Effect of Wall Slip on the Uniformity of Flow

7/27/2019 Effect of Wall Slip on the Uniformity of Flow

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40 J. Polym. Res., Vol. 2, No. l, 39-45, January 1995

proaches, such as the two-d imens ional flow analysis[13, 14] and the three- dime nsio nal flow analysis[15-18], can result in improved predictions, particu-larly for irregular regions or sharp corners that existinside the dies, but they invari ably require more com-plex preprocessing and greater computational expan-sion.

In the published literature, the flow velocity atthe solid boundary is assumed to be zero, i.e., thenonslip condition. But the assumption of nonslip atthe solid boundary is generally invalid for rheologi-cally complex fluids, such as polyme r melts and con-centrated polymer solutions, when certain criticalconditio ns are exceeded in flow-through conduits [19-24]. For example in the experime ntal results ofRamamurthy on linear low-density polyethylene(LDPE), as the wall shear stress exceeds a criticalvalue of approximately 0.1-0.14 MPa, wall slip occurs[20]. Although the mechanism involved in slip isunclear, according to the experimental data ofRamamurthy and of Jiang et al. [21 ], the slip velocityis described as a function of the wall shear stress.Yoshimura demonstrated this slip model with thedynamic oscillatory measurements and presented anew technique for Couette and parallel disk viscom-eters to identify the relationship between the slipvelocity and the wall shear stress [22, 23]. Flowanalysis for polymer melts involvin gwall slip is scarce;Thien [24] has studied the i nfluen ce of wall slip onextrudate swell, in which the functi on of slip velocitywas obtained through a curve fit from extensive dataof reference 20. His results showed that wall slipdecreased both the die swell ratio and the critricalWeisse nberg number. The thermal entrance probleminvo lvin g wall slip was studied by Sbih and coworkers[25, 26], and they obtained an analytic solution oftemperature by exp anding the temperature in a seriesof solutions.

Here we analyze the flow of power-law fluidsinside T-dies and coat-hange r dies and investig ate theinflu ence of wall slip on the flow uniformity. Themodel is based on the one-d imen sion al flow, and theslip velocity at the wall is assumed to be proportionalto the wall shear stress. The proportional coefficientis define d as slip coef fici ent 13. The cr itica l value ofwall shear stress described in reference 20 is regardedas zero here for simplification..

M a t h e m a t ic M o d e lFigures l(a) and l(b) show the geometry of

T-die and coat-hanger die respectively. In the T-die,the manifold is a cylindrical channel with constantcircula r cross-sectio n. The die land is a slit regionwhich has constant gap. In the coat-hanger die, the

manifold radius decreases along the width of the die.For power-law fluids the optimum variation ofmanifold radius under nonsl ip condition was derived{9]:

R ( y )=((2h)2"+1 r w ( 3 n + l ) ] n ) 3 n I l y 3 n n l

sin~ L2/t (2n + 1) (1 ) (I)I n s e r t i n g y = 0 i n t o t h e E q u a t i o n ( I ) g i v e s R 0; t h e r e -f o r e

R ( y ) = R o ( 1 _ Y )3.--"+, (2)

where R 0 is the m anifol d radius at the cent er of the die(y = 0), and n is flow index.Equati on (I) is the necessary and sufficient conditionto ensure uniform flow distribution across the diewidth in the linearly tapered coat-hanger die.

By symmetry, we limit the flow analysis to halfthe die. The flow problem can be stated as follows: asteady laminar flow of isothermal, incompressiblepolym er melts flow through the die, with the entranceand end effects neglected. The rheological propertiesof polymer melts are described by power-law model inwhich the viscosity is expressed as

n = K l ~ , r -~ ( 3)The partic ular boundary condi tion in this paper is thata slip velocity, Vs, exists at the wall of the die, itsmagnitude is assumed to be proportional to the wallshear stress, "c0~, i.e .

v ~ = [ 3 ~ ( 4 )Althrough this simple slip model may be too impre-cise to describe the real mechanis m of a slip model, itenables assessment of some influences of the wallslip condition on the flow behavior inside the die.The wall shear stress in the circular mani fold is writ-ten:

R d p ( 5 )"kv = 2 d7in which ~ is the axial direction of the manifold. Inthe slit the wall shear stress is

~w = - h d~ (6)Since the assumption of one-di mensiona l flow is ap-plied. Equation (6) can be integrate d


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h p% = T ( 7 )i n w h i c h p i s t h e p r e s s u r e i n t h e m a n i f o l d ( i . e . , e n -t r a n c e o f th e s l i t re g i o n ) a n d t h e p r e s s u r e a t t h e e x i to f t h e d i e i s a t m o s p h e r i c , 1 i s th e l e n g t h o f t h e s l i t. A st h e w a l l s l i p c o n d i t i o n i s c o n s i d e r e d , t h e p r e s s u r ed r o p / f l o w r a t e r e l a t i o n s h i p i n t h e m a n i f o l d i s d e -r i v e d :

n R 3 + m d p mQ = ~ R2 Vs + ~ [ ~ K ( - d ~ - ) ]i n w h i c h m = 1 . T h e w a l l s li p c o n d i t i o n u s e d t od e r i v e E q u a t i o n ( 8 ) i s

( 8 )

r = R , v = V ~ ( 9 )I n t h e s l i t r e g i o n t h e f l o w i s r e s t r ic t e d i n t h e x -

d i r e c t i o n , t h e f l o w r a t e p e r u n i t w i d t h q i s d e r i v e d2h 2+ m ) ]mq = 2 h V ~ + ~ t ( 1 ) ( p ( 1 0 )

T h e w a l l s l ip c o n d i t i o n u s e d t o d e r i v e E q u a t i o n ( 1 0 )i s

z = h , a g = V s ( 11 )

S i n c e t h e l o s s o f Q a l o n g t h e m a n i f o l d i s e q u a l t ot h e f l o w r a t e e n t e r i n g t h e s l i t, w e h a v ed Qd y - q ( 12 )

S u b s t i t u t i n g E q u a t i o n s ( 8 ) - ( 1 0 ) i n t o E q u a t i o n ( 1 2 ) ,a n d t o g e t h e r w i t h E q u a t i o n s ( 4 ) - ( 7 ) , w e o b t a i n

n13R3cos ~ m nR 3 + m cosmc~ d p ) m - 1 9[ 2 ( 2 K ) m ( 3 + m ) ( - d - y ] d ~ 2

3 x R 2 c o s tx d R d p n R 2 + m c o s m o ~ d R2 ( ~ - ) d y ( 2 K ) m ( ~ - )dp )m 2 h 2 + m p m ( - ~ = 2 h ~ P + ( 2 7 ~ K m ( T ) ( 1 3 )

T h e b o u n d a r y c o n d i t i o n s a r e

y = 0 , P = P0 /y _ - w : 0 J ( 1 4 )T o s o l v e t h e e q u a ti o n s w e i n t r o d u c e th e d i m e n s i o n -

l e s s v a r i a b l e s

18" = 13R o ( ~ ) mp = PP0 (15)

h e r e P o a n d R o a r e t h e p r e s s u r e a n d t h e r a d i u s a t y = 0r e s p e c t i v e l y . I n a d d i t i o n , t h e d i m e n s i o n a l p a r a m e t e r so f t h e e x t r u s i o n d i e a n d a x i a l c o o r d i n a t e s a r e a l s on o r m a l i z e d w i t h R o , i . e .

w Y = ~ o ow = R T oR * R _ z ( 1 6 )

T h e n w e o b t a i n a f t e r s u b s t i t u t i n g E q s . ( 1 5 ) a n d ( 1 6 )i n t o E q u a t i o n ( 1 3 )

[ n13*R*3cs ~ /rmR 3 + mcosm0~ d P ) m - 1 ] d2p2 + 2 ~ 7 3 - + m ) ( - ~ " d Y =

3nR*2~*cos cz dR* dP 7tR 2 + m c o s m ( x+ ~ ( ~ - ) ~ - 2 m ( dR* ) ( _ d P )m = 2H218*p 2H 2 + m p m" d Y ~ L + ~ ( E ) (1 7 )

a n d t h e b o u n d a r y c o n d i t i o n s a r e

Y = 0 , P = I }Y = W d P 0a y = ( 1 8 )

d R * e x p r e s s e s t h e v a r i a t i o n o fh e d e r i v a t i v e t er m ~ -m a n i f o l d r a d i u s a l o n g t h e w i d t h d i r e c t i o n . F o r T - d i e ,t h e m a g n i t u d e o f d R * i s z e r o , w h e r e a s f o r c o a t -h a n g e r d i e , i t b e c o m e s

2 + i1 1dR* - 1_d_y _ W ( ~ + m ) ( I _ Y ) 3 + m ( 1 9 )

E q u a t i o n s ( 1 7 ) a n d ( 1 8) f o r m t h e m a t h e m a t i c a l m o d e lt h a t d e s c r i b e s t h e p r e s s u r e d i s t r i b u t i o n i n t h e m a n i -f o l d u n d e r t h e c o n s i d e r a t i o n o f w a l l s l i p e f f e c t . T h i sm o d e l i s a n o n l i n e a r 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 f o ra g i v e n f lo w i n d e x n a n d g e o m e t r i c p a r a m e t e r s . W ec a n s o l v e t h i s m o d e l b y t h e s t a n d a r d f i n i t e - d i f f e r e n t


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m e t h o d c o m b i n e d w i t h t h e N e w t o n - R a p h s o n m e t h o d .O n c e P i s c a l c u l a t e d , t h e f l o w r a t e d i s t r i b u t i o n q ( y )c a n b e o b t a i n e d f r o m E q u a t i o n ( 1 0 ) . I n o r d e r t od e s c r i b e t h e f l o w r a t e d i s t r i b u t i o n q u a n t i t a t i v e l y , w ea l s o c a l c u l a t e t h e f l o w u n i f o r m i t y , U I , d e f i n e d i n t h ef o l l o w i n g e q u a t i o n

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h e r e q i r e p r e s e n t s t h e f l o w r a t e a t t h e i th n o d e , N i s t h et o t a l n u m b e r o f n o d e s u s e d f o r c a l c u l a t i o n , a n d q ave i st h e a v e r a g e v a l u e o f q f o r t h e w h o l e w i d t h . W h e n t h ef l o w r a t e d i s t r i b u t i o n i s p e r f e c t l y u n i f o r m a c r o s s t h ed i e w i d t h , t h e v a l u e o f U I i s e q u a l t o o n e , a n d t h ed i s t r i b u t i o n b e c o m e s w o r s e a s t h e v a l u e o f U I d e -c r e a s e s .

D i s c u s s i o nW e f i r st i n v e s t i g a t e t h e f lo w o f p o w e r - l a w f l u id s

i n a T - d i e w i t h H = 0 . 0 5 , W = 5 0 , a n d t h e l e n g t h o f t h ed i e l a n d is 2 5. F i g u r e 2 s h o w s t h e f l o w r a t e d i s tr i b u -t i o n a t t h e e x i t o f t h e T - d i e . T h e f l o w i n d e x i s 0. 5 . T h ef l o w r a t e d i s t r i b u t i o n f o r t h e c a s e o f n o n s l i p ( [3 * = 0 )i s m o r e e v e n t h a n t h a t f o r t h e c a s e o f s l i p ( [3* ~: 0 ) . A sw a l l s l i p o c c u r s , t h e fl o w d i s t r i b u t i o n w i l l b e u n e v e n .T h e f l o w r a t e n e a r t h e d i e c e n t e r ( y = 0 ) h i g h e r t h a nt h a t d ie e n d s . A s [3* b e c o m e s l a r g e r , t h e f l o w d i s t r i -b u t i o n w i ll b e m o r e u n e v e n . W h e r e a s t h e i n f l u e n c e o f

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w a l l s l i p a p p e a r s t o b e s l o w e r a s 13" i n c r e a s e s t o al a r g e r v a l u e . F i g u r e 3 p l o ts t h e f l o w u n i f o r m i t y U I asa f u n c t i o n o f f l o w i n d e x f o r v a r i o u s v a l u e s o f [ 3* .F o r t h e c a s e o f n o n s l i p , t h e v a l u e o f U I d e c r e a s e s w i t hd e c r e a s i n g v a l u e o f n , t h i s im p l i e s t h a t th e s h e a rt h i n n i n g p r o p e r t y o f p o l y m e r m e l t s i n d u c e s a n o n u n i -f o r m e f f l u x o b v i o u s l y . F o r th e c a s e o f s l ip , th e r es h o w s a n i n t e r e s t i n g a n d s i g n i f i c i a n t r e s u l t t h a t w h i l en b e c o m e s s m a l l e r , t h e m a g n i t u d e o f U I f o r a l l c a s e s([3 * = 0 . 0 0 1 , 0 . 0 1 , 0 . 1 ) w i l l a p p r o a c h t o a c o n s t a n tw i t h t h e v a l u e n e a r 0 .9 5 . I f w e e n l a r g e t h e d i e w i d t hw h i l e t h e g a p a n d l e n g t h o f t h e d i e l a n d a r e f i x e d t o t h ev a l u e o f 0 . 0 5 a n d 5 0 r e s p e c t i v e l y , t h e f lo w d i s t r i b u -t i o n at t h e e x i t b e c o m e s m o r e u n e v e n , a s e x p e c t e d .T h e m a g n i t u d e o f U I as a f u n c t i o n o f d i e w i d t h a r ep l o t t e d i n F i g u r e 4 . I t c a n b e s e e n t h a t a s t h e d ie w i d t hi n c r e a s e s , t h e U I f o r a l l c a s e s d e c r e a s e s . T h i s i m p l i e st h a t l o n g e r d i e l a n d i s n e e d e d t o e n s u r e t h e u n i f o r mf l o w d i s t r i b u t i o n f o r t h e w i d e r T - d i e . F i g u r e 5 s h o w st h e r e q u i r e d l e n g t h o f t h e d ie l a n d t h a t c a n r e g u l a t e t h ed e v i a t i o n o f f lo w d i s t r i b u t i o n w i t h i n _+5 % a c r o s s d i ew i d t h . S i n c e t h e e ff e c t o f w a l l s l ip m a k e a n o n u n i f o r md i s t r i b u t i o n o f fl o w r a t e , t h e r e q u i r e d l e n g t h o f t h e d i el a n d f o r t h e c a s e o f s l i p i s l o n g e r t h a n t h a t f o r t h e c a s eo f n o n s l i p . A s t h e v a l u e o f 13" i n c r e a s e s , t h e r e q u i r e dl e n g t h o f t h e d i e l a n d i n c r e a s e s t o o .

S e c o n d l y , W e i n v e s t i g a t e t h e f l o w o f p o w e r - l a wf l u id s i n a c o a t - h a n g e r d i e w i t h t h e s a m e d i m e n s i o n o fg a p a n d w i d t h ( i . e . , H = 0 . 0 5 , W = 5 0 ) , a n d t h e l e n g t ho f th e d i e l a n d i s z e r o . A c c o r d i n g t o E q u a t i o n ( 1 ), t h eo p t i m u m c o a t - h a n g e r a n g l e , c t, t h a t e n s u r e s t h e u n i -f o r m f l o w d i s t r i b u t i o n a t t h e e x it , i s a f u n c t i o n o f f l o wi n d e x . A s n = 0 . 5 , t h e o p t i m u m v a l u e o f cx i s 1 . 8 .F i g u r e 6 s h o w s t h e f l o w r a t e d i s t r i b u t i o n f o r p o w e r -l a w f l u i d s w i t h n = 0 . 5 a t th e e x i t o f th e c o a t - h a n g e r


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0.70 I I I I I I I I 1 I [10 30 5 0 70 90 110 130Dimensionless die width 2W

Figure 4. Uniformity vs. dimensionless die width for power-lawfluids with n = 0.5 in the T-die with H = 0.05, L0 = 25.

60

3 50 --

40

302O

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13"= 0' 11 ,. ~ ~0.0o.ool ~ " S / /0.0

10 20 30 40 50 60 70 80 90 100Dimensionless die width 2W

Figure 5. Dimensionless length of die land needed to uniform theflow of power-law fluids with n = 0.5 in the T-die with H = 0.05.

d i e . S i n c e t h e d e s i g n f o r m u l a i s b a s e d o n th e n o n s l i pm o d e l ( E q u a t i o n ( 1) ), t h e c u r v e f o r 1 3" = 0 i s v e r ye v e n a c r o s s t h e d i e w i d t h . A s w a l l s l i p o c c u r s , th ef l o w d i s t r i b u t i o n b e c o m e s u n e v e n r a p i d l y a n d o b v i -o u s l y . T h e f l o w r a t e n e a r t h e d i e c e n t e r (y = 0 ) ish i g h e r t h a n t h o s e a t d i e e n d s . A s I~* b e c o m e s a c r o s s ,t h e f lo w d i s t r i b u t i o n w i l l b e m o r e u n e v e n . C o m p a r i n gt h e c u r v e s i n F i g u r e 2 a n d t h e c u r v e s i n F i g u r e 6 , w eo b s e r v e t h a t t h e e f f e c t o f w a l l s l i p o n t h e f l o w d i s t r i -b u t i o n i n th e c o a t - h a n g e r d i e i s m o r e e v i d e n t t h a n t h a ti n t h e T - d i e . A d d i n g t o t h e d i e l a n d w i l l i m p r o v e f l o wu n i f o r m i t i e s , a n d t h e r e q u i r e d l e n g t h o f t h e d i e l a n d

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2.0

1.5q/qave

1.0

0.5

" ,0.01\\ // , o oo,

0.0 i I i I i I i I i0 10 20 30 40 50Dimensionless width direction

Figure 6, Flow rate distribution for power law fluids with n = 0.5in the coat- hanger die with a manifold designed by Equation (1) andH=0 .0 5 , W=5 0 , a= 1 .8 ,L0 =0 .

3

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40

30

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0 0

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p*= 0.I . /ooi I

I I i ~ 1 I10 20 30 40 50 60 70 80 90 100

Dimensionless die width 2W

Figure 7, Dimensionless length of die land needed to uniform theflow of power-l aw fluids with n = 0.5 in the coat-han ger die with h= 0.05.

t h a t c a n e v e n t h e fl o w d i s t r i b u t i o n u n d e r t h e e f f e c t o fw a l l s l i p i s sh o w n i n F i g u r e 7 . F o r t h e w i d e r d i e , t h er e q u i r e d l e n g t h o f t h e d i e l a n d i s l o n g e r . I n F i g u r e 7 ,w e a l s o f i n d t h a t w h e n t h e d i m e n s i o n l e s s d i e w i d t h i ss m a l l e r t h a n 2 0 , d i e l a n d i s n o t n e c e s s a r y a n d t h ee f f e c t o f w a l l s l i p i s l e s s o b v i o u s . W e a l s o i n v e s t i g a t et h e a p p l i c a b i l i t y o f c o a t - h a n g e r d ie . T h e c o a t - h a n g e rd i e d e s i g n e d o n t h e b a s i s o f n = 0 .5 i s u s e d t o e x t r u d a t ep o l y m e r m e l t s w h i c h h a s o t h e r v a lu e s o f n. F i g u r e 8


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F i g u r e 8 . U n i f o r m i t y v s f l o w i n d e x f o r p o w e r - l a w f l u i d s in th ec a n t - h a n g e r d i e w i t h H = 0 . 0 5 , W = 5 0 , o t = 1 8 .

p l o t s t h e u n i f o r m i t y a s a f u n c t i o n o f f l o w i n d e x f o rv a r i o u s v a l u e s o f 13 ". T h e i n t e r e s t i n g o b s e r v a t i o n i st h a t t h e U I c u r v e s f o r a l l c a s e s a r e s i m i l a r t o t h e s h a p eo f a m o u n t a i n . A t th e v e r te x , th e v a lu e o f U l i s n e a r1 a n d t h e f l o w d i s t r i b u t i o n i s v e r y u n i f o r m . F o r t h ec u r v e o f [ 3 = 0 , th e v e r t e x a p p e a r s a t t h e p o i n t o f n =0 . 5 . T h i s i s b e c a u s e t h e n = 0 . 5 i s t h e d e s i g n c o n d i t i o n .W h e n n is n o t e q u a l t o 0 .5 , U I d e c r e a s e s a n d t h e f l o wd i s t r i b u t i o n b e c o m e s u n e v e n . T h i s m e a n s t h a t t h e d i ed e s i g n e d f o r a s p e c i f i e d p o l y m e r m e l t s i s n o t a p p ro -p r i a t e f o r o t h e r s . F o r t h e c a s e s o f s li p , t he v e r t e x s h i f t st o w a r d t h e l a rg e r v a l u e o f n w i t h i n c r e a s i n g v a l u e o f13".

Conc l u s i onT h e f l o w o f p o w e r - l a w f l u i d s in t he s h e e t i n g

e x t r u s i o n d i e s u n d e r t h e c o n s i d e r a t i o n o f w a l l s l i p h a sb e e n i n v e s t i g a t e d . T h e f l o w d i s tr i b u t i o n a n d u n i fo r -m i t y a r e c a l c u l a t e d f o r b o t h o f s l i p a n d n o n s l i p m o d -e l s . T h e r e s u l t s s h o w t h a t a s t h e w a l l s l ip o c c u r s , t h ef l o w d i s t r i b u t io n b e c o m e s u n e v e n a n d t h e v al u e o f U Id e c r e a s e s . F o r t h e w i d e r d i e , t h e e f f e c t o f w a l l s l ip o nt h e f l o w u n i f o r m i t y i s s t r o n g e r. T h e r e f o r e , t h e e x t ru -s i o n d i e s d e s i g n e d a c c o r d i n g t o t h e t r a d i t i o n a l n o n s l i pm o d e l i s no t a p p r o p r i a t e f o r an e x t r u s i o n f l o w w i t hs l i p v e l o c i t y . H e n c e i t i s m o r e n e c e s s a r y t o i n c r e a s et h e l e n g t h o f t h e d i e l a n d t o e v e n t h e f l o w d i s t r i b u t i o n .

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C o f i n c i l o f t h e R e p u b l i c o f C h i n a f o r f i n a n c i a l a i dt h r o u g h t h e p r o j e c t N S C 8 3 - 0 1 1 7 - C - 1 5 1 - 0 0 3 - E .

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d i m e n s i o n l e s s h a l f o f th e s l it g a p d e f i n e d a c -c o r d i n g t o E q u a t i o n ( 1 6 )h a l f o f t h e s l i t g a pp a r a m e t e r o f th e p o w e r - l a w f l u i d d e f i n e d a c -c o r d i n g t o E q u t a i o n ( 3 )d i m e n s i o n l e s s l e n g t h o f t h e sl it d e f i n e d a c c o r d -i n g to E q u t a i o n ( 1 6 )d i m e n s i o n l e s s l e n g t h o f t h e d i e la n dl e n g t h o f t h e s l i t1nf l o w i n d e x o f a p o w e r - l a w f l u i d s

d i m e n s i o n l e s s p r e s s u r e d e f i n e d a c c o r d i n g t oE q u a t i o n ( 1 5 )p r e s s u r ep r e s s u r e a t t h e e n t r a n c eh a l f o f t h e t o t a l f l o w r a t ef l o w r a t e p e r u n i t w i d t ha v e r a g e v a l u e o f q f o r t h e w h o l e w i d t hm a n i f o l d r a d i u sm a n i f o l d r a d i u s a t t h e e n t r a n c ed i m e n s i o n l e s s m o n i f o l d r a d iu s d e f i n e d a c c o r d -i n g t o E q u a t i o n ( 1 6 )v e l o c i t y i n t h e m a c h i n e d i r e c t i o ns l i p v e l o c i t y a t t h e w a l ld i m e n s i o n l e s s h a l f o f t h e d i e w i d t h d e f i n e da c c o r d i n g t o E q u a t i o n ( 1 6)h a l f o f t h e d i e w i d t hd i m e n s i o n l e s s x - c o o r d i n a t e d e f i n e d a c c o r d i n gt o E q u t a i o n ( 1 6) c o o r d i n a t ex - c o o r d i n a t ed i m e n s i o n l e s s y - c o o r d i n a t e d e f i n e d a c c o r d i n gt o E q u a t i o n ( 1 6 )y - c o o r d i n a t ed i m e n s i o n l e s s z - c o o r d i n a t e d e f i n e d a c c o r d i n gt o E q u t a i o n ( 1 6 )z - c o o r d i n a t e

G r e e k S y m b o l so~ c o a t - h a n g e r a n g l e13 s l i p c o e f f i c i e n t~ ' s h e a r r a t er I p o w e r l a w v i s c o s i t y~ ,o w a l l s h e a r s t r e s s

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