8/13/2019 1-s2.0-073519339190082F-main

1/9

IN T . C OM M . H E A T M A S S T R A N S F E RVoL 18, pp. 705-713 , 1991Pergamon Press plc0735-1933/91 $3.00 + .00Printed in the United States

N O N L I N E A R T H E O R Y O F F I L M R U P T U R E W I T H V IS C OS IT Y V A R IA T IO N

M i n g -C h e n g W u a n d C h i -C h u a n H w a n gChung Y uan Chr i st ian Un ive r s it yDe pa r tm en t o f Mechan i ca l Eng inee r ingChung LL Ta iwan 32023Rep ub l i c o f Ch ina

(Communicated b y J .P. Hartnet t and W J. Minkowycz)

A B S T R A C TA nonl inear k inem at ic equa t ion for th in liqu id f ilm th ickness which takes in toaccoun t t he e f f ec ts o f v i scos it y va r i a ti on an d Lo ndo n /va n de r W aa l s a t t r a c ti onsis der ived to s tudy the mecha-i .~m of fi lm rupture . A f te r appro pr ia te resca l ingo f t ime va r i ab l e by a su i t ab l e f ac to r , we show tha t t h i s equa t i on cou ld becon verte d into a constant-viscosity equ ation. T he n w ithou t furth er analysis w ecan deduce d i rec t ly , by examina t ion of the resca l ing fac tor , tha t a cool ing(hea t ing) f rom the wal l wi l l increase ( reduce) the rupture t ime.

1 . I N T R O D U C T I O NT h e s t u d y o f t h e r u p t u r e o f t h in l i q u i d f i l m s has been mo t iva t ed by t he i r i ndus t r i a l

appl ica t ion in d i sperse and col lo id sys tem on one hand and the unders tanding of d iversebio logica l phenomena on the o ther [1 l - [5] . Ruckens te in and Ja in [6] s tudied the sponta-neous rup ture of a l iqu id f ilm on a so l id p lane . Th ey m odeled the l iqu id as a Navier -Stokescon t i nuum hav ing an ex t r a b ody for ce . Th i s bod y fo r ce ha s a po t en t i a l due t o t he van de rW aals in te rac t ion . S ince the wavelengths of the un s tab le d i s turb ance a re la rge compar ingto the th in f i lm th ickness . Ruckens te in and Ja in [6] used a lubr ica t ion approximat ion to

7 5

8/13/2019 1-s2.0-073519339190082F-main

2/9

706 M.-C. W u and C.-C. Hw ang Vol. 18, No. 5

o b t a i n d y n a m i c l i n e a r i n s t a b i li t y r e su l ts , b ' ~o m t h e i r r e s u l t s o n e c a n o b t a i n a r o u g h e s-t i m a t e o f t h e r u p t u r e t i m e n e e d e d f o r t h e film t o a t t a i n z e r o th i c k n e s s a t s o m e p o i n t.W i ll iam .q an d D av i s [ 7] ex t en d ed t h e an a l y s i s t o t h e n o n l i n e a r reg i o n s . T h ey f o r m a l i zeda l o n g - w a v e t h e o r y f o r t h e n o n l i n e a r d y n a m i c in s t a b i l i t y o f a l i q u i d f il m a n d d e r i v ed an o n l i n e a r e v o l u t i o n e q u a t i o n f o r t h e f i lm t h ic k n e s s. T h e n W i l li a m s a n d D a v i s [7] u s e d n u -m e r i c a l m e t h o d t o s o l v e t h e n o n l i n e a r e v o lu t i o n e q u a t i o n a s a n i u i ti a l -v a l u e p r o b l e m w i t hp e r i o d i c b o u n d a r y c o n d i t io n s . T h e i r r e s u lt s re v e a l t h a t t h e n o n l i n e a r i t i e s w il l a c c e l e r a tet h e r u p t u r e p r o c e s s . O n t h e o t h e r h a n d S h a r m a a n d R u c k e n s t e i n [8] d e v e l o p e d a p e r t u r b a -t i r e a n a l y s i s f o r th e n o n l i n e a r e v o l u t i o n e q u a t i o n . T h e i r a n a l y t i c a l r e s u l t s o b t a i n e d f o r th ed o m i n a n t w a v e l e n g t h a n d t h e t i m e o f r u p t u r e a g r e e w el l w i t h t h e n u m e r i c a l s i m u l a ti o n so f t h e p r ev i o u s s t u d i e s [ 7] .

A p p a r e n t l y , m o s t o f t h e ab o v e i n v es t i g a t i o n s [6 ]- [8 ] w e r e ad d r e s s ed t o t h e i s o t h e r m a lco n d i t i o n s . B u r e l b a ch e t a l . [ 9] co n s i d e r ed t h e n o n l i n e a r s t ab i l i t y an d r u p t u r e o f a li q u i df il m e v a p o r a t i n g ( c o n d e n s i n g ) o n a h o t ( c o ld ) h o r i z o n t a l s u r fa c e . I t i s a n o n l i n e a r p r o b l e mo f u l t r a t h i n f il m h e a t t r a n s f e r w i t h m a s s t r a n sf e r . T h e i r n u m e r i c a l r e s u l t s s h o w n t h a t i nt h e f in a l s ta g e s o f t h e c o n d e n s a t i v e f il m t h e g r o w t h o f t h e s u r f a c e w a v e w i ll b e c o m e m u c hr a p i d . T h e c o m p e t i t i o n s b e t w e e n s ta b i li z in g e ff e ct a n d d e s t a b i l iz i n g e f fe c t a r e t h a t t h ec o n d e n s a t i o n s lo w s d o w n t h e i n i t ia l g r o w t h o f th e w a v e b u t t h e l o n g - r a n g e f o rc e ea s il yw i n s i n t h e e n d . F u r t h e r m o r e , R e i s fe l d a n d B a n k o f f [1 0 ] h a v e e x t e n d e d t h e a n a ly s is b yi n c l u d in g a l i n e a r d e p e n d e n c e o f v i s c o si ty o n t e m p e r a t u r e . I f a v i s c o s i t y - t e m p e r a t u r ef a c t o r is d e f i n e d o n e c a n a r r iv e a n e v o l u t io n e q u a t i o n t h a t d i ff e rs f r o m a c o n s t a c t - v i sc o s i tye q u a t i o n b y a f a c t o r m u l t ip l i e d o n t h e sp a t i a l d e v a t iv e te r m s . H o w e v e r , a s t h e t e m p e r a t u r eg r a d i e n t t e n d s t o b e h i g h o r th e d e p e n d e n c e o f v i s c o si t y o n t e m p e r a t u r e b e c o m e st r o n g , t h el i n e a r d e p e n d e n c e o f v i s c o s it y o n t e m p e r a t u r e c a n h a r d l y b e ju s t i fi e d . I n s u c h s i tu a t i o n s, am u c h g e n e r a l m o d e l w i t h v i sc o s it y v a r i a t io n d e p e n d i n g e x p o n e n t i a l l y o n t e m p e r a t u r e c a nb e ad o p t ed . G o u s s i s e t a l . [ 1 1 ] , [1 2 ] an d H w an g e t a l. [ 1 3 ] h av e u s e d t h e g en e r a l m o d e l t os i m u l a t e t h e v i s co s i t y o f l i q u i d f i lm f o ll o w i n g d o w n a h o t ( co l d ) i n c l i n ed p l an e . I n w h i cht h e y s t u d i e d t h e l i n ea r an d n o n l i n ea r s t ab i l it i e s o f t h a t f a l l in g f i lm .

8/13/2019 1-s2.0-073519339190082F-main

3/9

V o L 18 , N o . 5 N O N L I N E A R T H E O R Y O F F I L M R U P T U R E 7 07

T h i s p a p e r s t u d i e s t h e n o n l i n e a r r u p t u r e p r o b l e m o f a t h i n l i q u i d f i lm w i t h v i sc o s it yv a r i a t io n d e p e n d i n g o n t e m p e r a t u r e a c c o r d in g t o t h e g e n e r a l m o d e l , t h a t i s, t h e A r r h e n i u s -t y p e r e l a t i o n [ 1 1] -[ 13 ]. T h e e f fe c ts o f su r f ac e t e n s io n a n d L o n d o n / v a n d e r W a a l s a t t r a c t i o n sa re a l so i nc l l nded i n t h i s s t u dy .

2 . T H E O R YC ons i de r i ng a t h i n l i qu i d f i l m on a ho t ( co l d ) ho r i zon t a l p l ane a s show n i n F IG .

1 , w he re t he l oca l f i l m t h i ckness i s sm a l l enough t ha t l ong - range m o l ecu l a r fo rce s a r eno t neg l ig i b le . A ssu m i ng t h a t t he l i qu i d i s a N ew t on i an f lu i d w i t h a l l p rope r t i e s be -i n g c o n s t a n t e x c e p t v i s c o s i t y t e m p e r a t u r e a c c o r d i n g t o A r r h e n i u s - t y p e r e l a t i o n , o r , p =# s e x p [ - A r ( T * - T , * ) /T j * ] . T h en by u s i ng t he fo l low i ng sca le s : l eng t h ( . ,, h* ) , t i m e(. . . h*2/vo), veloci ty ( , - . r , , / h * ) , p r e s s u r e ( ~ p t , 2 /T t 2 ) a n d d i m e n s i o n l e s s t e m p e r a t u r e( T . ., T * - T * ] T , ~ - 2 '* ) , one can fo rm u l a t e t he fo l low i ng t w o d i m ens i ona l gove rn i ng equa -t i ons

u , + u ~ , = O , 1 ). , .~ . + o ~ , = - P . - ~ . + 2 ~ r , . ) . + , p r ~ + ~ ) ) ~ . 2 )

Vt 4 It ~ . l ~USt = - P y - ~P~,-I- (eP T( v. q- try)) z -4- (2 efl Tv y) , , (3)1 TT . + u T . + v T , = f f~r( . . + Ty . ) . (4 )

w i t h t h e b o u n d a r y c o n d i t i o n s a t y = 0u = o = v , 5 . 1 )

T = 1 . 5 . 2 )a t y = h

(u , + v . ) (1 - h~ .) + 2h . (v , - u . ) = 0 . (6.1 )- P + 2 [(1 - h , )v y - h , ( u y + v ,) l( 1 + h l ) - ' = 3 S h , , ( 1 + h , ) T , 2' (6 .2)

8/13/2019 1-s2.0-073519339190082F-main

4/9

708 M.-C. W u and C.-C. Hw ang Vol. 18, No. 5

T~0

a n d t h e k i n e m a t i c c o n d i t i o n a t t h e f r e e su r f a c e is g i v e n b y(6.3)

h t + u h , = v a t y = h , (7)w h e r e ~ = A h - 3 is t h e v a n d e r W a a l s p o t e n t i a l . T h e d i m e n s i o n l e ss p a r a m e t e r s i n th ee q u a t i o n s ( 1 ) - ( 7 ) a r e

. _AA = A / 6 ~ r h * p v ~ , f l = ~ ( T ~ , - T : ) ,

p~ = ~ - , s = ~ * ~ / 3 p ~ , s )Vsw h e r e A ' i s t h e d i m e n s i o n a l H a m k e r c o n s t a n t , A r i s t h e A r r h e n i u s n u m b e r a n d a i s t h es u r f ace t en s i o n .

I n o r d e r t o s i m p l i f y t h e s o l u t i o n s o f t h e ab o v e eq u a t i o n s , t h e l o n g w av e t h e o r y [6 ]- [8 ] isu s e d h e r e , b y w h i c h o n e c a n e x p a n s e t h e d e p e n d e n t v a r i a b l e s i n t e r m s o f a s m a l l p a r a m e t e re ( w a v e n u m b e r ) a s

v = ~ (v o + ~ v l + . . . ) ,= ~ P o + ~ P 1 +. . . ) , (o)

T = T o + e T I + . . . .

F u r t h e r m o r e , b y s u b s t i t u t i n g e x p r e s s i o n ( 9 ) i n t o s y s t e m ( 1 ) - ( 6 ) , a l s o s c a l i n g t h e c o r r e -s p o n d i n g i n d e p e n d e n t v a r i a b l es a n d s o lv i n g t h e m o r d e r b y o r d e r , o n e c a n f i n a ll y o b t a i nt h e s o l u t i o n s o f u 0 an d v 0 a t y = h a s

1 f l + l ~ hu 01 ~= h = - ~ + ~ r - ~ - p ) ~,6 1 2 5

2 2 ~ 2+ [ ( ~ + N + ~ ) ~ - - N ] h 3 ~ '

8/13/2019 1-s2.0-073519339190082F-main

5/9

V oL 18, N o . 5 N O N L I N E A R T H E O R Y O F F I L M R U P T U R E 7 09w h e r e

= qo: 3 S h , : = .

B y l e t t i n g t h e z e r o t h o r d e r s o l u t i o n s (1 0 ) a s t h e a p p r o x i m a t i o n s o lu t io n s o f e q u a t io n s ( 1 ) -(6 ) , one can de r i ve t he evo l u t i on equa t i on by subs i t u t i ng so l u t i ons (10 ) i n t o t he equa t i on(7 ) , w h i ch r e su l t s i n

h t + f ( ~ ) [ A ( h - h . ) S ( h 3 h z z . ) ] . O , I I )w h e r e

- 3f / ~ ) = ~ - [ ~ + 2/~ + 2 ) : - 2].

I f w e i n t r o d u c e a r e s c a l i n g r a n s f o r m a t i o n a s

r = A 2 1 s ) t f ~ ) , (12)t h e n t h e e v o l u t io n e q u a t i o n c a n b e t r a n s f e rr e d i n t o t h e f o l lo w i n g f o r m

h , + [ (h -*h~ )~ + (h 3 h~ ) ] ~ = 0 . (13 )N o t e t h a t equ a t i on (13 ) i s i den t i c a l t o t ha t o f cons t an t v i s cos i ty ca se in [7 ] . A n d t hed i f fe r ence o f re sca l i ng t r ans fo rm a t i on be t w een t he p re sen t s t ud y w i t h t h e s t u dy i n [7] i st he no n l i nea r va r i ab l e v i s cos i ty f ac t o r , f (~ ) , show n i n exp re s s i on (12 ).

3 . R E S U L T S A N D D I S C U S S I O N S

E qua t i on (13 ) ha s been s t ud i ed by m any r e sea rche r s [6 ] - [8 ] t o p red i c t t he l i nea r andnon l i nea r ru p t u re t i m es o f t h i n l i qu id f i] rn . H ence , t he fo l low i ngs w i l l f ocuse s on t he e f fec to f v i s cos i ty va r i a t i on o n t he m e chan i sm o f rup t u re p roces s. F IG .2 show s t ha t t he v i scos i t yva r i a t i on f ac t o r , f( / 3 ) , i s sm a l l e r (l a rge r ) t han o ne w hen / ~ i s pos i t i ve (nega t ive ) . F romt h e t e m p o r a l r e s c a l in g f o r m , r = A 2 / ~ ) t J : ~ ) , i t i s a p p a r e n t t h a t t h e t r u e r u p t u r e t im e ,(A 2 / s ) t c , i s p rop o r t i on a l t o f -1 ( / 3 ) . T he re fo re w e can dedu ce d i r ec t l y t ha t a coo l i ng(hea t i ng ) f rom t he w a l l ,/ 3 > 0 ( /~ < 0 ) , w i ll dece l e r a te ( acce l e r a t e ) t he ru p t u re p roces s. I t

8/13/2019 1-s2.0-073519339190082F-main

6/9

710 M.-C. Wu and C. -C. Hw ang Vol . 18, No. 5

s h o u l d b e p o i n t e d o u t t h a t f ( f l ) is a n o n l i n e a r f u n c t i o n o f f l w h i c h i s d i ff e re n t f r o m t h a t o fR e i s f el d a n d B a n k o f f ' s s t u d y [ 10 ] w h e r e a l i n e a r m o d e l i s u s e d t o s i m u l a t e t h e d e p e n d e n c eo f v is c o s it y o n t e m p e r a t u r e .

I t i s k n o w n f r o m t h e p r e v i o u s s t u d i e s [ 6 ]- [8 ] t h t t h e d o m i n a n t u n s t a b l e d i s t u r b ra n c e so f t h e i s o t h e r m a l t h i n f il m s y s t e m s a r e lo n g w a v e m o d e s . W h i l e G o n s s i s e t a l. [ 1 2 ] h a v er e v e a l e d t h a t , i n t h e c a s e s o f co o l in g f r o m t h e p l a n e , t h e s h o r t w a v e m o d e s c o u l d b e th em a j o r u n s t a b l e d i s t u r b r a n c e s o f t h e u n i s o t h e r r a a l f a il in g f i lm s y s t e m s . S i n c e o u r r e su l t sa r e d e r i v e d f o r t h e l o n g w a v e s it u a t io n , w e w i ll s t u d y t h e r u p t u r e p r o b l e m w i t h s h o r t w a v ed i s t u r b r a n c e s i n t h e f u t u r e w o r k .

4 . C O N C L U S I O NI n t h i s w o r k w e c o n c e n t r a t e o n t h e e f f e ct s o f v i s c o s i ty v a r i a t i o n o n t h e m e c h a n i s m o f

r u p t u r e o f a th i n l iq u i d f il m . A f t e r d e r iv i n g a lo n g w a v e n o n l i n e a r e v o l u t io n e q u a t i o n a n di n t r o d u c i n g a s u i t a b l e r e s c a li n g f a c t o r , o n e c a n d e d u c e d i r e c t l y t h a t a c o U in g ( h e a t in g )f r o m t h e w a l l w i ll in c r e a s e ( r ed u c e ) t h e r u p t u r e t i m e f r o m o u r r e s u l ts .

5 . A C K N O W L E D G M E N T ST h e a n t h e r s w i s h t o ac k n o w l e d g e t h e fi n a n c ia l s u p p o r t ( G r a n t N o .

E 0 3 3 - 0 2 ) o f t h e N a t i o n a l S c ie n c e C o u n c i l o f t h e R e p u b l i c o f C h i n a .NSC 79-0401-

6 . N O M E N C L A T U R EA d i m e n s io n l e ss H ~ k e r c o n s t a n tA d i m e n s i o n a l H a m k e r c o n s t a n t

r A r r h e n i u s n u m b e rh f i lm th i cknessP p r e s u r eP r P r a n d t l n u m b e r

8/13/2019 1-s2.0-073519339190082F-main

7/9

V oL 18, N o . 5 N O N L I N E A R T H E O R Y O F F I L M R U P T U R E 71 1

S d i m ens i on l e s s su r f ace t en s i on ;T t e m p e r a t u r et t i m et c r u p t u r e t i m eu v e l o c it y i n t h e x - d i r e c t i o nv v e l o c i ty i n t h e y - d i r e c t i o nz , y spa t i a l coo rd i na t e sG r e e k S y m b o l sa t he rm a l d i i f~ s i v it y o f l i q u i d/~ pa ram e t e r i nd i ca t i ng t he g rad i en t o f v i s cos i ty

s m a l l w a v e n u m b e rre sca l i ng spa t i a l coo r i dna t ev i s c o s i t y

a s u r f a c e t e n s i o nr r e s c a l i n g d i m e n s i o n l e s s t i m e

S u b s c r i p t s

x , y , t d i f f e r e nt i a t i o n . r . t , x y ts quan t i t ie s a t t he f r ee su r f acew qua n t i t i e s a t t he p l ane

S u p e r s c r i p t s- m e a n s t a t e* d i m en s i ona l qua n t i t i e s

7 . R E F E R E N C E

[1] . A . Vr i j and J . Th . G . Overbeek , J . Amer . Chem. Soc . 90 , 3074 (1968) .[2] . B . U . Fe lderhof , J . Chem. Phys . 49 , 44 (1968) .

8/13/2019 1-s2.0-073519339190082F-main

8/9

712 M.-C. Wu and C.-C. Hw ang Vol. 18, No. 5

[3] . R. K . Jain, C. M aldarell i , mad E. Ruck enstein, A ICh E Syrup. Set . Biorheol . 74, 120(1978).

[4] . P. M. Bisch and A. Sanfeld, Bioelectrochem. Bi0energ. 5, 401 (1978) .[5] . C. Maldare l l i and R. K . Jain, J . Col loid Interface Sci. 90, 263 (1982) .[6] . E. Ru cke nstein and R . K. Jain, C hem. Soc. F ura dy Trans . 2, 70, 132 (1974).[7] . Malcolm B . W il l iams and Step hen H. D avis, J . Col loid Interface Sci. 90, 220 (1982) .[8] . A. S ha rn ~ and E. R ucken stein, J . Col loid Interface Sci. 113, 456 (1986).[9]. J . P . Bure lbach , Ph .D . thesi s , Chemica l Engineer ing De par tm ent , Nor thwes te rn Uni-

versi ty, Evanston, IL. (1988) .[10] . B. Reisfeld and S. G. Bankoff , AIChE Annual Mtg., San Francisco, CA. (1989) .[11]. D. Goussis and R. E. Kelly, Physics Fluids. 28, 3207 (1985).[12]. D. Goussls and R. E. Kelly, Physics Fluids. 30, 974 (1987).[13]. Chi-C hua n Hw ang and Cheng-I . Weng, Int . J . H eat M ass Trans. 31, 1975 (1988).

fh = l a

FIG. 1 The d i ag r am o f a t h i n l i qu i d f i l m sys t em

8/13/2019 1-s2.0-073519339190082F-main

9/9

V o l. 1 8, N o . 5 N O N L I N E A R T H E O R Y O F F IL M R U P T U R E 7 1 3

4 5t~43.5-3

~-25-.2-

1.5-1-

0.50 -2 - ; ~ - ~ - ~ .~ 6 : 5 i

FIG. 2 The relation betweenf{~) with