Computer Applications in Applied Polymer Science - American

15

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on March 30, 2016 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch015

Computer Simulation of Nip Flow in Roll Coating D. J. COYLE, C. W. MACOSKO, and L. E. SCRIVEN University of Minnesota, Department of Chemical Engineering & Materials Science, Minneapolis, MN 55455

The Galerkin finite element method is successfully applied to flow in a relatively simple element of roll coating; symmetric film-splitting in the nip region between smooth, rigid, counterrotating rolls. The calculated flow fields are solutions of the momentum and continuity equations together with appropriate boundary conditions at the liquid/gas interface. The computer code for the analysis accommodates effects of inertia, gravity, and shear-thinning rheology. The predicted locations of the line of film-splitting agree with available experimental data. Pressure profiles are also computed. For capillary number (N = μV/σ) less t h a n a b o u t 0.3, there is a pair of counterrotating eddies directly under the meniscus of the splitting film. Shear-thinning behavior by the liquid enlarges the eddies and moderates the pressure gradient in the nip region. Ca

A common method f o r a p p l y i n g a t h i n u n i f o r m f i l m o f a v i s c o u s l i q u i d o n t o a s o l i d s u b s t r a t e i s r o l l c o a t i n g , one v e r s i o n o f w h i c h i s d e p i c t e d i n F i g u r e 1. I n a l l s u c h c o a t i n g o p e r a t i o n s , i t i s d e s i r a b l e t o maximize t h e speed o f c o a t i n g w h i l e m a i n t a i n ing a h i g h degree o f u n i f o r m i t y i n t h e coated f i l m . Thus a b a s i c goal i n coating flow research i s t o understand the flow patterns and t h e mechanisms b y w h i c h n o n u n i f o r m i t i e s i n t h e f i n a l f i l m come a b o u t . A n o t h e r g o a l i s t o f u r n i s h t h e d e s i g n e n g i n e e r a means t o c a l c u l a t e a c c u r a t e t h e o r e t i c a l p r e d i c t i o n s o f how s u c h coating devices w i l l perform. Such t h e o r e t i c a l s i m u l a t i o n c a n b e e x t r e m e l y e f f e c t i v e f o r t h e d e s i g n a n d improvement o f c o a t i n g equipment. I n r o l l c o a t i n g a r r a n g e m e n t s o f t h e s o r t shown i n F i g u r e 1, l i q u i d i s c a r r i e d by t h e p i c k - u p r o l l i n t o t h e n i p r e g i o n b e tween i t and t h e a p p l i c a t o r r o l l . Beyond t h e n i p t h e l i q u i d i s , i n e f f e c t , s t r e t c h e d a n d s p l i t i n t o two f i l m s , e a c h c a r r i e d o f f

0097-6156/82/0197-0251$06.00/0 © 1982 American Chemical Society Provder; Computer Applications in Applied Polymer Science ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


252

COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE

by one o f t h e two r o l l s . I t i s the behavior i n t h i s r e g i o n , from n i p to s p l i t , t h a t determines the f l o w r a t e , the f i l m t h i c k n e s s on e a c h r o l l , t h e f o r c e s e x e r t e d on t h e r o l l s , and w h e t h e r o r n o t t h e f i n a l c o a t e d f i l m w i l l be u n i f o r m . The same i s t r u e o f t h e n i p f l o w b e t w e e n t h e a p p l i c a t o r r o l l and t h e web c a r r i e d on t h e b a c k - u p r o l l when t h e r o l l s t u r n as shown. More o f t e n i n p r a c t i c e , however, t h i s second n i p f l o w i s between c o - r o t a t i n g r o l l s , w h i c h a r e commonly r e f e r r e d t o as " r e v e r s e - r o l l s . " I n b o t h c a s e s , an a c c u r a t e t h e o r e t i c a l d e s c r i p t i o n o f t h e f l o w has y e t t o be d e v e l o p e d . T h i s paper d e a l s o n l y w i t h the c o n f i g u r a t i o n i l l u s t r a t e d i n F i g u r e 2, w h e r e t h e r o l l s a r e o f e q u a l r a d i i and t h e i r s u r f a c e s a r e m o v i n g w i t h e q u a l s p e e d and i n t h e same d i r e c t i o n a t t h e n i p . These r e s t r i c t i o n s b r i n g i n a symmetry t h a t a l l o w s one t o a r g u e t h a t t h e domain o f t h e s i m u l a t i o n c a n be h a l v e d , as shown. Once t h i s s i m p l e r c o n f i g u r a t i o n has b e e n a n a l y z e d , t h e t h e o r y c a n and i s b e i n g e x t e n d e d t o c a s e s where t h e c o n v e n i e n t symmetry i s l a c k i n g — and t h e n t o c a s e s o f r e v e r s e - r o l l c o a t i n g , w h i c h d i f f e r s somewhat i n t h e t y p e o f de f o r m a t i o n t h e l i q u i d s u f f e r s , b u t more i m p o r t a n t l y i n t h a t a w e t t i n g l i n e , o r dynamic c o n t a c t l i n e i s p r e s e n t near the n i p . C e r t a i n a s p e c t s o f t h e f l o w a n a l y z e d h e r e have b e e n e x a m i n e d before. P i t t s and G r e i l l e r (1) p r e s e n t e d one o f t h e e a r l i e s t e x p e r i m e n t a l and t h e o r e t i c a l a n a l y s e s : they s o l v e d approximately the biharmonic equation of c r e e p i n g Newtonian f l o w i n the f i l m s p l i t t i n g r e g i o n on t h e a s s u m p t i o n o f a p a r a b o l i c f r e e s u r f a c e s h a p e . W i l l i a m s o n (2) u s e d a f i n i t e d i f f e r e n c e scheme t o s o l v e t h e b i h a r m o n i c e q u a t i o n and e m p l o y e d a s i x t h - d e g r e e p o l y n o m i a l t o r e p r e s e n t the f r e e s u r f a c e . G r e e n e r and M i d d l e m a n (3) put f o r ward a s i m p l e l u b r i c a t i o n model w h i c h a g r e e s w e l l w i t h t h e i r f i l m - t h i c k n e s s d a t a , b u t r e l i e s on a p p r o x i m a t e , p r o b a b l y q u i t e o v e r s i m p l i f i e d , boundary c o n d i t i o n s a t the s t a g n a t i o n l i n e where the f i l m s p l i t s . W h i l e t h e s e e a r l i e r m o d e l s c a n be u s e f u l i n some s i t u a t i o n s , t h e y a l l a p p r o x i m a t e t h e b o u n d a r y c o n d i t i o n s a t t h e f r e e s u r f a c e and a r e n o t c a p a b l e o f d e s c r i b i n g t h e f l o w f i e l d accurately. In a d d i t i o n , they are a l l l i m i t e d to c r e e p i n g motion, i . e . t o l o w R e y n o l d s number f l o w i n w h i c h t h e i n e r t i a o f t h e l i q u i d i s n e g l i g i b l e , which i s not n e c e s s a r i l y a v a l i d approximation f o r high-speed coating operations. In order to s i m u l a t e r o l l c o a t i n g a c c u r a t e l y over a l l the r a n g e s o f p a r a m e t e r s o f p r a c t i c a l r e l e v a n c e , one w o u l d n e e d — f o r a Newtonian l i q u i d — to s o l v e the f u l l N a v i e r - S t o k e s equa t i o n s y s t e m i n c l u d i n g t h e c o r r e c t b o u n d a r y c o n d i t i o n s . Owing t o t h e shape o f t h e m e n i s c u s and t o t h e n o n l i n e a r e f f e c t s o f i t s f r e e n e s s and o f l i q u i d i n e r t i a , t h i s c a n n o t be done by c o n v e n t i o n a l m a t h e m a t i c a l m e t h o d s . However, c o m p u t e r - a i d e d t e c h n i q u e s em p l o y i n g G a l e r k i n s w e i g h t e d - r e s i d u a l method w i t h f i n i t e e l e m e n t b a s i s f u n c t i o n s proves very e f f e c t i v e . This approach to s o l v i n g v i s c o u s f r e e s u r f a c e f l o w p r o b l e m s , s u c h as t h o s e t h a t a r i s e i n c o a t i n g o p e r a t i o n s , has b e e n r e c e n t l y a p p l i e d t o r i m m i n g f l o w (4), e x t r u s i o n c o a t i n g 0 5 , 6 ) , s l i d e c o a t i n g ( 7 ) , and c u r t a i n 1

Provder; Computer Applications in Applied Polymer Science ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


Figure 1.

253

Nip Flow in Roll Coating

COYLE E T AL.

Typical forward (a) and reverse (b) role coating operations.

DOMAIN OF SIMULATION

Figure 2.

COATED FILM

Definition sketch for symmetric film-splitting analysis.



254


c o a t i n g ( 8 ) . In o r d e r to determine a p p r o p r i a t e boundary c o n d i t i o n s on t h e s i m p l e r e q u a t i o n s o f t h e l u b r i c a t i o n a p p r o x i m a t i o n a t t h e l i q u i d / g a s i n t e r f a c e where t h e f i l m s p l i t s i n r o l l c o a t i n g and s i m i l a r o p e r a t i o n s , R u s c h a k (9) r e c e n t l y a p p l i e d t h e method of matched a s y m p t o t i c e x p a n s i o n s a l o n g w i t h the f i n i t e element method t o e x a m i n e t h e f l o w b e t w e e n r o l l s . H i s a n a l y s i s , however, a p p e a r s t o be r e s t r i c t e d t o t h e l i m i t o f v i r t u a l l y p a r a l l e l r o l l s u r f a c e s at the f i l m - s p l i t t i n g r e g i o n . Thus t h e o b j e c t i v e h e r e i s a g e n e r a l l y a p p l i c a b l e s i m u l a t i o n of s t e a d y , t w o - d i m e n s i o n a l , i n c o m p r e s s i b l e f l o w between r i g i d r o l l s with f i l m s p l i t t i n g . The r e s u l t s r e p o r t e d a r e s o l u t i o n s o f the f u l l N a v i e r - S t o k e s system i n c l u d i n g the p h y s i c a l l y r e q u i r e d boundary c o n d i t i o n s . The a n a l y s i s i s a l s o e x t e n d e d t o a s h e a r thinning f l u i d . The s o l u t i o n s c o n s i s t o f v e l o c i t y and p r e s s u r e f i e l d s , f r e e s u r f a c e p o s i t i o n and s h a p e , and t h e s e n s i t i v i t i e s o f these v a r i a b l e s to parameter v a r i a t i o n s , v a l u a b l e i n f o r m a t i o n not r e a d i l y a v a i l a b l e from the c o n v e n t i o n a l approach (10). The r a t e o f - s t r a i n , v o r t i c i t y , and s t r e s s f i e l d s a r e a l s o a v a i l a b l e f r o m the s o l u t i o n s r e p o r t e d h e r e a l t h o u g h they are not portrayed. M o r e o v e r , t h e s t a b i l i t y o f t h e f l o w s t a t e s r e p r e s e n t e d by t h e s o l u t i o n s can a l s o be f o u n d by a d d i t i o n a l f i n i t e e l e m e n t t e c h n i q u e s ( 1 1 ) , and t h e r e s u l t s o f d o i n g so w i l l be r e p o r t e d i n t h e future. F i n i t e Element F o r m u l a t i o n P a r t i c u l a r a d v a n t a g e s o f t h e f i n i t e e l e m e n t method f o r v i s cous f r e e s u r f a c e f l o w problems a r e : p h y s i c a l boundary c o n d i t i o n s e n t e r i n t o t h e e q u a t i o n s i n a n a t u r a l way t h a t i s p a r t i c u l a r l y a d v a n t a g e o u s i n t h e c a s e o f f r e e b o u n d a r i e s ; c o m p l e x s h a p e s and curved boundaries are r e a d i l y handled; l o c a l l y steep gradients can be r e s o l v e d by i n c r e a s i n g t h e number o f e l e m e n t s i n t h e a p p r o p r i a t e r e g i o n o f t h e f l o w ; and a f u l l f u n c t i o n a l r e p r e s e n t a t i o n o f the s o l u t i o n i s o b t a i n e d . W i t h i n t h e f l o w i n g l i q u i d , mass and momentum must be c o n served: f o r steady, incompressible flow these requirements i n d i m e n s i o n l e s s form are V • u = 0

(1)

-V

• T + N_. V • uu - N f = 0 (2) Re — g~ where N = HQVP/JJ i s t h e R e y n o l d s number, t h e r a t i o o f i n e r t i a l t o v i s c o u s f o r c e s ; Ng = pgHQ /yV i s t h e g r a v i t y number, t h e r a t i o of g r a v i t y to v i s c o u s f o r c e s ; T i s the t o t a l s t r e s s tensor (pres s u r e and v i s c o u s s t r e s s ) , and f i s a u n i t v e c t o r i n t h e d i r e c t i o n in which g r a v i t y acts. The l o c a t i o n o f t h e f r e e s u r f a c e t h a t f o r m s p a r t o f t h e b o u n d a r y i s a l s o unknown, b u t w h e r e v e r i t l o c a t e s no l i q u i d c a n f l o w t h r o u g h i t ; h e n c e a t t h e f r e e s u r f a c e R e

2

n

• u = 0


(3)

15.

Nip Flow in Roll

COYLE ET AL.

Coating

255


At t h e i n l e t t o t h e f i n i t e element domain, t h e f l o w i s p a r a l l e l so t h e e q u a t i o n s o f t h e l u b r i c a t i o n a p p r o x i m a t i o n a r e used to s p e c i f y t h e i n l e t v e l o c i t y p r o f i l e . These e q u a t i o n s a r e i n t e g r a t e d f r o m -oo t o t h e i n l e t , g e n e r a t i n g a n e q u a t i o n r e l a t i n g t h e flow rate to the i n l e t pressure. The r e m a i n i n g b o u n d a r y c o n d i t i o n s a r e a s shown i n F i g u r e 3 . The o n l y c o m p l e x i t y h e r e i s t h a t t h e f l u i d t r a c t i o n , n • T, a t t h e f r e e s u r f a c e h a s t o be s p e c i f i e d a s a b o u n d a r y c o n d i t i o n o n t h e momentum e q u a t i o n . A force balance there g i v e s , i n dimensionless form,

n •T = i «

(4)

f + n p

N_ d s Ca

r

~ a

w h e r e N ^ = y v / a i s t h e c a p i l l a r y number, t h e r a t i o o f v i s c o u s to s u r f a c e t e n s i o n f o r c e s , and dt/ds i s thecurvature of the f r e e s u r f a c e ( r a t e o f change o f t h e u n i t t a n g e n t v e c t o r w i t h respect t o arc l e n g t h along the meniscus). E q u a t i o n (4) s i m p l y s t a t e s t h a t t h e t o t a l normal s t r e s s e x e r t e d on t h e g a s / l i q u i d i n t e r f a c e must e q u a l t h e c a p i l l a r y p r e s s u r e added t o t h e a m b i e n t gas p r e s s u r e , p . When t h e e f f e c t o f s u r f a c e t e n s i o n i s n e g l i g i b l e , N c -> a n d t h e c a p i l l a r y p r e s s u r e t e r m c a n be s t r u c k . Edge e f f e c t s a r e h e r e r e g a r d e d a s n e g l i g i b l e a n d so t h e s o l u t i o n s sought a r e two-dimensional f l o w s . I n G a l e r k i n ' s m e t h o d , t h e unknown v e l o c i t y c o m p o n e n t s , p r e s s u r e , a n d f r e e s u r f a c e l o c a t i o n a r e expanded i n a s u i t a b l e s e t o f b a s i s f u n c t i o n s , (j) a n d i ^ : a

a

00

a

1

u(x,y) = I u

1

(x,y),n(x,y)}

±

v(x,y) = E v. c j ) ^ (x,y),n(x,y)} 1

p(x,y) = I p

±

iJ^U

h(s) = U .

ct> {^(x,y)}

(5)

(x,y),n(x,y)}

1

Here t h e c o e f f i c i e n t s Uj_, v ^ , p ^ , a n d h-^ r e m a i n t o be d e t e r mined; £ and n a r e c o o r d i n a t e s i n t h e subdomains i n t o w h i c h t h e a c t u a l f l o w d o m a i n i s mapped i n o r d e r t o accommodate t h e c o m p l i c a t e d shape o f i t s c o m p l e t e b o u n d a r y . N e x t , t h e weak f o r m s o f t h e mass a n d momentum c o n s e r v a t i o n e q u a t i o n s ( 1 ) a n d ( 2 ) a r e w r i t t e n u s i n g t h e s e same b a s i s f u n c t i o n s as weighting f u n c t i o n s : M. = / ^ ( - V A 1

= /(Vcj) A

1

• T + N V • uu - N f ) d A Re ~~ g~ • T + N K

e

V • u u - N f ) d A - f \ ' T d s = 0 ~~ ~ 3A 8


(6)

256


C.

/ A

K.

/ (f) !! • uds F.S.


l

V • udA = 1

0 =

(7) 0

(8)

F i n i t e element b a s i s f u n c t i o n s are employed, each of which i s a s i m p l e p o l y n o m i a l on a s m a l l s u b d o m a i n and z e r o e v e r y w h e r e else. B e c a u s e t h e s e f u n c t i o n s a r e so d e s i g n e d t h a t e a c h t a k e s on t h e v a l u e o f u n i t y a t j u s t one node and i s z e r o a t a l l o t h e r nodes, the c o e f f i c i e n t s i n the expansions, equations ( 5 ) , are a l s o t h e v a l u e s o f t h e unknown v a r i a b l e s a t t h e n o d e s . The r e s u l t s r e p o r t e d b e l o w w e r e o b t a i n e d w i t h c o n v e n t i o n a l " m i x e d i n t e r p o l a t i o n " on i s o p a r a m e t r i c r e c t a n g l e s ( 1 2 ) , u s i n g nine-node b i q u a d r a t i c b a s i s f u n c t i o n s (f) f o r t h e v e l o c i t y com p o n e n t s and f o u r - n o d e b i l i n e a r b a s i s f u n c t i o n s i j ; f o r the p r e s sure. The f r e e s u r f a c e shape was r e p r e s e n t e d by t h e i s o p a r a m e t r i c map, t h e p o s i t i o n b e i n g d e t e r m i n e d by t h e c o e f f i c i e n t s h^ i n t h e g e n e r a l " f r e e - s p i n e " r e p r e s e n t a t i o n r e c e n t l y d i v i s e d by K i s t l e r (13). A t y p i c a l s u b d i v i s i o n o f t h e f l o w domain i n t o f i n i t e e l e ments u s e d i n t h i s i n v e s t i g a t i o n i s shown i n F i g u r e 4. The e n t i r e mesh i s s h i f t e d w i t h t h e f r e e s u r f a c e p o s i t i o n so t h a t no e l e m e n t becomes s e v e r e l y d e f o r m e d (and so c e r t a i n i n a c c u r a c i e s a r e a v o i d e d ) , and t h e n o d e s r e m a i n c o n c e n t r a t e d i n t h e r e g i o n where t h e f l o w c h a n g e s most r a p i d l y . E q u a t i o n s ( 6 ) , ( 7 ) , (8) r e p r e s e n t a l a r g e s e t o f n o n l i n e a r a l g e b r a i c e q u a t i o n s w h i c h a r e s o l v e d by Newton's method. This method c o n v e r g e s q u a d r a t i c a l l y — a g r e a t a d v a n t a g e — o v e r t h e e n t i r e r a n g e o f p a r a m e t e r v a l u e s so f a r e x a m i n e d . The l o c a l n a t u r e of the b a s i s f u n c t i o n s d r a s t i c a l l y r e d u c e s the work r e q u i r e d t o e v a l u a t e t h e G a l e r k i n i n t e g r a l s , w h i c h must be done n u m e r i c a l l y , and i t g i v e s t h e J a c o b i a n m a t r i x i n Newton's method a banded s t r u c t u r e t h a t l e s s o n s t h e c o m p u t a t i o n a l w o r k r e q u i r e d i n t h e ma t r i x s o l u t i o n procedure. The n o n l i n e a r a l g e b r a i c e q u a t i o n s r e p r e s e n t e d by e q u a t i o n s ( 6 ) , ( 7 ) , (8) w e r e g e n e r a t e d and s o l v e d on a CDC CYBER 74 comput er. S o l u t i o n s t y p i c a l l y t o o k 4-5 i t e r a t i o n s t o c o n v e r g e t o an o r d e r o f 10"^, and f o r t h e 440 e q u a t i o n s c i t e d i n F i g u r e 4 e a c h i t e r a t i o n took approximately 7 seconds. 1

1

R e s u l t s f o r Newtonian F l u i d I n a N e w t o n i a n f l u i d , t h e s t r e s s i s a l i n e a r , homogeneous, i s o t r o p i c f u n c t i o n of the r a t e - o f - s t r a i n p a r t of the v e l o c i t y gradient. I n s e r t i n g t h i s c o n s t i t u t i v e equation i n t o equations (6) - (8) c l o s e s t h e s e t o f e q u a t i o n s f o r n o d a l v e l o c i t i e s and p r e s s u r e s and f o r f r e e s u r f a c e l o c a t i o n s . T h i s s e t c a n t h e n be s o l v e d i n t h e manner d e s c r i b e d a b o v e . F i g u r e 5 shows t h e com puted flow f i e l d f o r parameter values r e p r e s e n t a t i v e of o p e r a t i o n



COYLE ET AL.

Nip Flow in Roll Coating

257

PARALLEL FLOW Figure 3.

Figure

4.

Typical

Boundary conditions for symmetric film-splitting.

subdivision of the flow domain into finite elements: 39 elements and 440 equations.



258


of the authors' l a b o r a t o r y t w o - r o l l apparatus. The most s t r i k i n g f e a t u r e i s the p a i r of l a r g e , r e l a t i v e l y s l o w l y c i r c u l a t i n g eddies, one on e a c h s i d e o f t h e symmetry p l a n e , l o c a t e d j u s t u n d e r t h e s t a g n a t i o n l i n e at which the f i l m s p l i t s . F i g u r e 6 shows how t h e e d d i e s s h r i n k and f i n a l l y d i s a p p e a r as t h e c a p i l l a r y number i s i n c r e a s e d , i . e . as the v i s c o u s normal s t r e s s overwhelms the cap i l l a r y p r e s s u r e a t and n e a r t h e c e n t r a l s t a g n a t i o n l i n e . When the eddies are absent, t h e r e i s o n l y the c e n t r a l s t a g n a t i o n l i n e ; b u t a s t h e y grow, t h r e e s t a g n a t i o n l i n e s move away f r o m t h e c e n t r a l one: one down t h e symmetry p l a n e and one t o e a c h s i d e on the f r e e s u r f a c e . T h e s e r e s u l t s show c l e a r l y how a l u b r i c a t i o n a n a l y s i s f a i l s . The l u b r i c a t i o n a p p r o x i m a t i o n r e q u i r e s n e a r l y s t r a i g h t s t r e a m l i n e s and i s e n t i r e l y i n a p p r o p r i a t e f o r t h e r e c i r c u l a t o r y f l o w when t h e e d d i e s a r e p r e s e n t . And when e d d i e s a r e p r e s e n t t h e f i l m does n o t s p l i t a t t h e f i r s t s t a g n a t i o n p o i n t d o w n s t r e a m o f t h e n i p , as assumed i n some l u b r i c a t i o n a n a l y s e s (_3), b u t r a t h e r t h e s e c o n d . When t h e c a p i l l a r y number i s h i g h enough t h a t t h e eddies are absent, l i q u i d i n e r t i a l e f f e c t s that are neglected i n t h e l u b r i c a t i o n a p p r o x i m a t i o n may become a p p r e c i a b l e . F i g u r e 7 r e v e a l s t h a t t h e f r e e s u r f a c e p o s i t i o n and s h a p e a r e q u i t e s e n s i t i v e t o the c o m p e t i t i o n between v i s c o u s normal s t r e s s and t h e c u r v a t u r e - d e p e n d e n t c a p i l l a r y p r e s s u r e . As t h e r a t i o o f t h e f o r m e r t o t h e l a t t e r , i . e . Ca = y v / a , r i s e s , t h e m e n i s c u s r e c e d e s t o w a r d t h e n i p and c u r v e s more s h a r p l y . The s a m p l e r e s u l t s i n F i g u r e s 5-10 a r e f o r c a s e s i n w h i c h t h e e f f e c t s o f g r a v i t y and l i q u i d i n e r t i a a r e n e g l i g i b l e . The c o m p u t e r p r o gram u s e d h e r e i s c o n s t r u c t e d t o i n c l u d e t h e s e e f f e c t s a s i t s o l v e s e q u a t i o n s (6) - ( 8 ) ; c o m p r e h e n s i v e c a s e s t u d i e s and com p l e t e p o r t r a i t s o f v e l o c i t y , p r e s s u r e , s t r e s s , and v o r t i c i t y f i e l d s w i l l be r e p o r t e d e l s e w h e r e ( 1 4 ) . As w i t h any t h e o r e t i c a l p r e d i c t i o n , t h e c a l c u l a t i o n i t s e l f must be v a l i d a t e d , and t h e p r e d i c t i o n has t o be h e l d up a g a i n s t as c l o s e l y c o m p a r a b l e an e x p e r i m e n t as i s a v a i l a b l e . Comprehen s i v e v a l i d a t i o n o f t h e p r e s e n t c a l c u l a t i o n s w i l l be d e t a i l e d e l s e w h e r e . As f o r e x p e r i m e n t s , u n f o r t u n a t e l y few d e t a i l s o f t h e flow f i e l d are f u l l y d e s c r i b e d i n the l i t e r a t u r e . P i t t s and G r e i l l e r (1) d e t e c t e d t h e e d d i e s , b u t d i d n o t i n d i c a t e o v e r what p a r a m e t e r r a n g e s e d d i e s w e r e o r were n o t p r e s e n t . They d i d , how e v e r , measure where t h e f i l m s p l i t s i n a number o f experiments. The t h e o r e t i c a l p r e d i c t i o n s c a l c u l a t e d by t h e G a l e r k i n f i n i t e e l e m e n t method compare w e l l w i t h t h e i r d a t a , as shown i n F i g u r e 8. C a r e f u l measurements o v e r a w i d e r r a n g e o f p a r a m e t e r s a r e n e e d e d . The agreement o v e r P i t t s and G r e i l l e r s r a n g e i s e n c o u r a g i n g b e c a u s e t h e t h e o r y shows ( F i g u r e 7) t h e p o s i t i o n o f t h e f r e e s u r f a c e i s q u i t e s e n s i t i v e t o v a r y i n g p a r a m e t e r s , more so t h a n f l o w r a t e o r f i l m t h i c k n e s s , w h i c h can be m e a s u r e d as w e l l . L o a d i n g o f t h e r o l l s and t h e p o s s i b i l i t y o f c a v i t a t i o n down s t r e a m o f t h e n i p depend on t h e p r e s s u r e p r o f i l e , w h i c h i s d i f f i c u l t to measure. I t i s r e a d i l y p r e d i c t e d w i t h the G a l e r k i n ?


COYLE ET AL.

Nip Flow in Roll

259

Coating


15.

Figure 6.

Effect of N

Ca

on flow field. Key for N : Ca

a, 0.1; b, 0.2; c, 0.5.




260


15.

COYLE ET AL.

Nip

Flow in Roll

Coating

261

f i n i t e e l e m e n t method, as shown i n F i g u r e 9. The p r o f i l e s , w h i c h are f o r a n i p completely flooded upstream at atmospheric pres s u r e , i l l u s t r a t e how t h e p r e s s u r e v a r i a t i o n i n c r e a s e s as t h e c a p i l l a r y number i s i n c r e a s e d a t v a n i s h i n g R e y n o l d s number. Shear s t r e s s d i s t r i b u t i o n on t h e r o l l s i s r e a d i l y p r e d i c t e d as w e l l .


Results for shear-thinning

fluids

Many i m p o r t a n t c o a t i n g p r o c e s s e s a r e o f l i q u i d s t h a t a r e n o t N e w t o n i a n , and so t h e e f f e c t s o f n o n - N e w t o n i a n r h e o l o g y on f l o w between r o l l s i s o f g r e a t i n t e r e s t . The code u s e d h e r e has b e e n a p p l i e d t o the s i m p l e s t non-Newtonian model, namely the p u r e l y viscous, shear-thinning f l u i d . V i s c o e l a s t i c i t y , though a l s o im p o r t a n t , i s more d i f f i c u l t t o t r e a t and i s n o t c o n s i d e r e d h e r e . One o f t h e b e t t e r m o d e l s a v a i l a b l e f o r f i t t i n g n o n - N e w t o n i a n v i s c o s i t y (n) as a f u n c t i o n o f s h e a r r a t e i s t h a t o f C a r r e a u ( 1 5 ) :

= O

2

( 1 -

(AY) } ^

d o )

00

where X and n are e m p i r i c a l constants. T h i s m o d e l has a New t o n i a n p l a t e a u , n = n , a t l o w s h e a r r a t e s and a "power l a w " r e g i o n of s h e a r - t h i n n i n g at higher shear r a t e s . This v i s c o s i t y f u n c t i o n makes t h e s t r e s s a n o n l i n e a r f u n c t i o n o f t h e r a t e o f s t r a i n b u t d o e s n o t change t h e f u n c t i o n a l r e l a t i o n s h i p s i n t h e equations. Inasmuch as t h e unknown f r e e s u r f a c e and t h e i n e r t i a l t e r m s a r e a l r e a d y n o n l i n e a r , t h e G a l e r k i n f i n i t e e l e m e n t code i s a l m o s t t h e same as f o r a N e w t o n i a n f l u i d ; n o n l i n e a r t e r m s r e p r e s e n t i n g v i s c o u s s t r e s s r e p l a c e l i n e a r o n e s . The method o f s o l v i n g the e q u a t i o n s e t remains unchanged. Some p r e l i m i n a r y r e s u l t s a r e shown i n F i g u r e 10, and r e v e a l that s h e a r - t h i n n i n g behavior causes the f i l m to s p l i t f u r t h e r d o w n s t r e a m o f t h e n i p and t o d e v e l o p l a r g e r e d d i e s . The sheart h i n n i n g a l s o r e d u c e s s u b s t a n t i a l l y t h e m a g n i t u d e s o f t h e maximum and minimum p r e s s u r e s i n t h e n i p r e g i o n , as t h o u g h t h e f l o w w e r e b e i n g l u b r i c a t e d by t h e l o w - v i s c o s i t y z o n e s t h a t d e v e l o p w h e r e the shear r a t e i s h i g h e r . A more d e t a i l e d a n a l y s i s i s i n p r o g ress. Q

Summary Computer-aided a n a l y s i s employing the G a l e r k i n f i n i t e e l e ment method p r o v i d e s t h e means o f m a k i n g a c c u r a t e t h e o r e t i c a l p r e d i c t i o n s of complicated v i s c o u s f r e e surface flows without r e s o r t i n g to s i m p l i f y i n g assumptions, which are g e n e r a l l y q u i t e restrictive. The a p p r o a c h has b e e n s u c c e s s f u l l y a p p l i e d t o f l o w i n a r e l a t i v e l y simple element of r o l l c o a t i n g , symmetric f i l m s p l i t t i n g i n the n i p r e g i o n between smooth, r i g i d c y l i n d r i c a l r o l l s o f e q u a l r a d i i t u r n i n g a t e q u a l s p e e d s l o w enough t h a t i n e r t i a l e f f e c t s are i n s i g n i f i c a n t . The p r e d i c t e d l o c a t i o n s o f t h e



262


Figure 9.

Pressure profiles as a junction oj N - Conditions: N li — 10 poise,

Computer Applications in Applied Polymer Science - American

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