9 we g e t , w h e r e a g a i n the u s u a l n o t a t i o n for t h e p h y s i c a l c o m p o n e n t s of the s t r e s s in c y l i n d r i c a l p o l a r c o - o r d i n a t e s i s u s e d , ( r r ) - p = N - M - r 2 q'^M 0 0 2 ( 2 . 5 1 ) (eé) - p = N + r* q'^N - M o 2 0

{£t) - p = N - M

0 0 ( r ö ) = r q ' ( N + M^) ( r Z ) - (Z 6) = 0 w h e r e M. and N. a r e f u n c t i o n s of r . 1 1 T h e e q u a t i o n s of m o t i o n , a s s u m i n g t h e r e a r e no body f o r c e s o t h e r t h a n g r a v i t y , a r e , r ^ (A-) + ^ ^ ( P e ) + ^ ( r ' z ) + l ( ( P r ) - ( 6 6 ) ) = - rq»(r) p 3 r r 3 e a Z r \ /

A (e-r) -. 1 ^(6%) -. - ^ (e^z) + 2(£eL = 0 (2.52)

3r r a e a z r

-9- (Zr) + i ^ (Z6) + A ( ^ ) + i ^ = _ gp

a r r cjo dZ r w h e r e p i s t h e d e n s i t y of t h e l i q u i d . I n s e r t i n g ( 2 . 51) i n t o ( 2 . 52), we g e t ,

I f ' - I T [^'^"^2]- 7[r2q'2Jj^N +M^j = -q»pr (2.53)

tf + ^ 1 1- q'<N +M ) 1 + 2 q'JN +M = 0 (2.54)

36 3 r (_ 1 1 J (, 1 'J ^ ' = - P g ( 2 . 5 5 ) w h e r e p ' = p + N - M . •^ "^ o o

Since ÖQ = O, equation (2. 54) determines q(r) as a function of r when

n is known. This value of q(r) enables p' to be determined. A knowledge of the

boundary conditions will then enable the stress at any point to be determined. Let us consider the special case when n is given by (2.6). Integrating

(2.54) and using the boundary conditions that q(r^) = w^ and q(r2) = w , we find,

q(r) = A - ^ hence q'(r) = ^ (2.56) r r

where ^"(0 -v\ u r^ r^ (w -u )

A = -^—^ S_2 and B = -^ ^ ^ '— ( 2 . 5 7 ) _{2 2 2 2} r - r r„ - r^ _{1 2 2 1}

Using this value of q(r) we can integrate (2. 53) to give,

p' = — r 3 M^ - N^' I -pj ^ - 2AB log r - — j - pgZ + constant

(2,58)

where M ' and N ' are given by (2. 7) and (2. 8). This gives,

(ZZ) = 2_ SM*^ - K\ '^ P\' T ~ "*" ^ ^ ^ ^°^ '^ ''• ~ 2 r " ^^ "^ constant

r ' *- J y. " 2 r

(2.59)

Z = - (ZZ) represents the normal forces on a boundary Z = constant which would be required to maintain the assumed state of flow. We see that Z* is composed of three parts: the first part is due to the non-Newtonian nature of the fluid, the second due to its centripetal acceleration and the third part gives the usual variation of pressure with height in a liquid. For non-Newtonian fluids the first

part may be considerably greater than the second part. In this case, if SM^^ > N^

then Z* is smaller near the centre than it is at the circumference of the gap; hence, if this force were removed, the liquid would tend to fall in the centre and rise at the circumference. However, if "iM^ < N^ the reverse would happen: on removing

the force the liquid would tend to rise at the centre and fall at the circumference.

The second effect is known as the positive Weissenberg effect and the first as the negative Weissenberg effect.

16

-Discussion

In general, equations of state may be divided into two c l a s s e s : those which have microrheology as their b a s i s , such as those of Lodge (1956) and Oldroyd (1950), and those which are derived from phenomenological considerations, (Coleman and Noll, 1961 ; Rivlin and Ericson, 1955). The rheological equation of state put forward in this paper belongs mainly to the second class and is an attempt to extend the phenomenological theory of large elastic deformations to a fluid.

The basis of this equation of state is the use of a time dependent function n which is a generalisation of W, the stored energy for a purely elastic deformation. The physical significance of Q is not well understood. Because of the nature of the generalisation p r o c e s s , n dt will still have the dimensions of W, energy per unit volume, but must be connected both with the recoverable energy at time t and with the rate of energy dissipation at this time. Thus, even the phenomenological

significance of n is not clear. The liquid characterised by ( 1 . 8) is in a sense a composite relaxing solid composed of many relaxation p r o c e s s e s , the elastic modulus for each process being characteristic of a general nonlinear elastic deformation. The energy associated with these p r o c e s s e s is gradually dissipated, in some unspecified way, in a fashion determined by the distribution of the

relaxation p r o c e s s e s and the s t r e s s and strain history of the sample. In this way n is a time dependent distribution of energies among the relaxation p r o c e s s e s . Since this equation has been derived from considerations of a thermodynamic stored energy function, it is to be hoped that further investigation will produce a t h e r m o -dynamic justification for it.

Although the form of Q is not specified, the advantage of using this generalised form is that it can be altered to fit any experimental results for a large class of liquids. It can be seen, from analysis of simple shear flow, that the normal components of the s t r e s s can be varied independently by a suitable choice of 0 , and in this sense all possible experimental r e s u l t s can be analysed in t e r m s of this equation of s t a t e . Such a procedure might throw some light on the nature of n ; its general form may be specialised to give mathematically simple relations which, if they agree with experiment, may help to place it on a sound physical b a s i s .

Experimental m e a s u r e m e n t s in this field do not, in general, cover a sufficiently wide range of shear r a t e s to enable an unequivocal form of n to be chosen. However, an examination of the work of Brodnyan et al (1957) on polyisobutylene solutions

shows that a form of n which is a function of J^ only is sufficient to explain the experimental r e s u l t s . Many more experimental m e a s u r e m e n t s on other liquids over a s i m i l a r range of shear r a t e s a r e highly d e s i r a b l e .

A comparison of this equation of state with other equations of state can, at this e a r l y stage, only be discussed briefly. Since it includes Lodge's equation of

state as a special case, all phenomena which can be explained by his equation can, of course, be equally well explained by this; in addition, this equation will cover the c a s e s in which the normal s t r e s s e s in simple s h e a r a r e all different. It would be of interest to enquire whether this equation of state describes a simple fluid in the sense of Coleman and Noll and whether it is included in their general formalism.

Certain types of liquids a r e obviously not described by this equation of state: liquids which change volume on applying a s t r e s s are not included because we have assumed incompressibility; the equation is isotropic and therefore anisotropic

liquids a r e not included; and since the most general form of n (2.20) predicts a viscosity which is independent of t i m e , the equation will not describe

thixotropic liquids. However, it should be possible, in this last c a s e , to make the S'. of equation (1.10) functions of t nnd t' r a t h e r than of (t - t ' ) , and

J

18 -References 1. 2. 3 . 4 . 5 . 6. Brodnyan, J . G . , Gaskins, F . H . , Philippoff. W. Coleman, B . D . , Noll, W. E r i c k s e n , J . C. Lodge, A . S . Lodge, A . S . Markowitz, H. (1957) (1961) (1956) (1956) (1960) (1962) 7. 8. 9 . 0. 1. Oldroyd, J . G . Oldroyd, J . G . Rivlin, R . S . Rivlin, R . S . E r i c k s e n , J . L. T r e l o a r , L . R . G . (1950) (1958) (1956) (1955) (1958) T r a n s . Soc. Rheol. V o l . 1 , p. 109.

Ann. New York Acad. Sci. , Vol.89, pp 672-714.

Q.Appl. Maths. Vol.14, p. 299.

T r a n s . F a r a d a y S o c , Vol.52, p. 120. Colloque International de Rhelogie, P a r i s . Int. Symposium on 2nd O r d e r Effects in Elasticity, Plasticity and Fluid Dynamics, Haifa, 1962.

P r o c . Roy. Soc. A, Vol.200, p . 523. P r o c . Roy. Soc. A, Vol.245, p . 278. Rheology, Vol.1 (Academic P r e s s Inc.) p . 3 6 2 .

J . Rational Mech. & Anal. Vol.4, p . 3 2 3 .

APPENDIX I

Proof that equation ( 1 . 8) can be expressed in the form (1.11)

We first note that,

, a x' 3x' 3x, ax.

ik kj ax. ax, 3x' 3x' " ij ^ ' 1 k jS' /?

This provides a justification for the use of the notation S.. . Let S be the matrix whose element belonging to the i row and j column is S.. and similarly

-1 -1 ^•' let the matrix of S.. be S . The Cayley-Hamilton theorem states that

0 (A. 2) s ' - J s ' + J^ S

where I is the matrix of 6 .. .

13 F r o m A2 we obtain, S J^ - s' = J^ l_ - ^3 L - ^3 l ' (A. 3) F o r incompressible m a t e r i a l s J = 1,

Now by inserting equation A. 3 into the equation of state ( 1 . 8) we obtain the form (1.11), where p' is given by (1.13).

20

-APPENDIX II

The equation of state in t e r m s of convected co-ordinates

At the current time t the position of the particle can be represented by rectangular Cartesian co-ordinates x. or by a system of orthogonal curvilinear co-ordinates 6 . where,

1

X. = X. ( 6 . ) ( A . 4 )

and ds is a small element of length given at tinae t by,

ds"* = dx^ + dx^ + dx^ = hf , dö^ + hf , de^ + h ' , d e ^ (A. 5)

1 2 3 ( l ) 1 ( 2 ) a ( 3 ) 3

We now "name" the particle by the co-ordinates 6 . , and, for all t i m e s t ' , 6 . becomes a convected set of co-ordinates, which are of course only orthogonal at time t. Therefore we must have,

X.' = x!(6.) (A. 6)

1 1 1

At the time t the physical components of the s t r e s s are given by p . . with respect to the rectangular Cartesian axes x.. The physical components of the s t r e s s at the point 6 ., with respect to rectangular Cartesian axes which are locally coincident with the mutually orthogonal lines 6 . = constant, a r e given by,

where,

where i is not summed.

F r o m (A. 7), (A. 8) and the equations of state (1.11), t cr . -0-6.. = f 13 13 / " i J ^ ^ p = I. ip \ ) t. p

3q pq

' \ 1

' \ ^ i )

ax

p

36.

(A. 7) (A. 8) hence, an / a e . a e . ax ' 3 J , l ^ ^ i ) N j ) a X p 3x^ 3 x ; „ an / 1 ^^p ' \ " a j , Vh/.vh,.. 3 6 . 36. 2 \ (1) (3) 1 3 90 / ^ ^ ^^i l ! l \ 9 J V (i) (3) 9x'^ ax^ ) 9x ,

3-J

ax' 3x' .-, a a \ 1 3x ax j p q ^ J an / 1 a J i h,..h.., 2 \ (1) (3) dt'

ae .

1 d t ' (A. 9) which is the required r e s u l t .

S.., r e f e r r e d to the orthogonal set of curvilinear co-ordinates, is given by, J % -13 ^ i ) ^ j ) 1 ^ i ) %)

ae.

1 3x' _a

ax'

a 36 . 1

ae .

3x' a 3 6 . 3 (A. 10)

Il

3?o

FIGURE I . GRAPH OF ^ AGAINST — FOR ELONGATIONAL

FLOW WITH W = W C l j ) 5 —

1l

39» 3 — 2 — O -OS -I -IS -2 -25 -3 -35 -4 .45 .5 55

FIGURE n . GRAPH OF ^ AGAINST — FOR ELONGATIONAL

3fo <\ FLOW WITH W - W Cli)