flow fo Oldryod-8 constant fluid in a converging channel

7/25/2019 flow fo Oldryod-8 constant fluid in a converging channel

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Acta Mechanica 148, 117-127 (2001)

C T M E C H N I C

9 Springer-V erlag 2001

F l o w o f a n O l d r o y d 8 c o n s t a n t f lu i d in a

c o n v e r g e n t c h a n n e l

S . B a rl ~ , I s t a n b u l , T u r k e y

(Received January 13, 2000; revised Feb ruar y 28, 2000)

Summary The pro blem considered is the steady flow of an Oldroyd 8-constant fluid in a convergent

channel. U sing series expansions in terms of decreasing pow ers of r given by Strauss [8] for the stream

function and stress components, the governing equations of the pro blem are reduced to ordinary differen-

tial equations. The resulting differential equations have been solved by employing a numerical technique.

It is shown th at the streamline patterns are strongly dependent on the non-Newtonian parameters.

1 I n t r o d u c t i o n

T h e p r o b l e m o f a s t e a d y t w o - d i m e n t i o n a l f l o w o f v i s c o u s, i n c o m p r e s s i b l e f lu i d in a c h a n n e l

b o u n d e d b y t w o i n f i n i te n o n - p a r a l l e l p l a n e w a l l s w i t h a l in e s o u r c e o r s i n k a t t h e i n t e r s e c t io n

o f t h e w a l ls w a s f i r s t d i s c o v e r e d b y J e f f r e y [ 1 ], a n d H a m e l [ 2 ]. T h e y f o u n d t h a t a p u r e l y r a d i a l

f l o w c a n b e o b t a i n e d w h i c h s a t is fi e s e x a c t ly b o t h t h e e q u a t io n s o f m o t i o n a n d t h e b o u n d a r y

c o n d i ti o n s . S u b s e q u e n t l y , m a n y w r i te r s w o r k e d o n s p e c i al a s pe c t s o f th e p r o b l e m b u t t h e

m o s t c o m p r e h e n s i v e t r e a t m e n t s h a v e b e e n g i v e n b y R o s e n h e a d [ 3] , a n d M i l l s ap s a n d P o h l -

h a u s e n [ 4 ]. A n e x p l i ci t e x a c t s o l u t i o n c a n b e o b t a i n e d i n t h e e v e n t o f n o n - i n e r t i a l f l o w ( s e e

B i r k h o f f a n d Z a r a n t a n e l l o [ 5 ]) . M o f f a t t a n d D u f f y [6] h a v e a n a l y z e d t h e b r e a k d o w n o f th e

J e f f r e y - H a m e l s i m i la r it y t r a n s f o r m a t i o n s .

T h e f l o w o f t h e N e w t o n i a n f l u i d b e t w e e n i n t e rs e c t i n g p l a n e s h a s b e e n e x t e n d e d w i t h i n t h e

c o n t e x t o f v a r i o u s d i f f e r en t n o n - N e w t o n i a n f lu i d m o d e l s b e c a u s e o f t h e w i d e r a n g e o f e n g i-

n e e r i n g a p p l i c a t i o n s . F o r e x a m p l e , i t w i l l h e l p p r o v i d e v a l u a b l e i n f o r m a t i o n o n t h e d e s i g n o f

e x t r u s i o n d i e s o f i n d u s t r i a l i m p o r t a n c e . T h e o r e t i c a l r e s e a r c h o n s t e a d y f l o w o f t h is t y p e w a s

i n i t ia t e d b y L a n g l o i s a n d R i v l i n [7 ] w h o c a r r i e d o u t p e r t u r b a t i o n a n a l y s e s a b o u t s l o w fl o w o f

a N e w t o n i a n f lu i d t h r o u g h a w e d g e a n d a c o n e . T h e y f o u n d t h a t t h e s t re s se s d e v e lo p e d b y t h e

v i s c o e la s t ic p r o p e r t i e s o f th e f l u i d w e r e i n c o m p a t i b l e w i t h r a d i a l f l o w a n d v o r t i c e s w e r e p r e -

d i c t e d i n t h e p e r t u r b a t i o n s o l u t io n s . S t r a u s s [ 8] w a s t h e f i r s t to p r e s e n t t h e s o l u t i o n f o r t h e

s t e a d y , t w o - d i m e n s i o n a l , a n d i n e r t i a l f l o w o f a n i n c o m p r e s s i b l e M a x w e l l f l u id b e t w e e n i n t e r -

s e c t in g p l a n e s b y u s i n g a s e r ie s e x p a n s i o n i n t e r m s o f d e c r e a s i n g p o w e r s o f r . I n h i s s u b s e -

q u e n t s t u d y [ 9 ] , h e c o n s i d e r e d t h e s t a b i l i t y o f t h e s a m e f l o w p r o b l e m . H a n a n d D r e x l e r [1 0]

h a v e s t u d i e d n o n - i n e r t i a l c o n v e r g i n g f l o w , u s i n g a m o d i f i e d s e c o n d - o r d e r f l u i d m o d e l w h i c h

a s s u m e s t h a t a l l t h r e e m a t e r i a l f u n c t i o n s d e p e n d o n t h e s e c o n d i n v a r i a n t o f t h e r a t e o f d e f o r -

m a t i o n . T h e y a l s o m a d e a c o m p a r i s o n t h e t h e o r e t i c a l l y p r e d i c t e d v e l o c i t y p r o f i l e s a n d s t r e s s

d i s t r i b u t i o n s w i t h t h e e x p e r i m e n t a l l y d e t e r m i n e d o n e s . Y o o a n d H a n [ 1 1] c a r r i e d o u t e x p e r i -

m e n t s o n t h e c o n v e r g i n g s l o w fl o w o f a p o l y m e r b e t w e e n i n t e r s e c ti n g p l a n e s a n d t r i e d t o


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118 S. Ban~

e x p l a in t h e d a t a i n t e r ms o f t h e s e c o n d - g r a d e f lu id mo d e l . T h e y f o u n d t h a t t h e t h e o r e t i c a l

a n a ly si s c o r r o b o r a t e s q u a l i t a ti v e ly e x p e r ime n ta l l y d e t e r min e d s t re s s d i s t r i b u t i o n s .

I n t h e c a s e o f mo s t n o n - N e w to n i a n f l u id s a p u r e ly r a d i a l f l o w i s n o t p o s s ib l e i f in e r t ia l

t e r ms a r e t o b e r e t a in e d in t h e e q u a t i o n s o f mo t io n . K a lo n i a n d K a m e l [ 1 2 ] h a v e s h o w n th a t

t h e r e c a n n o t b e a p u r e ly r a d i a l f l o w o f C o s s e r a t f l u id s i n c o n v e r g e n t c h a n n e l s . L a t e r , H u l l

[13] s tud ied the non- ine r t ia l f low of a gene ra l l inea r v iscoe las t ic f lu id in th is geometry . He

s h o w e d t h a t r a d i a l f l o w i s o b t a in e d f o r a w e d g e o f 9 0 ~ a n d n o o th e r s . T h e s imi l a r r es u l ts a r e

valid for the Rivlin-Ericksen f luids (e .g. , [14, 15]) .

M a n s u t t i a n d R a j a g o p a l [ 1 4 ] s t u d i e d t h e n o n - in e r t i a l fl o w o f a s h e a r t h in n in g f l u id

b e tw e e n i n t e r s e c t i n g p l a n e s . T h e y s h o w e d t h a t s h a r p a n d p r o n o u n c e d b o u n d a r y l a y e r s

d e v e lo p a d j a c e n t t o t h e s o li d b o u n d a r i e s , e v e n a t z e ro R e y n o ld s n u m b e r . R e c e n t l y , B h a tn a g a r

e t a l. [15] have ex tende d the ana lys is o f S t rauss [8] to a n O ldroy d-B f lu id which i s cha ra c te r -

i z ed b y a v i s c o s it y a n d tw o ma te r i a l c o n s t a n t s w i th u n i t s o f time . I n t h e i r w o r k , t h e e f fe c t s o f

mu c h h ig h e r v a lu e s o f t h e R e y n o ld s n u m b e r t h a n S t r a u s s' w o r k [ 8] a n d t h e e l a s t ic p a r a m e te r

c~ 0 on the s t r eaml ine pa t te rn s have been d iscussed .

I n t h i s p a p e r , t h e f l o w o f a n O ld r o y d 8 - c o n s t a n t f l u id i n a c o n v e r g e n t c h a n n e l i s e x a m in e d

u s in g s e r i e s e x p a n s io n s p r o p o s e d b y S t r a u s s [ 8 ] f o r s t r e a m f u n c t i o n a n d s t r e s s c o mp o n e n t s .

T h e s i g n if l ci a n t c o n t r i b u t i o n o f a n e w n o n - N e w to n i a n p a r a m e te r , i .e . 7 , w h ic h d o e s n o t

app ea r in the prev ious s tud ies , to the f low pa t te rn s ( so lu t ions) is ca re fu l ly de l inea ted . Our

resu l t s a re s imi la r to those of S t rauss [8], and Bh a tna ga r e t a l. [15] bu t d i f f e r in de ta i l. Also , i t

i s a s expec ted poss ib le to e s tab l i sh r e la t ion s wi th the i r works .

Formulation of the problem

C o n s id e r t h e s t e a d y , tw o - d in e n t i o n a l , i n c o mp r e s s ib l e , l a min a r f l o w o f t h e O ld r o y d 8 - c o n s t a n t

f l u id t h r o u g h a c o n v e r g in g c h a n n e l b o u n d e d b y tw o n o n p a r a l l e l p l a n e s i s s c h e ma t i c al l y il lu s-

t r a ted in F ig . 1 .

T h e C a u c h y s t r es s t e n s o r T f o r t h e O ld r o y d 8 - c o n s t a n t f l u id i s r e l a t e d t o t h e mo t io n i n t h e f o l-

low ing m an ne r (e .g., [16] , [17])

T : - p I + S , (2 .1)

S + A I ~ - + ~ - ( S . A , + A I . S ) + - - ( tr S ) A x + - - [ t r ( S . A x ) ] I

( D A 1 , A z )

: A1 -+-/~2 ~ - t/~ 4A 12 -[- ~- [tr (A12)] I , (2.2)

0~ O~

/

Fig. 1. Definition sketch for a convergent channel


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F low of an O ld royd 8 -cons t an t f lu id i n a conv e rgen t channe l 119

w h e r e p i s t h e p r e s s u r e , I i s t h e i d e n t i t y t e n s o r , S i s t h e e x t r a s t r e s s t e n s o r , a n d

, A 1, A 2, A 3, A 4, A s , A 6, A 7 a r e t h e m a t e r i a l c o n s t a n t s . A 1 i s th e f i r s t R i v l i n - E r i c k s e n t e n s o r a n d

l ) / D t t h e c o n t r a v a r i a n t c o n v e c t e d d e r i v a t i v e d e f i n e d a s f o l l o w s ( s ee G i e s e k u s [1 8]):

A 1 = V v z _ V v T ( 2.. 3)

a n d

~ S O S

- t- v . V S - S - V v - V v T - S , 2 . 4)

D t O t

w h e r e v b e i n g t h e v e l o c i t y v e c t o r , V i s t h e g r a d i e n t o p e r a t o r , t h e s u p e r s c r i p t T d e n o t e s a

t r a n s p o s e o p e r a t i o n .

I t s h o u l d b e n o t e d t h a t t h i s m o d e l i n c l u d e s t h e c l a s s i c a l l i n e a r l y v i s c o u s N a v i e r - S t o k e s

f lu id a s a spec i a l ca se fo r A1 = A 2 = A a = A 4 = A 5 = A 6 = A 7 = 0 .

I n a d d i t i o n t o E q s . ( 2 .1 ) a n d ( 2 .2 ) , t h e f ie l d e q u a t i o n s c o n s i s t o f th e e q u a t i o n s o f m o t i o n

a n d t h e c o n t i n u i t y e q u a t i o n . I n t h e c a s e o f s t e a d y f l o w , t h e f o r m e r e q u a t i o n s i n t h e a b s e n c e o f

b o d y f o r ce s t a k e s th e f o r m

t ) ( v . V v ) = V . T , ( 2 .5 )

w h e r e t) i s t h e ( c o n s t a n t ) d e n s i t y . T h e c o n t i n u i t y e q u a t i o n i s

t r A 1 = 0 . 2 . 6 )

W e s h a l l a s s u m e a v e l o c i t y f ie l d in a p l a n e p o l a r c o o r d i n a t e s y s t e m ( r , O) o f th e f r o m

v ( r , 0 ) = u ( r , 0 ) e r -~- v r , O) eo , (2 . 7 )

w h e r e u a n d v d e n o t e t h e v e l o c i ty c o m p o n e n t s i n t h e d i r e c ti o n s o f r a n d 0 , r e s p e c t iv e l y .

W e s h a l l n o w w r i t e t h e fi e ld e q u a t i o n s i n t e r m s o f a s e t o f d im e n s i o n l e s s v a r i a b l e s a n d , f o r

t h i s p u r p o s e , w e s h a l l c h o o s e A 1, > a n d Q a s c h a r a c t e r i s t i c u n i t s . I f f i s u s e d t o d e n o t e t h e

d i m e n t i o n l e s s f o r m o f a q u a n t i t y f , i t f o l l o w s t h a t ( s e e [ 15 ]) :

r ~ A ~ A - - A

r - - ~ V ~ I ' ~ = u , v = v , p = ] 9, S i j S i j ,

(2.8)

w h e r e Q i s t h e v o l u m e f l u x p e r u n i t d is t a n c e n o r m a l t o t h e p l a n e o f fl o w a n d h a s d i m e n s i o n s

o f L 2 T - 1 . W e s u p p o s e t h a t t h e f l o w i s d r i v e n s t e a d i l y w i t h f l u x Q . T h u s , E q s . ( 2 . 1 ) , ( 2 .2 ) ,

( 2 .5 ) , a n d ( 2 . 6 ) i n n o n d i m e n s i o n a l f o r m b e c o m e :

= - f I + S , 2 . 9 )

2 5 6 [ tr ( S . A 1 ) ] I =

_l_ ~ _ ~__ ~ D e3 (S - ~k @ J~k 9S) -[-

( t r S ) A 1 + ~ -

~ ) A I e 7 [ t r A 1 2 ) ] I ,

i l + s 2 ~ + r 2 + ~ -

R e ( V . V v = V - T ,

tr 2~kl = 0 ,

w h e r e

R e = ~ Q ,

2 . 1 o )

2 . 1 1 )

2 . 1 2 )

A2 Aa A 4 A5 A6 A7

s ~ ~ 1 1 ~ s = ~ s = ~ 1 1 s ~ - - ~ I s = - - . ~ i s = - - , ) ~ i

2 . 1 3 )


4/11

1 2 0 S . B a n ~

w h e r e t h e - A1 s a t i s fi e s a b o v e d i m e n s i o n l e s s e q u a t i o n s o b t a i n e d f r o m E q . ( 2 .3 ) b y r e p l a c i n g A ,

b y A 1 - H e n c e f o r t h f o r co n v e n i e n c e , w e s h a l l d r o p t h e b a r s t h a t a p p e a r o v e r t h e v a r i o u s q u a n -

t i t i e s .

W e n o w t u r n o u r a t t e n t i o n t o t h e e q u a t i o n s o f m o t i o n ( 2 .1 1 ) . A f t e r t h e p r e s s u r e i s e l i m i -

n a t e d b y c r o s s - d i f f e r e n t ia t i n g E q s . ( 2 .1 1 ) , o n e o b t a i n s t h e f o l l o w i n g g o v e r n i n g e q u a t i o n :

q - v - -

r 002 arc O SJ r cOrcqO{ r ( S ~ - Se e) } - O r f o r ( r 2S e ) q r 0 82 ( 2 . 1 4 )

W e s h a l l e x p r e s s s t r e a m f u n c t i o n a n d e x t r a s tr e s s c o m p o n e n t s a s a se r ie s e x p a n s i o n o f t h e

f o l l o w i n g f o r m ( s e e [ 8[ )

> r, 8) = ~ r 2.15)

,Fn

n = 0

S ,.~. (r , 8 = ~ a~(8 )) s~O(r, 8 ) = ~ b ~ ( 8 ) sO O (r ,O ) ~ c~(O)

( 2 . 1 6 )

r n ~ n ~ gn

n = 0 n = 0 n = 0

H e r e , w e t a k e i n t o a c c o u n t t h e f i rs t fi v e t e r m s o f th e s e r i es e x p a n s i o n ( 2 .1 5 ) .

B y d e f i n i n g a s t r e a m f u n c t i o n ~ b (r , 8 ) , s u c h t h a t

1 0 r 0 r

u - - v - - ( 2 . 1 7 )

r 08 Or

t h e c o n t i n u i t y e q u a t i o n i s s a t i sf i e d a u t o m a t i c a l l y . N o w i s t h e t im e t o e x p r e s s th e f u n c t i o n s

a~(8), b.,~(8), a n d c ~ (8 ) i n t e r m s o f t h e r f u n c t i o n s in t h e s e r i e s e x p a n s i o n (2 .1 5 ) . W e d o

t h i s b y s u b s t i t u t i n g E q s . ( 2 . 1 6 ) a n d ( 2 .1 7 ) i n t o E q s . ( 2 .1 0 ) . E q u a t i n g p o w e r s o f r - ~ u p t o

n = 6 , a n d t h u s

a ~ = b ~ = ~ .,~ = 0 , ( n = 0 , 1 ) ( 2 . 1 8 )

a 2 = - 2 r b 2 = r , c 2 = 2 r ( 2 . 1 9 )

a 3 = - 4 r b 3 = r - 3 r , c a = 4 r , ( 2 . 2 0 )

a 4 = - 6 r + 2 f l r ' '2 + r ] ( r ' '2 + 4 r ( 2 . 2 1 )

b 4 = r - - 8 r q - 4 f l r ( 2 . 2 2 )

c 4 = 6 ~ 2 ' + 8 f l 9 o ' 2 + r /( ga o 2 + 4 r ( 2 . 2 3 )

a 5 = - - 8 r q - 4 f l ( r 1 6 2 r 1 6 2 + 2 7 ]( 8 r - 3 r 1 6 2 q - r 1 6 2 , ( 2 . 2 4 )

b5 = r - 1 5 r q - f l ( - 3 r 1 6 2 - k 6 r 1 6 2 q - 5 r 1 6 2 - r 1 6 2 ( 2 .2 5 )

c 5 = 8 r - - S f l ( 4 r 1 6 2 - r 1 6 2 q - 2 r ] ( 8 r 1 6 2 - 3 r 1 6 2 q - r 1 6 2 , ( 2 .2 6 )

6 ~ , , _ r . ~ / . , 2 r , ,

a 6 = - - 1 0 ~ ) 4 ' q - 2 f l ( 2 r '2 - - 2 r 1 6 2 - - V 0 V 2 ~ - D r 0 V 0 - - r 1 6 2 / -} - q - 2 r , r ) + r1 (9 r

- c 1 6 @ 1 I 2 - ~ 2 4 r 1 6 2 - 1 6 r 1 6 2 - - 6 ~ Jd l@ l Zq - r - }- 2 r 1 6 2 / ) - - 2 T( 4 ~ /) 0 , 3 @ r 1 6 2 ( 2 . 2 7 )

b 6 = r - 2 4 r q - f l ( - 1 6 r 1 6 2 q - 6 r 1 6 2 q - 2 4 g ; o I 'r '2 q - 8 r 1 6 2 - - 7 r J r 6 r

- 2 r 1 6 2 - r 1 6 2 - 2 7 ( 4 r '2 + r ' '2 ) ~bo / , ( 2 . 2 8 )


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F low of an O ld ro yd 8 -con s t an t f lu id i n a conv e rgen t channe l 121

e6 = 10r + 2 f l (9 r 2 24r ' a + 14r q - 26 r16 2 - 10 r16 2 - 5 r16 2 + r /(9@ 12 q - 16~ )1 2

4- 2 4 r 1 6 2 - - 1 6 r 1 6 2 - - 6 r 1 6 2 q - r 2 q - 2 r 1 6 2 q - 2 ~ ( 4 r '2 + r 2 ) r ( 2 . 2 9 )

w h e r e f l , ~ , % % a n d t~ a r e d i m e n s i o n l e s s c o n s t a n t s w h i c h o n l y d e p e n d o n t h e m a t e r i a l c o n -

s t a n t s o f t h e v i s c o e l a st i c fl u id c o n s i d e r e d h e r e . T h e y c a n b e r e p r e s e n t e d i n th e f o l l o w i n g w a y :

f ] : c 4 q - c 7 - - ~ 3 - - c6

= ,(~ 3 + C5 + 1) + f l (C5 + ~ 6) ,

% = ( ~ 3 +

C5) (~ + ~)

- -

, ,

= - ( ~ + ~ ) ( , + 9 ) + 3~ - 9 ( ~ + ~ ) .

(2 .30 )

(2 .31 )

( 2 . 3 2 )

( 2 . 3 3 )

(2 .34 )

N e x t , i n s e r t i n g

a ~ ' s , b r J s ,

a n d c ~ ' 8 ( u p t o n = 6 ) f r o m E q s . ( 2 . 1 8 ) - ( 2 . 2 9 ) i n t o E q . ( 2 . 1 6 ) a n d

s u b s t i t u t i n g t h e s e e x p r e s s i o n s f o r

S T , c o ~ ' ~

a n d

S ~ 1 7 6

n t o E q . ( 2 .1 4 ) , a v e r y l o n g a n d t e d i o u s c a l -

c u l a t i o n y i e l d s th e e q u a t i o n s a t v a r i o u s o r d e r s o f r - ~ . H e r e , w e c a r r y o u t o u r a n a l y s i s u p t o

o r d e r r ~ = 4 . E q u a t i n g t h e c o e f f i c i e n t s o f t ~ r - 1 , r - 2 , 7 a , a n d r 4 t o z e r o , w e g e t t h e f o l l o w i n g

d i f f e r e n t i a l e q u a t i o n s , r e s p e c t i v e l y ,

r IV @

4 r + 2 R e r 1 6 2 = 0 , ( 2 .3 5 )

r I v + ( 1 0 + 3 R e r r + ( 2 R e r r + ( 9 + R e { 3 r - r r = 0 , ( 2 . 3 6 )

r I v + ( 2 0 + 4 R e r r + ( 2 R e r r + ( 6 4 + R e { 1 6 r - 2 r r

= - 4 f l ( r ~ v + 4 r r + R e ( r - 2 r 1 6 2 - 3 ~ 1 ' ~ ) , ( 2 . 3 7)

r IV q- ( 3 4 + 5 R e r % + ( 2 R e r r @ ( 2 2 5 + R e { 4 5 r - 3 r r

= f l ( - - 4 5 r 16 2 - - 16r , - - 50 r 1 62 q - 4 r 162 - - 4 r 162 ' - - 5 r162 -1- r 16 2

-4- R e ( - - 1 4 r 1 6 2 q - r 1 6 2 - - 3 r 1 6 2 r - - 4 r q - 2 r 1 6 2 q - r 1 6 2 ,

( 2 . 3 8 )

r ( 5 2 + 6 R e r G ~ + ( 2 R e r r + ( 5 7 6 + R e { 9 6 r - 4 r r

= R e ( - 8 r 1 6 2 - 4 r + 2 r 1 6 2 - 4 2 r + 6 r - 3 r - 5 r 1 6 2 + 3 ~ z %

+ r 1 6 2 + f l ( - 3 8 4 r 1 6 2 - 1 4 4 r 1 6 2 '2 - 1 6 r 1 6 2 + 1 2 r 3 - 1 2 0 r 1 6 2 + 8 r 1 6 2

- 2 4 r 1 6 2 1 6 2 - 2 4 ~ o I V r '2 - 4 r ' - 6 ~ o ' r T + 2 r 1 62 - 3 6 r 1 6 2 - 5 0 r 1 6 2

+ 1 0 r ~ - .5~ r1 62 I v + r 1 6 2 v ) + 2 2 / ( 1 4 4 r 1 6 2 ~2 + 4 r 3 + 6 4 r 1 6 2 + 6 r 1 6 2 m 2

q- 4 r162 ' 2 q - 3 r162 .

T h e a d h e r e n c e b o u n d a r y c o n d i t io n s o f t h e p r o b l e m a s f o ll o w s:

u ( r , i ~ / = 0 , v ( r , ~ ) = 0

w h i c h b y v i r t u e o f E q s . ( 2 . 1 5 ) a n d ( 2 . 1 7 ) i m p l i e s t h a t

r ~) = o , ( ,~ - - o , 1 , 2 , . . . )

r 5 ) = o , (n = 1 , 2 , 3 , . . . ) .

(2 .a 9 )

(2 .40 )

(2 .41 )

(2 .42 /


6/11

122 S. Ban~

I n o r d e r t o so l v e 0 o w e n e ed t o sp e c i fy t w o a d d i t i o n a l b o u n d a r y c o n d i t i o n s . F o r t h is r e a -

so n , t h e v o l u m e t r i c f l o w r a te t h r o u g h t h e c h a n n e l is u se d :

~ d~

~ o

= 0 ~ , + ~ ) - 0 ~ , - ~ ) : - 1 . 2 . 4 3 )

H e r e t h e m i n u s s i g n d e n o t e s t h e f l o w o f a c o n v e r g e n t c h a n n e l . A ssu m i n g t h e f l o w is sy m -

m e t r i c a b o u t 0 = 0 , t h e n

]

0( r , + a ) = :F ~ (2 .44)

and us ing the se r i e s expans ion (2 .15) , we have

1

00 ( = :F ~ . (2 .45)

0 o ( 0 ) i s t h e J e f f r e y - H a m e l so l u t i o n f o r t h e c o r r e sp o n d i n g f l o w o f a N e w t o n i a n f l u i d w h i c h

can b e expres sed exac t ly in t e rm s of e l l i p t ica l func t ion s ( see [19] ).

E q . ( 2. 36 ) i s a li n e a r h o m o g e n e o u s o r d i n a r y d i f f e r e n t i a l e q u a t i o n su b j e c t to b o u n d a r y c o n -

d i t i ons (2 .41) a nd (2 .42) an d i t s so lu t ions i s

0 1 0 ) _= o . 2 . 4 6 )

On subs t i t u t ing E q . (2 .46) in to Eqs . (2 .37) an d (2 .38), t he t e rm h avin g Re fac tor i n Eq . (2 .37)

v a n i sh e s a n d t h e t e r m s o n t h e r i g h t - h a n d s i de o f E q . ( 2 .3 8 ) d o . W e t h u s f i n d t h a t

~ 3 0 ) - o . 2 . 4 7 )

By v i r tue of Eqs . (2 .46) an d (2 .47), t he s t re am fun c t ion red uces to

0 o ) - - 0 0 0 ) + + + . . 2 . 4 s )

a n d t h e e x t r a s t r e s s c o m p o n e n t s a r e :

s, T , ol = . . . , 2 . 4 9 1

b 2 o ) b 4 o ) + v o o ) 2 . 5 0 )

s T ~ ~ , o ) = ~ - + T 4 - ~ 6 .. . ,

c4(0) + c6(0) (2.51)

s~ 17 6 o) = c~o_~) + ~4 ~6 . . . .

W e n o w p a y sp e c i a l a t t e n t i o n t o t h e c a se i n w h i c h i n e r t i a l e f f e c t s a r e n e g l e c t e d . F o r su c h

f lows, the r igh t -ha nd s ide of Eq . (2 .37) i s equa l t o ze ro due to Eq . (2 .35). T hen i t bec om es a

l in e a r h o m o g e n e o u s o r d i n a r y d i f f e re n t ia l e q u a t i o n a n d g i ve s a z e ro s o l u t i o n u n d e r t h e b o u n d -

a ry con di t ion s (2 .41) and (2 .42), i. e. 02(0) -= 0 . Thu s , t he re su l t i ng equ a t ion s a re now in the

f o l l o w i n g f o r m :

Oo

T

+ 400 " = O, (2.52)

04

T

+ 5204" + 57604 = 2f l ( - -7200"0 0 ' 2 + 600 f'3 - - 120 0 '00 "06 " - - 1200IV00 2)

- -27 (14400 "00 '2 406 "a + 64 00 '0~ 00 " + 6 0 0 0 0 2 + 400IV00'2 + 300IV00"2) 9 (2.53)


7/11

Flow of an Oldroyd 8-constant fluid in a convergent channel 123

Eq. (2 .52) wh ich s a t i sf i e s bo un da ry co ndi t ion s (2 .41) and (2 .45) i s so lved by the fo l lowing

s i m p l e a n a l y t i c a l e x p r e s s i o n

20 cos 2c~ - sin 20

~0 (0) = 2 sin 2c~ - 4c~ co s 2c~ ' (2.54 )

Us in g the so lu t ion g iven abo ve for ~0(0) in Eq . (2 .53), i t i s s impl i f i ed to y i e ld

~ 4 I V - I - 52~4' t -F 576~4 = 1 6 ( 3 /3 + 87 + (16-y - 3 /3) co s4 a '~

- - / 2 0 2 .55 )

Th e gene ra l so lu t ion o f Eq . (2 .55) s a t i s fy ing the bo un da ry con di t ion s (2 .41) and (2 .42) i s

~4 (0) = ( 3/3 + 82 /+ (167 - 3/3) cos 4c~ ~ (cos 2c~ - cos 20) 2 s in 20. (2.56)

\ 6 ( 3 + 2 c o s 4 c ~ )( s in 2 5 - 2 5 c o s 2 a ) 3 J

Resul ts and d iscuss ion

W e s o l v e d t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s a s a s y s t e m o f f ir s t - o r d e r d i f f e r e n t ia l e q u a t i o n s

f o r R e ~ 0, u s in g t h e s h o o t i n g m e t h o d . F o r g i v e n v a lu e s o f p a r a m e t e r s , t h e c o n d i t i o n s

~ I ' ( - a ) a n d ~ 1 ~ ( - c ~ ) a r e r o u g h l y e st i m a t e d a n d d i ff e re n t ia l e q u a t i o n s a r e p r o c e s s e d b y

u s i n g th e f o u r t h - o r d e r R u n g e - K u t t a p r o c e d u r e . T h e m a t h e m a t i c a l p r o b l e m i s t o f in d t h e c o r-

r e c t v a l u e s o f ~ (-c~) a n d ~ ' ( - a ) w h i ch yi el d t he k n o w n v a lu e s o f ~ ( + a ) a n d ~ ' ( + c ~ )

a t t e r m i n a l p o i n t . S i n ce f o r R e = 0 t h e a n a l y t i c s o l u t i o n p r o v i d e s e x a c t i n i ti a l v a l u e s f o r

~ ( - a ) a n d % ' ( - a ) t h e n s u cc e ss iv e n u m e r i c a l s o l u ti o n c a n b e g e n e r a t e d a s R e i s i n c re a se d .

T h e s y s t e m a t i c w a y u s e d h e r e f o r f i n d i n g v a l u e s o f th e m i s s in g i n i ti a l c o n d i t i o n s i s e q u i v a l e n t

t o a m o d i f i e d N e w t o n ' s m e t h o d , f o r f i n d in g t h e r o o t s o f e q u a t i o n s i n se v e r a l v a r i a b l e s. T h e

a c c u r a c y o f m i s s i n g in i ti a l c o n d i t i o n s a t 0 - - c ~ w h i c h y i e ld t h e k n o w n v a l u e s a t t h e te r m i n a l

po int (0 = +c~) is 10 5 a t leas t .

T h e p r e d i c t i o n s b a s e d o n t h e f o r e g o i n g a n a l y s i s a r e d i s p l a y e d g r a p h i c a l l y i n F i g s . 2 - 6 .

T h e c u r v e s i n th e s e f ig u r e s a r e o b t a i n e d f o r t h e c o n s t a n t s t r e a m f u n c t i o n o f t h e f o r m :

~2(0) ~4(0) (3.1)

~ r ,0 ) = ~ 0 0 ) + W ~ - ~ r~

T h e s t r e a m l i n e p a t t e r n s a r e i n d i c a t iv e o f th e n a t u r e o f th e s e c o n d a r y f l o w n e a r t h e c o r n e r ,

u n l i k e t h e c a s e o f N e w t o n i a n f l u id .

T h e s a m e p r o b l e m a s t h a t i n v e s t i g a t e d i n t h e p r e s e n t p a p e r h a s b e e n s o l v e d p r e v i o u s l y

b y S t r a u s s [8 ] f o r t h e M a x w e l l f l u id , a n d B h a t n a g a r e t al . [1 5 ] f o r t h e O l d r o y d - B f l u i d . I n

t h e s p e c i a l c a se s c o r r e s p o n d i n g t o M a x w e l l f l u i d (e 2 = 7 = 0 ) a n d O l d r o y d - B f l u id

(5 2 r 0 , 7 = 0 ) , th e r e is a s e x p e c t e d a n o v e r l a p b e t w e e n t h e i r g o v e r n i n g e q u a t i o n s a n d o u r s

( s ee E q s. ( 2 . 3 5 ) - ( 2 . 3 9 ) ) . N o t e t h a t s o m e t e rm s , w h i c h a r e m e r e l y r e l e v a n t t o ~ 1 , o n t h e

r i g h t - h a n d s id e s o f t h e E q s . ( 3 .6 ) a n d ( 3 .1 3 ) i n [ 15 ] w e r e w r i t t e n w r o n g , b u t t h e s e d o n o t

c a u s e a n y c h a n g e i n t h e s t r e a m l i n e p a t t e r n s ( s o l u t i o n s ) . T h e r e s u l t s p r e s e n t e d h e r e a r e i n

c o m p l e t e a g r e e m e n t w i t h t h o s e g i v e n b y t h e p r e s e n t a u t o r s [ 8 ], [ 1 5 ], f o r t h e s p e c if i c v a l u e s o f

t h e R e y n o l d s n u m b e r R e a n d t h e e l a s t i c p a r a m e t e r 5 2 f o r w h i c h t h e y h a v e g i v e n r e s u l t s . T h i s

g i v e s u s c o n f i d e n c e r e g a r d i n g t h e n u m e r i c a l w o r k .

F i g u r e 2 a a n d b d e p i c t s t h e s t re a m l i n e p a t t e r n s f o r t h e n o n - i n e r t i a l f lo w s o f th e M a x w e l l

a n d O l d r o y d - B f l u id , r e s p e c ti v e l y . W e o b s e r v e f r o m t h is f i g u r e t h a t t h e e f f e c t o f 52 is to d i m i n i s h

the s i ze of the c i rcu la t ing ce ll s. F o r the O ldro yd -8 co ns ta n t f lu id , t he e f fec t o f 7 i s i ns ign i f i can t

o n t h e s t r e a m l i n e p a t t e r n s , c o m p a r e d w i t h O l d r o y d - B f l u id , s o it i s n o t p r e s e n t e d h e r e .


8/11

124 S . Ba n~

9-~ !i . . . . . . !j /

0.8 i 0.8

0.6 0.6

Y Y

0.4 t .... 0.4

.2 0.2 ......

-0 .6 -0 .3 0 0 .3 0 .6 -0 .6 -0 .3 0 0 .3 0 .6

a b

Fi g . 2 . Co n t o u r p l o t s o f t he s t r e a ml i ne r~) f o r Re = 0 : a = 60 o a nd " / = 0 ; a c2 = 0 , b c2 = 0 . 75

0.8

0.6

0.4

0.2

9 .

0.8

Y 0 '6 1

0.4

0.2

-0 .6 -0 .3 0 0 .3 0 .6 -0 .6 -0 .3 0 0 .3 0 .6

x

b

0 8 0 8

0.6 0.6

Y Y

0.4 0.4

0.2 0.2

0 . 6 -0 .3 0 0 .3 0 .6 -0 .6 -0 .3 0 0 .3

c d

Fig. 3 . C on to ur p lo ts of the s t rea ml i ne ~b for e2 = ")' = 0 ;

a = 3 0 0 , c R e = 1 0 0 , a = 3 0 0 , t i R e = 5 0 , c ~ = 6 0 o

0.6

a R e = 1 1 . 4 2 , c ~ = 3 0 ~ b R e = 1 9 . 6 7 ,

I n t h e c a s e o f i n e r t i a l f l o w , F i g s. 3 t o 5 p r o v i d e t h e s t r e a m l i n e p a t t e r n s i n t h e f lo w d o m a i n

0 < r < 1 f o r M a x w e l l , O l d r o y d - B , a n d O l d r o y d - 8 c o n s t a n t fl u i d , r e s p e c t iv e l y . O u r m a i n p u r -

p o s e i s t o d e l i n e a t e t h e e ff e c t o f t h e p a r a m e t e r 7 , w h i c h d o e s n o t a p p e a r i n t h e p r e v i o u s s t u -

d i e s, o n t h e f lo w p a t t e r n s . I n F i g . 5 , w e h a v e p l o t t e d t h e s t r e a m l i n e s r e l e v a n t t o t h e O l d r o y d - 8

c o n s t a n t f l u i d w i t h t h e i n t e n t i o n o f i n v e s t ig a t i n g th e c o n t r i b u t i o n o f th i s n e w p a r a m e t e r t o t h e

f l o w f i el d . F r o m F i g . 5 i t i s e v i d e n t t h a t t h e p a r a m e t e r 7 d o e s a f f e c t t h e s t r e a m l i n e s o f th e s e c -

o n d a r y f l o w n e a r t h e c o r n e r i n a s i g n if i c a n t w a y .


9/11

Fl ow o f a n O l d r oy d 8 - c ons t a n t f l u id i n a c onve r ge n t c ha nne l 125

0.8 i '. 0.8

0.6 ' . .. 0.6

y Y

0.4 ' ' 0.4

0.2

0.2

-0 .6 -0 .3 0 0 .3 0 .6 -0 .6 -0 .3 0 0 .3 0 .6

x

b

. . . . \

t i t l l

, . .

0.6 0.6

y i

.4 0.4

0.2 0.2

-0 .6 -0 .3 0 0.3 0 .6 -0 .6 -0 .3 0 0.3 0 .6

x

e d

F i g. 4 . C o n t o u r o f t h e s tr ea m l in e ~ fo r ~ , = 0 ; a R e = 1 1 . 4 2 , a = 3 0 0 , c 2 = 0 . 5 , b R e = 1 9 . 6 7 ,

a = 3 0 ~ c 2 = 0 . 7 5 , c R e = 1 0 0 , a = 3 0 ~ e 2 = 0 . 5 , fl R e = 5 0 , c ~ = 6 0 ~ e 2 = 0 . 7 5

A s t h e p a r a m e t e r 7 i s i n c r e a s e d f r o m 0 t o 0 . 01 , w h i l e k e e p i n g R e f i x e d a t 1 1 .4 2 a n d

c 2 = 0 . 5 , f lo w h a s f o u r - c e l l s t r u c t u r e l i k e O l d r o y d - B f l u i d b u t t h e s i z e o f c i r c u l a t i n g c e l ls

a d j a c e n t t o t h e s t a t i o n a r y p l a t e s b e i n g b i g g e r a n d t h e o t h e r s s m a l l e r ( se e F i g s. 4 a a n d 5 a ). I n

F i g . 5 b , w e p o r t r a y t h e f l o w f i e l d w h e n R e - - 1 9 . 6 7 , c~ = 3 0 0 , c 2 = 0 . 7 5 , a n d 7 = 0 . 0 1 . I n

t h is c a s e , i t i s c o n c l u d e d t h a t t h e s h a p e o f c i r c u l a t in g c e ll s i s m a r k e d l y d i f f e re n t f r o m t h o s e

i n F i g s . 3 b a n d 4 b . O n c o m p a r i n g F i g . 5 c w i t h F i g s . 3 c a n d 4 c ( o r F ig . 5 d w i t h F i g s . 3 d a n d

4 d ) , w e a r r iv e a t t h e c o n c l u s i o n t h a t t h e s t r e a m l i n e s o f th e s e c o n d a r y f lo w n e a r t h e c o r n e r

c o r r e s p o n d i n g t o 7 r 0 c h a n g e s f r o m t w o - c e l l s t r u c t u r e i n t o f o u r - c e l l s t r u c t u r e .

F i n a l l y w e s h a l l d i s c u s s r e li a b i l it y o f t h e s o l u t i o n s n e a r t h e a p e x o f t h e w e d g e . T h e d i f fe r -

e n t i a l c o n t i t u t i v e e q u a t i o n s u s e d i n t h is p a p e r a r e n o t l i m i t e d t o s m a l l , s l o w l y c h a n g i n g d e f o r -

m a t i o n r a t e s a s is t h e R i v l i n - E r i c k s e n f l u i d s , a s u b c l a s s o f th e f l u i d s o f t h e d i f f e r e n t i a l t y p e .

N o t e t h a t t h e f lo w i n a c o n v e r g i n g c h a n n e l i s c o n s i d e r e d t o b e a r a p i d m o t i o n a n d t h e g r a d i -

e n t s o f v e l o c it y b e c o m e v e r y l a r g e n e a r t h e c o r n e r . H o w e v e r , S t r a u s s ' s e ri e s e x p a n s i o n i s n o t

a p p r o p r i a t e t o a p e r t u r b a t i o n f o r r < 1 . T h i s is w h y t h e s o l u t i o n s b a s e d u p o n h i s a p p r o x i m a -

t i o n c a n n o t b e r e l i a b l e w h e n r < 1 . O f c o u r s e , th i s a l s o d e p e n d s o n t h e n a t u r e o f th e f u n c t i o n s

~ b~ (0 ), t h a t i s , i f ~ ( 0 ) a r e n o t i d e n t i c a l l y z e r o f o r l a r g e n , t h e s o l u t i o n c a n n o t b e t r u s t e d a s

b e i n g m e a n i n g f u l f o r r < 1 . I n a d d i t i o n t o , a s a r e s u l t o f s i n g u l a r i t y a t r = 0 , t h e s o l u t i o n s a r e

e x p e c t e d t o b e v a l i d o n l y i n c o n v e r g i n g ( o r d i v e r g i n g ) n o z z le s r a t h e r t h a n i n b e t w e e n t w o

i n t e r s e c t i n g p l a n e s h a v i n g a s o u r c e ( o r s in k ) a t r = 0 . T h e s e a r e a s e x p e c t e d ; h o w e v e r i t s h o u l d

b e m e n t i o n e d t h a t t h e e x p a n s i o n s g i v e n in E q s . ( 2 .1 5 ) a n d ( 2 .1 6 ) r e su l t i n a s y m p t o t i c d i v e r g i n g


10/11

12 6 S. Bar1@

0.8

0.6

0.4

0.2

,

0.8

0.6

0.4

0.2

-0.6 -0.3 0 0.3 0.6 -0.6 -0.3 0 0.3 0.6

X X

b

~ ~ ~ ~ .

. '

,

0 8 0 8 i

0.6 0.6

Y Y

0.4 0.4

0.2 0.2

-0.6 -0.3 0 0.3 0.6 -0.6 -0.3 0 0.3 0.6

x X

e d

Fig. 5. Co nto ur plo ts of the streamline ~b a Re = 11.42, c~ = 30 o , c2 = 0 .5, q, = 0.01, b Re = 19 .67,

c ~ = 3 0 ~ c 2 = 0 . 7 5 , 7 = 0 . 0 1 , c R e 1 00 , c e = 3 0 ~ c 2 = 0 . 5 , ~ = 0 . 0 1 8 , d R e = 5 0 , c ~ = 6 0 ~

e2 = 0.75, ~ = 0.1

1 .5 .y .

1.3

Y

1.1

0.9

-0. 8 -0.4 0.4 018

a

Fig . 6 . Co n to u r p lo t s o f t h e

b Re = 19 .67

0911131 5 / ~ / / / /l / /

-0.8 -0.4 0 0.4 0.8

X

b

stream line ga w he n c~ = 30~ e2 7 = 0 for 0 .8 < r < 1 .6 ; a Re = 11 .42 ,

se r ie s see [13] ) . I t i s ev iden t tha t fo r so lu t ions re l iab le nea r the corn e r , se r ies expans ion s in

te rms of a scending p owe rs o f r a re r equi red . But , a s sho wn in [20], the re i s a s ign i f ican t d i f f i-

c u l t y in d o in g t h i s d u e t o t h e e f f e c t o f t h e u p s t r e a m h i s to r y o f d e f o r m a t io n .

On the o the r h and , for r > 1 , s ince the e f fec t o f successive te rms a re less s ign i f ican t , the

so lu t io n i s p ro bab ly qui te r e l iab le a s r~ inc reases. This g ives us adeq ua te i n fo rm at io n in the

f low dom ain r < 1 f rom the tendenc ies sugges ted by the re su l t s a t r _> 1 . Fo r ins tance , the

s t reamlines d epic ted in F ig . 6a ind ica te the presen ce of two-ce l l s in F ig . 3a , whereas F ig . 6b

sugges ts a four -ce l l s t ruc ture in F ig . 3b . O f course , the s t r eaml ines o f the seco nda ry f low i llus-

t r a ted in F igs . 2 to 5 for r < 1 m ay no t have the prec ise s t ruc ture .


11/11

Fl ow o f a n O l d r o yd 8 - c ons t a n t f l u id i n a c onve r ge n t c ha nne l 127

e f e r en c e s

[1] Jef f rey , G. B . : The tw o-dim ensio nal s teady mot io n of a v i scous f lu id . Phi l . Mag . 29, 455 -46 5 1915).

[2 ] H a m e l , G . : Sp i r a l f 6 r m i ge B e w e gunge n z /i he r F l fi s s igke i t e n . J a h r . D e n t s c h . M a t h . V e r . 25 , 34 - 60

1917).

[ 3] R os e nhe a d , L . : The s t e a dy t w o- d i m e ns i ona l r a d i a l f low o f a v i s c ous f lu i d be t w e e n t w o i nc l i ne d w a l l s.

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Author s addr ess:

Se r da r B a r l ~ , Fa c u l t y o f Me c ha n i c a l Eng i ne e r i ng , I s t a nbu l Te c hn i c a l U n i ve r s i t y ,

Gf imf i~suyu 80191, I s tanbul , Tu rkey

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