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J O UR , A t orC O M P U T A T I O N A L A N DAPPLIED MATHEMATICS
E L SE V IE R J o u r n a l o f C o m p u t a t i o n a l a n d A p p l i e d M a t h e m a t i c s 6 0 ( 19 95 ) 3 3 1 - 3 4 6
E l e c t r o h y d r o d y n a m i c K e l v i n - H e l m h o l t z i ns ta b il it y fo r a v e lo c i t ys tra t i f i ed f lu id
A. A. Mo h a m e d , E . F . E l s h e h a we y , M. F . E 1 - S a y e d *Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt
R e c e i v e d 3 0 A u g u s t 1 9 9 3 ; r e v i s e d 1 2 J a n u a r y 1 9 94
Abstract
T h e e l e c t r o h y d r o d y n a m i c s t a b i l i t y o f a c o n d u c t i n g i n c o m p r e s s i b l e s t r a ti f ie d f l u id t o p p i n g a d i e le c t r ic f lu i d l a y e r i ss t u d i e d . T h e s t a b i l i t y o f t h e s y s t e m i s d i s c u s s e d t h e o r e t i c a l l y a n d n u m e r i c a l ly . I t i s f o u n d t h a t t h e n o r m a l e l e c t r ic f ie l d h a sa d e s t a b i l i s i n g e f f ec t w h i l e t h e i n c r e a s e o f t h e t h i c k n e s s o f t h e l a y e r h a s a s t a b i l i s in g i n f l u e n ce . T h e s p e c i a l c a s e o f th eR a y l e i g h - T a y l o r i n s t a b i l i t y i s a l s o e x a m i n e d .Keywords: L i n e a r h y d r o d y n a m i c s ta b i li ty ; E l e c tr o h y d r o d y n a m i c s ; K e l v i n H e l m h o l t z a n d R a y l e ig h T a y l o r i n s ta b il it ie s ;D i e l e c t r i c a n d c o n d u c t i n g f l u i d s ; S t r a t i f i e d f l u i d s ; C o n f l u e n t h y p e r g e o m e t r i c f u n c t i o n ; S t u r m ' s f u n c t i o n s
1 . I n t r o d u c t i o n
T h e s t a b i l it y o f a n i n t e r fa c e s e p a r a t i n g t w o s t r e a m i n g f lu i d s h a s r e c e i v e d c o n s i d e r a b l e a t t e n t i o no f i n v e s t i g a t o r s d u e t o i t s p h y s ic a l a p p l i c a t i o n s . A r e v i e w o f c l a ss ic a l K e l v i n - H e l m h o l t z t h e o r y isp r o v i d e d i n [ 5 ] . T h e t h e o r y i s l im i t e d i n t h a t i n m o s t p h y s ic a l s it u a ti o n s t h e t w o f lu i d c o m p o n e n t sa r e n o t m o v i n g w i t h c o n s t a n t v e l o c i ti e s [ 1, 9, 2 9 ] . T h u s o n e h a s t o c o n s i d e r f lo w s p o s s e s s i n gv e l o c i t y s t r a t i f i c a t i o n a n d t h e r e s u l t s o f t h e t h e o r y o f s t a b i li t y o f p a r a l l e l f l o w I -3 , 6 , 2 7 ] a r e b r o u g h ti n t o a c t i o n d u e t o t h e v e l o c i t y s t ra t if ic a t i o n s . R e c e n t w o r k s o n t h e s t a b i li ty o f s u p e r p o s e d f l u id st h a t a r e i n i t i a ll y s t r e a m i n g w i t h v a r i a b l e v e l o c i ti e s [ 1 5 , 2 0 - 2 5 ] s h o w d i f fe r e n t r e s u lt s t h a n t h o s e o ft h e c la s s ic a l K e l v i n - H e l m h o l t z i n s ta b i l it y .O n t h e o t h e r h a n d , i n c r e a s i n g i n t e r e s t i n t h e e l e c t r o h y d r o d y n a m i c s t a b il it y is d u e t o t h ei m p o r t a n t r o l e p l a y e d b y e l e c t r ic f ie ld s in b i o p h y s i c s 1 -3 2], c h e m i c a l e n g i n e e r i n g [ 1 3 , 1 4 ] , a n d o t h e rd o m a i n s o f i n t e r e s t [- 4, 8, 1 0 , 1 1, 2 3, 2 8, 3 0 ] . T h e p r e s e n c e o f a n e l e c t r i c f i e ld p r o d u c e s e l e c t r ics t re s s e s o n t h e i n t e r f a c e s e p a r a t i n g t w o d i e l e c t r i c f lu i d s . I n t h e l i n e a r s t a b i l it y t h e o r y , t h e t a n g e n t i a lf ie l d h a s a s t a b i l i s i n g e f fe c t [- 1 3 ], w h i l e t h e n o r m a l f ie l d h a s a d e s t a b i l i s i n g i n f lu e n c e [ 1 3 , 1 6 , 1 9 ].* C o r r e s p o n d i n g a u t h o r .037 % 0427 / 95 / $09 .50 1995 E l s e v i e r Sc i e nc e B .V. Al l r i gh t s r e s e r ve dSSDI 0 3 7 7 - 0 4 2 7 ( 9 4 ) 0 0 0 4 8 - 6
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33 2 A.A. Mohamed et al./Journal of Computational and Applied Mathematics 60 (1995) 331 346H o w e v e r , r e c e n t s t u d ie s o f t h e n o n l i n e a r e l e c t r o h y d r o d y n a m i c s t a b i li ty s h o w e d t h a t b o t h t h en o r m a l f ie l d a n d t h e t a n g e n t i a l f i e l d p l a y r o l e s i n s ta b i l i ty c r i t e r i o n [ 17 , 1 8 ].
T h e a i m o f t h i s w o r k i s t o s t u d y t h e e f f e c t o f a n o r m a l e l e c t ri c f ie l d o n a d i e l e c t r ic f l u id l a y e rt o p p e d b y a s t r e a m i n g c o n d u c t i n g f lu i d w i t h s i n u s o i d a l b o u n d a r y w a v e p r o fi le . T h e l in e a r s t a b il i t yo f t h e i n te r f a c e w i ll b e e x a m i n e d .
2 . F o r m u l a t i o n o f t h e p r o b le mT h e b a s i c f l o w i s a s s u m e d t o b e t h e s t e a d y f l o w o f t w o i n v is c i d in c o m p r e s s i b l e f l u id s in
a g r a v i t a t i o n a l f ie ld . A r e c t a n g u l a r c o o r d i n a t e s y s t e m is u s e d , t h e c o o r d i n a t e a x e s x - y a s s h o w n inF i g . 1 , w i t h o r i g i n a t t h e i n t e r f a c e .
T h e u p p e r f l ui d is o f d e n s i t y P 2 (Y ) a n d e x t e n d s t o i n f i n it y a n d i t i s a s s u m e d t o b e a c o n d u c t i n gi n v is c i d i n c o m p r e s s i b l e f l u i d w h o s e d e n s i t y a n d v e l o c i ty in t h e s t a t i o n a r y s t a te a r e g i v e n b y
p~)2) = P2(Y ) = P0 e-B y, b/~ ) = U 2 ( y ) = Ay a n d U~ ) = 0 ,w h e r e u a n d v a re t h e v e l o c i t y c o m p o n e n t s i n th e d i r e c t i o n s o f i n c r ea s i n g x a n d y a n d P o , / / a n dA a r e p o s i t i v e c o n s t a n t s .
T h e l o w e r f l u id is o f d e n s i t y P l a n d d e p t h L , a n d i t i s a s s u m e d t o b e a d i e l e c tr i c in v i s c idi n c o m p r e s s i b l e f l u i d a t r e s t . T h u s
p ~ l) = P l = c o n s t a n t , u~) ) = U l = 0 a n d /)~1) = 0 .T h e lo w e r f lu i d is b o u n d e d f r o m b e l o w b y a r i g id c o n d u c t i n g p l a n e h a v i n g p o t e n t i a l V o a t y = - L .
j ,
u ( 2 } o = yv ! 2 ) = 0p ( 2 ) = p o e - 6 y
T { 2 )
x
I E o" ! 1 ) : 1! l ) = oP ~ I ) = o 1
( i )
/ 1 1 1 1 / / / / / / / / I / I I l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
Fig. 1 . De finit ion sketch of the problem.
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A.A . M oham ed e t aL /Journa l o f Com puta tiona l and App l ied Mathem at ics 60 (1995) 331-346 33 3I n t h e s t a t i o n a r y s t a t e , t h e i n t e r f a c e i s g i v e n b y y = 0 , a n d t h e l o w e r f l u id o n l y i s s u b j e c t e d t o
a c o n s t a n t e l e c t r ic f ie l d E o a n d d i r e c t e d p e r p e n d i c u l a r t o t h e i n t e r f a c e , i.e . E o = E o j , w h e r eE o = V o / L . A l s o t h e g r a v i t a t i o n a l f o r c e is t a k e n i n t o a c c o u n t i n th e t w o r e g i o n s , a n d t h e r e e x is ts u r f a c e c h a r g e s o n t h e in te r fa c e . T h e e q u a t i o n s g o v e r n i n g t h e m o t i o n a r e t h e u s u a l e q u a t i o n s o ff lu i d m e c h a n i c s a l o n g w i t h t h e e q u a t i o n s g o v e r n i n g t h e e l e ct ri c fi el d.
T h e e q u a t i o n s o f m o t i o n a re :[ c~u"~ c~u"~ v" l ~u "~ 7 c~/7" 'p (l ) L - ~ + U(t) ~ X + --T-S-,, _j = d x ' (2 .1 )
p(l) F ~v(I) tOY(t) ~ v( l)]L- - + U l) ~ X "~- V l) --~-y 3 ~- OFt(l)~y p( / ) g , ( 2 . 2 )w h e r e t h e s u p e r s c r i p t s I = 1 , 2 r e f e r t o t h e l o w e r a n d u p p e r f l u id , r e s p e c t i v e l y, a n d t h e t o t a l p r e s s u r ei s d e f i n e d b y [ 1 2 ]
1 E 2 0~/7 = p - ~ 7pp ~p ' ( 2 . 3 )w h e r e t is t h e t i m e , p is p r e s s u r e a n d t h e s u b s c r i p t ~ m e a n s t h a t t h e d e r i v a t i o n is e v a l u a t e d a tc o n s t a n t t e m p e r a t u r e .
T h e e q u a t i o n o f c o n t i n u i t y is0u"~ 0v"~- - + - - = 0 . ( 2 . 4 )c~x c~y
B o t h t h e u p p e r a n d l o w e r f l u i d s a r e a s s u m e d t o b e i n c o m p r e s s i b l e , s o t h a tc~p.) c~p.) ~p(t)- - c q t + u " ) - - ~ - x + v " ) c ~ y = 0 . ( 2 . 5 )
I t is a s s u m e d t h a t t h e q u a s i - s ta t i c a p p r o x i m a t i o n i s v a l id f o r t h e p r o b l e m o n h a n d , a n d t h e r e f o r eM a x w e l l ' s e q u a t i o n s r e d u c e t o
V E = 0 , V . [ e E ] = 0 , ( 2 .6 )w h e r e e is t h e d i e l e c t r i c c o n s t a n t i n t h e l o w e r f l u i d.
3 . P e r t u r b a t i o n e q u a t i o n sA s u s u a l, t h e m o t i o n i s r e s o l v e d i n t o p r i m a r y m o t i o n a n d p e r t u r b a t i o n . H e r e t h e s u ff ix 0 is u s e d
t o d e n o t e v a l u e s in t h e u n d i s t u r b e d f lo w , w h i l e t h e s u b s c r i p t 1 w i ll re f er t o t h e p e r t u r b a t i o n s i nv a r i o u s q u a n t i t i e s , w e m a y a s s u m e :
u = u g ' + u t v + v t ' , 1 -1 tp(2) = p~o ) + p]2 ), E = Eo + E 1
i n w h i c h H ~ ) is t h e t o t a l p r e s s u r e f o r t h e p r i m a r y f lo w .
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3 3 4 A . A . M o h a m e d e t a l . / J o u r n a l o f C o m p u t a t io n a l a n d A p p l i e d M a t h e m a t i c s 6 0 (1 9 9 5 ) 3 3 1 - 3 4 6
T h e l i n e a r i z e d e q u a t i o n o f c o n t i n u i t y a l l o w s t h e u s e o f a s t r e a m f u n c t i o n q J(o , i n t e r m s o f w h i c hu~ a n d v~ s o l v i n g (2 .4 ) c a n b e e x p r e s s e d a s f o ll o w s :
u ( l l ) = i ll (1 ) ' U i/ )= _ ~u(0 , (3 .1 )w i t h s u b s c r i p t s i n d i c a t i n g p a r t i a l d i f f e r e n t ia t i o n .
W e a s s u m e a n e x p o n e n t i a l t im e f a c t o r fo r a ll p e r t u r b a t i o n q u a n t i t ie s , s u c h th a t[w (o ) - i (0 , ,(2 )n = [01(Y) , f t (Y) , h E ( y ) ] e i~(x-a) (3 .2 )t ,~ t l , V I J
i n w h i c h e = 2 n / 2 i s t h e w a v e n u m b e r , 2 b e i n g t h e w a v e l e n g t h , a n d c = c r + ic i i s t h e w a v e v e l o c i ty .T h e s t a b i l i t y o r i n s t a b i l i ty i s t h e n d e c i d e d b y t h e s i g n o f c i [ 1 9 ].
F o l l o w i n g t h e u s u a l p r o c e d u r e [ 5 ] , f r o m E q s . ( 3.1 ), (3 .2 ) a n d t h e l i n e a r i z e d f o r m o f E q s . (2 .1 ), (2 .2 )a n d (2 .5 ) w e o b t a i n t h e f o l l o w i n g t w o e q u a t i o n s f o r t h e u p p e r a n d l o w e r f lu i d , r e s p e ct iv e l y:
d 2 O 2 ( y ) / ~ d O E (y ) + _ ~2 + _ _ + 0 2 (y ) = O, (3 .3 )d y 2 d y ( A y - c ) ( A y - - C ) 2d 2 0 1 ( Y ) e 2 0 1 ( y ) = 0 . ( 3.4 )d y 2
W e c a n n o w i m a g i n e t h a t t h e e q u i l i b r i u m i n t e rf a c e s e p a r a t in g t h e t w o f lu id s is p e r t u r b e d a n d t h es u r fa c e o f t h e d e f o r m e d i n te r f a c e i s g i v e n b y
y = ~ = 6 e i ~ l x - a l ( 3 . 5 )w h e r e 6 is a s m a l l n e s s p a r a m e t e r h a v i n g t h e d i m e n s i o n s o f a le n g t h .
I f n i s t h e u n i t n o r m a l v e c t o r t o t h e d i s t u r b e d i n t e r f a c e y = 3 , t h e n t o t h e f ir s t o r d e r t e r m s , n isg i v e n b y
n = - i ~ i + j . ( 3 . 6 )F r o m t h e l i n e a r i z e d f o r m o f E q . (2 .6 ), E 1 i s a n i r r o t a t i o n a l v e c t o r , a n d t h e r e f o r e t h e r e e x i st s a ne l e c t r i c p o t e n t i a l q51 s u c h t h a t E ~ = - V ~b~ a n d
V2 tk l = 0 , (3 .7)w i t h t h e s o l u t i o n
s i n h e ( y + L )~bx = E o (3.8)s i n h ( e L )T h e r e f o r e , t h e t o t a l e l e c t r i c f i e l d i s g i v e n b y
E = E o { - k z ~ i + [ 1 - c t ~ c s h ~ ( y + L ) l }s i n h ( ~ L ) j , ( 3 .9 )w h e r e t h e p o t e n t i a l t h t v a n i s h e s a t t h e ri g id b o u n d a r y y = - L .
U s i n g t h e i n t e r f a c i a l c o n d i t i o n s :(i) T h e n o r m a l c o m p o n e n t o f t h e v e l o c i ty is c o n t i n u o u s a t t h e i n te r f a c e y = 3 , t h e n i t is r e q u i r e d
t h a t0 1 (0 ) = 0 2 (0 ). ( 3 . 1 0 )
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A . A . M o h a m e d e t a l . / J o u r n a l o f C o m p u ta t io n a l a n d A p p l i e d M a t he m a t ic s 6 0 ( 1 99 5 ) 3 3 1 - 3 4 6 335F r o m t h e l i n e a r i z e d k i n e m a t i c c o n d i t i o n s a t th e i n t e rf a c e y = 4 ,
0 4 a4v ] ') = W r + c t U '~ x x a t y = O , ( 3 .1 1 )i t f o l l o w s t h a t
4 = 0 1 (0__~) l, Ix_ c,). (3 .1 2)C(ii) T h e n o r m a l c o m p o n e n t o f t h e s t r e s s t e n s o r i s d i s c o n t i n u o u s a c r o s s t h e s u r fa c e o f s e p a r a t i o n
y = 4 b y t h e s u r f a c e t e n s i o n T . T h e n , t o t h e f i r s t - o r d e r t e r m s , w e o b t a i nC pl I /I ~ ( 0 ) - - P O [ C @ 2 ( 0 ) + A ~ /2 ( 0 ) ] + [ ( P o - P , ) ,q - - 0~2 T + a e E o c o t h ( ~ L ) ] ~ 1 (0 ) _ 0 . (3 .13)c
T h e v a n i s h i n g o f t h e n o r m a l c o m p o n e n t o f t h e v e lo c i t y v~1) a t th e r i g id b o u n d a r y y - - - Li m p l i e s t h a t
$ 1 ( - L ) = 0 . ( 3 .1 4 )
4 . T h e d i s p e r s io n e q u a t io nT h e s o l u t i o n o f E q s . (3 .4 ) a n d ( 3 .1 4 ) c a n b e w r i t t e n i n t h e f o r m
~ 1 (Y ) -- G s i n h c ~ (y + L ) ,w h e r e G is a n a r b i t r a r y c o n s t a n t .
T o o b t a i n t h e s o l u t i o n o f E q . (3 .3 ), w e m a y u s e t h e t r a n s f o r m a t i o n [ 5 ]@ 2 ( Y ) = W (Y ) e (1 /2 )~ y .
T h e n E q . (3 .3 ) c a n b e w r i t t e n i n t h e f o r mf 1 k - - m 2- - d 2 W ( O + k - ~ + ~ + - - - ~ ) W ( O = 0 ,
2
w h e r e
(4 .1 )
( 4 . 2 )
( 4 . 3 )
( 1 1 2 ( A )= (40~ 2 + f l 2 )1 / 2 ' m = _ Q , ( = y _ ( 4~ x2 + f l 2 ) 1 / 2a n d t h e R i c h a r d s o n n u m b e r Q is Q = g f l / A 2 .
W e r e c o g n i z e i n E q . ( 4.3 ) W h i t t a k e r ' s s t a n d a r d f o r m o f t h e e q u a t i o n f o r th e c o n f l u e n t h y p e r -g e o m e t r i c f u n c t i o n . T h e c o n d i t i o n a t i n f in i ty re q u i r e s t h a t t h e s o l u t i o n o f E q . ( 4.3 ) a p p r o p r i a t e t ot h e p r o b l e m o n h a n d b e W h i t t a k e r ' s f u n c t io n .
W = H W k , m (~), (4 .4)
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336 A.A. Mohamed et al./Journal of Compuhztional and Applied Mathematics 60 (1995) 331 - 346
w h e r e H is a n a r b i t r a r y c o n s t a n t . T h i s s o l u t i o n h a s t h e a s y m p t o t i c b e h a v i o u r [ 2 1, 3 1 ]:Wk,m(~) ~" (ke - (1 / 2) ; , (~ ~ ) .
F r o m E q s . (4 .2 ) a n d ( 4 .4 ), w e g e t~ t2 (y) = H e (a /E)ar W k,, , (~) . (4 .5 )
U s i n g E q s . ( 4 . 1 ) a n d ( 4 . 5 ) , t h e n t h e b o u n d a r y c o n d i t i o n ( 3 . 1 0 ) g i v e sG s in h ( e L ) - H W k , m - - -~ (4e 2 + 32 ) 1 /2 = 0 (4 .6 )
a n d t h e b o u n d a r y c o n d i t i o n ( 3 . 1 3 ) g i v e sG { c p x e c o s h ( e L ) + [ ( p o - p l ) g - e 2 T + e eE Z c o th ( eL ) ] s in h ! e L ) }
- H ( A + f i C ) p o W k , , . - ~ ( 4 ~ 2 + )1/2
, i t 1 } : o ,p o c ( 4 ~ 2 + 32 ) 1/2 W k, ,~ _ A ( 4 e 2 + )1 /2w i t h r e s p e c t t o t h e u n k n o w n s G a n d H , t h e n t h e s y s t e m o f E q s. (4 .6 ) a n d (4 .7) h a s a s o l u t i o nd i f fe r e n t f r o m z e r o , if t h e d e t e r m i n a n t o f c o e f fi c ie n t s i s e q u a l t o z e r o , a n d i t th e n f o l lo w s t h a t
- p o c ( 4 e 2 + 32 ) 1/5 s i n h ( e L ) W k , ,, - - ~ (4 e 2 + )1 /2
+ { c p l ~ C O s h ( e L ) + [ (Po - - P l )O -- e 2 T - ( A + C ) poC+ e e E o c o t h ( e L ) ] s in _ e L ) W k , ,, - - ~ ( 4 e 2 + 32 ) 1/2 = 0. (4 .8)
E q . ( 4 . 8 ) i s t h e d i s p e r s i o n e q u a t i o n c h a r a c t e r i s i n g c . A c c o r d i n g l y t h e s t a b i l i t y o r i n s t a b i l i t y i sd e t e r m i n e d t h r o u g h t h e s o l u t io n s f o r c r e s u lt in g f r o m t h e d i s p e r s io n e q u a t i o n . U n f o r t u n a t e l y , t h ea b o v e r e l a t i o n is r a t h e r c o m p l i c a t e d i m p l i c i t t r a n s c e n d e n t a l e q u a t i o n . T h e r e l a t i o n c a n b e s i m p li fi -e d c o n s i d e r a b l y f o r l a rg e w a v e n u m b e r s . I t is w e l l k n o w n t h a t t h e b e h a v i o u r f o r t h e W h i t t a k e r ' sf u n c t i o n f o r l a r g e a r g u m e n t s h a s t h e p r o p e r t y [ 2 5 ] :
W L , , ( Z ) ~ - W k , , , ( Z ) , la rg Z I < ~2rt (4 .9 )a n d t h e r e f o r e , t h e d i s p e r s i o n e q u a t i o n ( 4 . 8 ) r e d u c e s t o t h e s i m p l i f i e d f o r m
c 2 [ P o ( 4 ~ 2 + f l2)1/2 + p l ~ c o t h ( e L ) - f l P o ] - c A p o+ [ ( P o - - P x ) g -- ~ 2 T + e ~ E ~ c o t h (~ L ) ] = 0 . (4 .10)
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A.A. Mo hamed et al . /Journal o f Computational and Applied Mathematics 60 (1995) 331 346 33 7E q . (4 .1 0 ) is q u a d r a t i c i n c . F r o m t h e n a t u r e o f t h e r o o t s , w e f i n d t h a t t h e p r i n c i p a l o f
o v e r s t a b i l i t y is v a li d . T h e m a r g i n a l s t a t e is g iv e n b y2 * = [2 p o( 4C 2 + f l2 )~ /2 + 4 p l ~ c o t h ( 0 c L ) - 2 / ~ p o ] [ ( p o - P , ) g - ~ 2 T
+ c te Eo c o t h ( ~ L ) ] - A Z p ~ = O. (4 . 1 1 )T h e s y s t e m is s t a b l e i f 2 " < ~0 . T h e e q u a l i t y o f th e r e l a t i o n ( 4 .1 1 ) c a n b e e x p r e s s e d i n t h e
f o r mE ~ 2 = E o _ t a n h ( ~ L ) { A Z p ~
u e [ 2 p o x / 4 u 2 + / ~ 2 + 4 p l cZ c o t h (c c L ) - 2 / ~ po ]
- [ ( p o - p , ) g - T ] } . ( 4 . 1 2 )F o r v a l u e s o f t h e f i e l d s u c h t h a t E o ~< E ~ ', t h e s y s t e m is s t a b l e . T h e v a l u e s o f t h e c r i t ic a l
f ie ld E * d e p e n d s o n t h e r a t io o f t h e d e n s it ie s , t h e w a v e n u m b e r a n d t h e t h i c k n e s s o f t h e l a y e r L .T h e s u r f a c e t e n s i o n p l a y s a s t a b i l is i n g r o l e . I f P o > P l , t h e n s t a b i l i t y is p o s s i b l e i f E o < E~ 'u p t o a c r i t i c a l v a l u e o f t h e d i f f e r e n c e (P o - P t ). I f t h e d i f f e r e n c e (P o - P 1 i s v e r y l a r g e , t h e n t h et e r m - ( Po - P l ) i n E q . (4 .1 2 ) d o m i n a t e s a n d s t a b i l i t y i s n o t p o s s i b l e f o r a n y v a l u e o f t h e e l e c t r i cf i e l d . T h u s i f
( p o - p , ) > [ g { 2 p oA 2 p ~ + - - ,
x / 4 ~ 2 + / ~ 2 + 4 p , ~ c o t h ( ~ L ) - 2 /3 po }t h e s y s t e m is u n s t a b l e r e g a r d l e s s o f t h e v a l u e o f E o . I f t h e l o w e r f lu i d is d e n s e r , t h e n ( Po - P l ) isn e g a t i v e a n d o n e c a n a l w a y s f in d a n e l e c t r ic p o t e n t i a l s u c h t h a t E o < E ~' a n d s t a b i l i ty is a c h ie v e d .
5 . N u m e r i c a l d is c u s s i o n
F r o m E q . (4 .1 2 ) w e c a l c u l a t e th e v a l u e s o f t h e e l e c tr i c f ie ld s q u a r e E o c o r r e s p o n d i n g t o s o m eg i v e n v a lu e s o f t h e w a v e n u m b e r ~ f ro m ~ = 1 0 - 2 t o ~ = 3 0 00 f o r v a r io u s c o n s t a n t v a l u es o f t h et h i c k n e s s L o f t h e l o w e r f lu i d. T h e o t h e r p a r a m e t e r s a r e t a k e n t o b e : g = 9 .8 m / s 2, P o = 1 .2 k g / m 3,p l = 9 9 8 . 2 k g / m 3 , T = 0 . 0 7 2 8 N / m , e = 7 8 .5 4 , A = 1 .0 a n d f l = 1 .1 x 1 0 - 6 .
T h e r e l a t i o n ( 4 .1 2 ) is d r a w n b e t w e e n t h e e l e c t r ic f ie l d s q u a r e E ~ a n d t h e w a v e n u m b e r ~ f o r g i v e nv a l u e s o f t h e t h i c k n e s s L o f t h e l o w e r f l u id . T h e r e s u l t i n g c u r v e s in F i g s . 2 - 4 r e p r e s e n t t h e n e u t r a lc u r v e s o r t h e m a r g i n a l s t a te s e p a r a t i n g t h e s t a b le a n d u n s t a b l e r e g io n s . F o r a g iv e n c u r v e , w eo b s e r v e t h a t t h e s t a b l e r e g i o n is d e c r e a s e d b y t h e i n c r e a s e o f ~ ti ll a c r i ti c a l p o i n t ( ~c , E ~ c ) a f t e rw h i c h t h e s t a b l e r e g i o n i s i n c r e a s e d b y t h e i n c r e a s e o f 7, e .g . f o r L > / 0 .0 3 ; ~ ~ 3 6 6 .1 4 a n dEoZ ~ 0 . 6 71 4 9 5 8 , a n d a s L d e c r e a s e s t h e v a l u e s ~ a n d E o2 a l s o d e c r e a s e . W e o b s e r v e f r o m t h ef i g u re s t h a t t h e r e a r e s t a b l e r e g i o n s u n d e r t h e c u r v e s . T h u s t h e r e a r e s m a l l v a lu e s o f E o f o r w h i c hi n s t a b i l i t y is n o t p o s s i b le : a s E (] i n c r e a s e s a n u n s t a b l e r e g i o n i s r e a c h e d ; a n d t h e s y s t e m c a n b eb r o u g h t i n t o t h e u n s t a b l e s t a t e f o r v a l u e s o f E o > E oZ c. W e a l s o o b s e r v e t h a t t h e s t a b l e a r e a s
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33 8 A.A. M ohamed et a l . / Jou rn a l o f Computa t iona l and A pp l ied M athemat i cs 60 (1995) 331 -34 6
E o 2
1 0 0 0 , 6
8 0 0
6 0 0
4 0 0
2 0 0
' ' I . . . . I , , ' ,(o )
, , , , I . . . . I . i i i
1 2
. . . . I ' ' ' , I ' ' ' '
( b )
0 0 , , , , I , , , , I . . . .0 3 0 5 0 0 1 0 0 0 1 5 0 0
F ig . 2 . V a r ia t io n o f E 0 w i th th e w a v e n u m b er ~ fo r v a r io u s v a lu es o f th e l iq u id d ep th L fo r a sy s tem h a v in g O = 9 .8 m /s 2 ,P o = 1 .2 k g / m 3 , P l = 9 9 8 . 2 k g / m 3 , T = 0 . 0 7 2 8 N /m , s = 7 8 . 5 4 , A = 1 .0 a n d f l = 1 .1 x 1 0 - 6. U a n d S d en o te u n s ta b le a n ds ta b le reg io n s a b o v e a n d u n d er ea ch cu rv e , resp ec t iv e ly , w h ere (a ) th e so l id , d a sh ed , d o t ted -d a sh , d o t ted , 3 d o t ted -d a shcu rv es co rres p o n d to th e v a lu es L = 2 , 4 , 6 , 8 , 1 0 , resp ec t iv e ly , (b) a ll th e a b o v e cu rv es co in c id e
8 0 ' ' ' ' I ' ' ' ' I . . . . I . . . .
60
Eo2 4 0
2 0
\" . . . " \
" . \
Si l l l
0 5 1 0 1 5 2 0t3 t
F ig . 3 . F o r th e sy s tem co n s id ered in F ig . 2 , b u t w i th th e so l id , d a sh ed , d o t ted -d a sh , d o t ted , 3 d o t ted -d a sh cu rv esco rr esp o n d in g to th e v a lu es L = 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 , resp ec t iv e ly , a n d b y in crea s in g ~ , w e ca n g e t a cu rv e s im i la r to th a t inFig . 2 (b) .
S i n c r e a s e w i t h t h e i n c r e a s e o f t h e t h i c k n e s s L . T h u s t h e i n c r e a s e o f th e t h i c k n e s s y i e l d s a s t a b i l i s i n gi n f l u e n c e . T h e u p p e r b o u n d s o f t h e e le c t r i c f i e ld r e q u i r e d f o r s ta b i l i ty a r e in c r e a s e d b y t h e in c r e a s eo f L f o r f i x e d ~ . T h e u p p e r v a l u e s o f t h e e l e c tr i c f i el d a n d t h e c o r r e s p o n d i n g v a l u e s o f L a r e g i v e n i nT a b l e 1 f o r ~ = 1 0 - 2 .
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A.A. Mo hamed et al . /Journal o f Computational and Applied Mathematics 60 (1995) 331 -346 3 3 9
Eo2
6 , 3
0 S ' ~b0 2 0 0
I I ( o )
I i I i4 0 0 6 0 0 8 0 0
. . . . I . . . . I . . . .
u9/ /~ . ~ ; , i ~ J / / S. . . . I . . . . I , i i i1 0 0 0 2 0 0 0 , 3 0 0 0
F i g . 4 . F o r t h e s y s t e m c o n s i d e r e d i n F i g . 2, b u t w i t h t h e s o l id , d a s h e d , d o t t e d - d a s h , d o t t e d , 3 d o t t e d - d a s h c u r v e sc o r r e s p o n d i n g t o t h e v a l u e s L = 0 . 0 1 , 0 .0 2 , 0 .0 3 , 0 .0 4 , 0 .0 5 , r e s p e c t i v e l y , i n (a ) , a n d t h e v a l u e s L = 0 . 0 0 1 , 0 . 0 0 2 , 0. 00 3 , 0 .0 0 4 ,0 .0 0 5, r e s p e c t i v e l y , i n ( b) , a n d i t s h o w s t h e d e s t a b i l i s i n g i n f l u e n c e o f t h e e l e c t r i c f i e ld a n d t h e s t a b i l i s i n g e ff e c t o f t h e l i q u i dd e p t h .
T a b l e 1T h e t a b l e s h o w s t h e u p p e r v a l u e s o f a s t a b l e f ie l d E 2 c o r r e s -p o n d i n g t o t h e t h i c k n e s s o f t h e f l u id l a y e r L f o r a g i v e n~ = 1 0 - 2L E o 2 1 ~ = ~ 0 2 L E Z l ~ = l o 2
0 . 0 0 0 0 1 0 . 0 0 1 2 4 4 0 0 . 0 5 6 . 2 2 0 1 4 2 00 . 0 0 0 5 0 . 0 6 2 2 0 1 4 0 . 1 1 2 . 4 4 0 2 80 90.001 0 .1244029 0 .2 24 .88053890 . 0 0 2 0 . 2 4 8 8 0 5 7 0 . 3 3 7 .3 2 0 7 4 2 80 . 0 0 3 0 . 3 7 3 2 0 8 6 0 . 4 4 9 . 7 6 0 8 7 3 40 . 0 0 4 0 . 4 9 7 6 1 1 4 0 .5 6 2 . 2 0 0 9 0 9 40 . 0 0 5 0 . 6 2 2 0 1 4 2 2 . 0 2 4 8 .7 7 2 5 83 00 . 0 1 1 . 2 4 4 0 2 85 4 . 0 4 9 7 . 3 4 6 2 4 0 20 . 0 2 2 . 4 88 0 5 6 9 6 . 0 7 4 5 . 5 2 2 9 4 9 20 . 0 3 3 . 7 3 2 0 8 5 4 8 .0 9 9 3 . 1 0 5 4 6 880 . 0 4 4 . 9 7 6 1 1 3 5 1 0 . 0 1 2 3 9 . 89 8 6 3 2 8
T h e a n a l y s i s b e c o m e s m o r e o b v i o u s w h e n w e d i sc u s s th e s y s t e m i n t h e L - ~ p l a n e f o r f ix e d E 2 .T h e c u r v e is m u l t i b r a n c h e d a n d t h e d e t e r m i n a t i o n o f t h e a s y m p t o t e t o t h e c u r v e i s n e c es s ar y. T h ep r o c e d u r e i s t o s o l v e t h e e q u a l i t y o f (4 .1 2) f o r c o t h ( ~ L ) a n d h e n c e f o r L . T h e a s y m p t o t e s a r eo b t a i n e d b y l e t ti n g c o t h ( ~ L ) = 1 w h i c h r e s u lt s in t h e s i x t h - o r d e r a l g e b r a ic e q u a t i o n :
A l o ~ 6 + A 2 ~ 5 + A 3 ~ 4 + A 4 ~ 3 + A 5 ~ 2 + A 6 ~ + A 7 = 0 , ( 5 .1 )
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340 A.A. Mohamed et al . /Journa l o f Computational and Applied M athematics 60 (1995) 331 -346w h e r e
A ~ = 4 T 2 ( 0 2 - 0 2 ) ,A 2 = 4 T { [ 3 T p o p ~ - 2 e E g ( p ~ - 0 2 ) } ,A 3 = - - 8 T ~ P o P , 8 E 2 - 4 ( 0 2 - P 2 ) { 2 T ( p o - P , ) g - 8 2 E g } ,A 4 = 8 0 8 E ~ ( p o - 0 , ) 2 ( 0 0 + 0 1 ) - 2 T A 2 p 2 p ~ - 4 ~ p o p ~ { 2 T ( 0 o - 01)0 - ~2Eo4},A s = 4 0 2 ( 0 0 - 0 , ) 3 ( 0 0 + P 1 ) + T A 2 0 ~ + 2 0 o0 1 ~E 2 o { A 2 0 o + 4 1 ~( 0o - - P l ) 0 } ,A 6 = 2 p o p l g ( p o - p l ) { A 2 p o + 213( po - P l ) g } - ~ A 2 R 3 o e E ~ ,A 7 = - A 2 p ~ { A 2 p o + 4 / ~( p o - 0 1 ) 0 } .
T h e a b o v e e q u a t i o n a d m i t s t w o p o s i t iv e re a l r o o t s , a n d T a b l e 2 s h o w s t h e c o m p u t e d v a l u e s o ft h e r o o t s e = eA , , UA2 ( c o r r e s p o n d i n g t o t h e a s y m p t o t e s ) f o r a g i v e n v a l u e o f E o . T h e o t h e rp a r a m e t e r s o f t h e s y s t e m a r e a s t a b u l a t e d b e f o re . F ig s . 5 a n d 6 s h o w t h e s y s t e m in t h e L - e p l a n e f o rd i f f er e n t v a l u e s o f E g , a n d f r o m w h i c h , w e o b s e r v e t h a t f o r a g i v e n v a l u e o f E o , t h e s t a b l e r e g i o n S isr e d u c e d ( o r i n c r e a s e d ) b y t h e i n c r e a s e o f u < CA1 ( o r e > C A 2). F o r e > e A , a n d e < C A, s t a b i l i t y i sn o t p o s s i b l e f o r t h i s v a l u e o f t h e e l e c t r i c f i el d , a n d t h e v a l u e o f CA , ( o r C A 2) d e c r e a s e s ( o r i n c r e a s e s )w i t h t h e i n c r ea s e o f E o a l l o w i n g m o r e u n s t a b l e r e g i o n s t o a p p e a r a s s h o w n i n F ig s . 5 a n d 6 . T h i se m p h a s i s e s t h e d e s t a b i l i s in g i n f l u e n c e o f t h e e l e c t r i c f i el d .
W e o b s e r v e f r o m t h e f i g u re s th a t t h e r e a r e u n s t a b l e r e g i o n s u n d e r t h e c u r v es , a n d t h e s e re g i o n sa r e n o t t h e s a m e f o r th e s a m e v a l u e o f E o , b u t i t d e p e n d s o n w h e t h e r e < C a, o r e > C a2 , w h e r e t h ec u r v e s c o r r e s p o n d i n g t o a g i v e n v a l u e o f E 2 w i t h e > CA2 a r e t o o c l o s e t o t h e e - a x i s m o r e t h a n t h e
Table 2The table g iv es the pos i t ive rea lEq . (5 .1) for som e given va lue s of E ] roots of
E~ (~At ~A21.0 143.4865 935.35972.0 64.0885 2093.58623.0 42.0298 3194.52514.0 31.3328 4284.05655.0 25.0065 5366.9495
10 .0 1 2 . 4 5 8 0 1 0 7 7 6 . 0 0 7 820.0 6 . 2 2 4 4 2 1 5 7 0 . 7 0 1 63 0.0 4 . 1 4 7 3 3 2 3 7 1 . 7 9 2 64 0.0 3 . 1 1 0 3 4 3 1 5 0 . 7 3 8 55 0.0 2 . 4 8 7 7 5 3 9 3 4 . 6 9 1 16 0.0 2 . 0 7 2 5 6 4 7 2 3 . 2 6 5 170 .0 1 . 7 7 6 1 7 5 5 1 2 . 7 9 5 280 .0 1 . 5 5 3 9 8 6 2 9 2 . 3 2 2 39 0.0 1 . 3 8 0 8 9 7 0 6 8 . 8 3 5 11 00 .0 1 . 2 4 2 7 1 0 7 8 6 8 . 3 4 1 4
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A.A . M ohamed e t al . / Jou r na l o f Compu ta t iona l and App l ied M athemat ics 60 (1995 ) 331 -3 46 3 4 10 . 0 6
0 . 0 4
0 . 0 2
S Ii. I / I
: I,: I. . . " / '
/. , / '
' 1 , i , , , I ' ' ' ,, (o )Ii
4 x 1 0 - 4
Yu
I , , , , I . . . .
, 50 1 O0 150
2 x 1 0 - 4
0 ' ' ' ' 00 0
I ; ' I I
( b )
" S
\ \ \ " ' . . ' ~ , .\ , . . ,\
2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 04
F i g . 5 . S t a b i l i t y d i a g r a m i n t h e ( L - c 0 p l a n e f o r v a r i o u s v a l u e s o f t h e e l e c t r i c f i e l d s q u a r e E o f o r a s y s t e m h a v i n g0 = 9 . 8 m / s 2 , P 0 = 1 .2 k g / m 3 , P l = 9 9 8 - 2 k g / m 3 , T = 0 . 0 7 2 8 N / m , ~ = 7 8 . 5 4 , A = 1 . 0 a n d /~ = 1 .1 1 0 - 6 . T h e s o l i d ,d a s h e d , d o t t e d - d a s h , d o t t e d , 3 d o t t e d - d a s h c u r v e s c o r r e s p o n d t o t h e v a l u e s E 02 = 1 , 2 , 3 , 4 , 5, re s p e c t iv e l y , w h e r e ( a)r e p r e s e n t s t h e c u r v e s i n t h e r e g i o n ~ < ~ a , (b e f o r e t h e f ir s t a s y m p t o t e t o e a c h c u r v e ) , a n d ( b ) r e p r e s e n t s t h e s a m e c u r v e s i nt h e r e g i o n ~ > ~ a2 ( a f te r t h e s e c o n d a s y m p t o t e s ) , w h e r e t h e v a l u e s ~ A , a n d ~ a2 a r e g i v e n i n T a b l e 2 .
0 . 6
0 .4
0 . 2
II
iI
S iI : I. . " i. , " / "
//
' [ ' ,
( o )
SU
I , , , ,
1 0
6 x 1 0 - 5
4 x 1 0 - s
2 x 1 0 - 5
0 5 15
I I ' ! I ' I', i ( b >L iI i 5
~ ' ~ .\ " ~ ' . '
\ \ . ' ,
I t I i I2 x 1 0 4 4 x 1 0 4 6 x 1 0 4
F i g . 6. S t a b i l i t y d i a g r a m f o r t h e s a m e s y s t e m c o n s i d e r e d i n F i g . 5 , b u t w i t h t h e s o l i d , d a s h e d , d o t t e d - d a s h , d o t t e d ,3 d o t t e d - d a s h c u r v e s c o r r e s p o n d i n g t o E o = 1 0 , 2 0 , 3 0 , 4 0 , 5 0 , a n d f i g u r e s ( a ) a n d ( b ) a r e d r a w n f o r t h e v a l u e s o f ~ < C~A,a n d ~ > ~ a 2, r e s p e c ti v e l y , a n d t h e y s h o w a l s o t h a t t h e e l e c t r ic f ie l d h a s a d e s t a b i l i s i n g ef fe c t a n d t h e l i q u i d d e p t h h a sa s t a b i l i s i n g e f f e c t .
s a m e c u r v e s c o r r e s p o n d i n g t o t h e s a m e v a l u e o f E o w i t h ~ < ~ a , . T h u s t h e r e a r e s m a l l v a l u e s o fL f o r w h i c h s t a b i l i t y i s n o t p o s s i b l e , a n d a s L i n c r e a s e s a s t a b l e r e g i o n i s r e a c h e d ( f o r ~ > ~A 2 m o r eq u i c k l y t h a n f o r ~ < ~ a , w i t h t h e s a m e v a l u e o f E 0Z ); a n d t h e s y s t e m c a n b e b r o u g h t i n t o a s t a b l es t a t e f o r a g i v e n ~ < ~ a , o r ~ > ~ A2 "
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3 4 2 A.A. M oham ed e t a l . / Jou r na l o f C omputa t iona l and App l i ed Mathemat i c s 60 (1995) 33 1-3 466 . The Rayle igh-Taylor ins tab i l i ty
I t i s i n t e r e s t i n g t o e x a m i n e t h e a b o v e s y s t e m w h e n t h e u p p e r f l u i d i s a t r e s t ( i . e . t h eR a y l e i g h - T a y l o r i n s ta b i li ty ) , a n d t h i s c a n b e d o n e b y le t t in g A ~ 0 . U n f o r t u n a t e l y , t h e t r a n s f o r m a -t i o n fo r ( b r e a k s d o w n w h e n A - . 0 a n d c o n s e q u e n t l y t h e W h i t t a k e r e q u a t i o n a s g iv e n b y E q . (4 .3 )is n o t a p p r o p r i a t e f o r t h e p r o b l e m . T h u s t h e d i s p e r s io n e q u a t i o n ( 4 .8) a n d i ts c o n s e q u e n c e s a r e n o tv a l i d f o r t h e R a y l e i g h - T a y l o r i n s t a b i l i t y .H o w e v e r , t h e a n a l y s i s c a n b e r e s t a r t e d a t a s t a g e e a r l i e r t o E q . ( 4 .2 ). I f w e le t A ~ 0 i n E q . ( 3 .3 ) , w eg e t
d 2 ~ 2 ( y ) f l d ~ 2 ( y ) ~ 2 ( 1 g f l }d y E d ~ ( ~ c ) 2 ~ 2 ( Y ) - - 0 . ( 6.1 )T h e g e n e r a l s o l u t i o n o f E q . ( 6.1 ) is [ 2 6 ]
~2 (Y) -- H 1 e ~ '+y + HE e m-y, y > 0, (6.2)w h e r e H 1 a n d H 2 ar e a r b i t r a r y c o n s t a n t s , a n d
m + = + + : 1 ' : 2_ _ ( c ) 2 j
B o u n d a r y c o n d i t i o n s d i s a l l o w d i s t u r b a n c e s w h i c h i n c re a s e e x p o n e n t i a ll y a s th e o u t e r b o u n d o f t h ef lu i d is a p p r o a c h e d . T h u s ,~O2(y) = H 2 e m-r , y > 0 (6.3)
w i t h t h e r e q u i r e m e n t t h a t+ (Z 2 1 - - g f l " ( 1 1 / 2 f l (6.4)
( c ) 2 j j / >
u s i n g E q s . (4 .1 ) a n d ( 6 .3 ) , t h e n t h e b o u n d a r y c o n d i t i o n ( 3.1 0 ) g i v e sG s i n h ( ~ L ) = H 2 ( 6.5 )
a n d t h e b o u n d a r y c o n d i t i o n ( 3.1 3) gi v esG [ p l c c c o s h ( ~ L ) + { ( p o - P l ) g - c c 2 T + c c e E 2 o o t h ( c c L ) } si n h - ( c ~ L )
= H a P o c m - , (6.6)w i t h r e s p e c t t o t h e u n k n o w n s G a n d H a . T h e s y s t e m o f E q s . (6 .5 ) a n d ( 6 .6 ) h a s a s o l u t i o n d i f f er e n tf r o m z e r o i f t h e d e t e r m i n a n t o f c o e f fi c ie n t s is e q u a l t o z e r o , a n d i t t h e n f o l lo w s t h a t
'~ 1 5 - - ) ' 2 3 - - " ~3 2 " ~- 2 4 C + '~ '5 = 0 , (6.7)
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A . A . M o h a m e d e t a l ./ J o u r n a l o f C o m p u ta t io n a l a n d A p p l i e d M a t h e m a t i c s 6 0 ( 19 95 ) 3 3 1 - 3 4 6 343
w h e r e21 ~-2 2 =2 3 =2 4 =25 =-
{ P o f l - 2 p 1 ~ c o t h ( ~ L ) } 2,4 { P o f l - 2 p 1 ~ c o t h ( a L ) } { (P o - P l ) g - ~ 2 T + ~ eE 2o c o t h ( aL ) },po2(4a2 + f12) ,4 { ( p o - P , ) g - ~ 2 T + ~ e E o 2 c o t h (a L ) } 2 ,4 p ~ g f l .
T h e s t a b i l i t y o f t h e s y s t e m d e p e n d s o n t h e p r o p e r t i e s o f t h e r o o t s o f E q . ( 6.7) . F o r s t a b i l it y it isn e c e s s a r y a n d s u f f ic i e n t t h a t a l l t h e r o o t s o f E q . (6 .7) s h o u l d b e r e a l a n d d i s t i n c t [ 5 ] . I n o r d e r t oe x a m i n e th e n a t u r e o f t h e r o o t s w e c o n s t ru c t t h e S t u r m f u n c t io n s [ 2 ] f ( c ) , f t ( c) , f z ( c ) , f 3 ( c ) , f 4 ( c )a n d f 5 ( c) ( se e t h e A p p e n d i x ) .T h e r o o t s o f a n e q u a t i o n o f o r d e r n a r e r e a l a n d d i s t in c t , i f a n d o n l y if t w o c o n d i t i o n s a r es a t i s f i e d :
(i) T h e n u m b e r o f S t u r m ' s f u n c t i o n s m u s t b e (n + 1).( ii) T h e l e a d i n g c o e f f i c i e n t s o f a l l t h e s e f u n c t i o n s m u s t b e p o s i t iv e .N o w , f o r t h e E q . ( 6 .7) , t h e f i r s t c o n d i t i o n ( i) i s s a t is f ie d , a n d t h e s e c o n d c o n d i t i o n ( ii) l e a d s t o a s e to f in e q u a l i ti e s . T h e d e t a i ls o f t h e s e i n e q u a l i ti e s a r e v e r y le n g t h y [ 7 ] a n d w i ll n o t b e i n c l u d e d h e r e( a n d t h e y a r e a v a i l a b l e f r o m t h e a u t h o r o n r e q u e s t ) . T h e f ir s t i n e q u a l i t y is t r i v ia l l y s a t is f ie d , a n d t h es e c o n d o n e g i v e s
{ P o f l - 2 p t a c o t h ( a L ) } { ( P o - P l ) g - ~ 2 T + a e E ~ c o t h ( a L ) } > 0 . ( 6.8 )E i t h e r
t a n h ( ~ L )E o < { ( P t - P o ) g + o ~ 2 T } (6.9)a n d
f l < 2 ( p l ~ l P o ) C o t h ( a L ) (6 .10)o r
t a n h ( ~ L )E o < { ( P , - P o )g + ~ 2 T } (6 .11)a n d
f l > 2 ( p t ~ / p o ) c o t h ( ~ L ) . (6 .12)T h e l a t te r i s i n c o n s i s t a n t w i t h c o n d i t i o n (6 .4 ), a n d t h e r e f o r e t h e s t a b i li ty i s o n l y g o v e r n e d b y E q .( 6.9 ) o r E q . (6 .1 0 ). F o r t h e c l a s s ic a l R a y l e i g h - T a y l o r i n s t a b i l i t y w h e r e th e u p p e r d e n s i t y i s c o n s t a n t ,i . e . f l = 0 , t h e s t a b i l i t y i s g o v e r n e d b y E q . ( 6 . 9 ) o n l y .
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344 A.A. Mohamed et al./Journal of Computational and Applied Mathematics 60 (1995) 331-3467. Summary and conclusions
F r o m t h e p r e c e d i n g , i t i s q u i t e c l e a r t h a t w e h a v e s t u d i e d t h e e ff ec t o f a n o r m a l e l e c tr ic f ie ld o na d i e le c t r i c f lu i d l a y e r t o p p e d b y a s t r e a m i n g c o n d u c t i n g f lu i d w i t h s i n u s o i d a l b o u n d a r y w a v ep r o f il e . T h e l i n e a r s t a b i l i t y o f t h e i n t e r f a c e b e t w e e n t h e t w o f l u i d s is e x a m i n e d . I n S e c t i o n 2 , w ef o r m u l a t e d th e p r o b l e m a n d w r o t e d o w n t h e e q u a t i o n s o f m o t i o n . I n S e c t io n 3, w e p u t t h e s ee q u a t i o n s i n t h e p e r t u r b e d f o r m a n d b y i n t r o d u c i n g t h e s t re a m f u n c t i o n 7 /, w e o b t a i n e d t w od i ff e r en t ia l e q u a t i o n s ( o n e o f t h e m is W h i t t a k e r ' s s t a n d a r d f o r m o f t h e e q u a t i o n f o r t h e c o n f l u e n th y p e r g e o m e t r i c f u n c t io n ) fo r t h e u p p e r a n d l o w e r f lu id s . W e t h e n e x p r e s s e d t h e f o r m o f t h e n o r m a le le c tr ic fi el d E a n d w r o t e d o w n t h e p e r t u r b e d f o r m o f th e r e l e v a n t b o u n d a r y c o n d i t i o n s to o u rm o d e l . I n S e c t i o n 4, w e d e r i v e d t h e d i s p e r s i o n e q u a t i o n i n a s i m p l e f o r m b y u s i n g th e p r o p e r t i e s o ft h e W h i t t a k e r f u n c t i o n , a n d f r o m t h e n a t u r e o f t h e r o o t s o f t h is e q u a t i o n , w e s t u d i e d t h e s t a b i li ty o ft h e s y s t e m t h e o r e t i c a ll y a n d n u m e r i c a l ly . T h e r e s u l ts o b t a i n e d f r o m t h e p r e s e n t s t u d y c a n b es u m m a r i z e d a s f o ll ow s :
(i) T h e i n c r e a s e o f t h e n o r m a l e l e c t ri c fi e ld h a s a d e s t a b i l i s i n g ef fe c t, w h i l e t h e i n c r e a s e o f t h et h i c k n e s s L o f th e l o w e r l a y e r y i e ld s a s t a b i l i s i n g i n f lu e n c e . T h e r e a r e s m a l l v a l u e s o f L a n d E 2 f o rw h i c h i n s t a b i l i t y i s n o t p o s s i b l e . T h e v a l u e o f t h e c r i t i c a l fi e ld E ~' ( a b o v e w h i c h t h e s y s t e m i su n s t a b l e ) d e p e n d s o n t h e r a t i o o f t h e d e n s i ti e s, t h e w a v e n u m b e r , a n d t h e t h i c k n e s s o f t h e l o w e rl a y e r .
(ii) I n t h e ~ - E 2 p l a n e , t h e s t a b l e r e g i o n ( f o r a g i v e n c u r v e ) i s d e c r e a s e d b y t h e i n c r e a s e o f ~ ti lla c r i t i c a l p o i n t (o ct, E 2 c) a f t e r w h i c h t h e s t a b l e r e g i o n i s in c r e a s e d b y t h e i n c r e a s e o f ~ .
(iii) F o r a g i v e n v a l u e o f E 2 in t h e ~ - L p l a n e , t h e v a l u e s ~ a l , ~A2 c o r r e s p o n d t o t h e a s y m p -t o t e s , a n d t h e s t a b l e r e g i o n i s r e d u c e d o r i n c r e a s e d b y t h e i n c r e a s e o f ~ < ~a~ o r ~ > (ZA 2. O u t s i d et h e s e r e g i o n s s t a b i l i t y is n o t p o s s i b l e f o r t h i s v a l u e o f th e e l e c t r i c f ie l d, a n d t h e u n s t a b l e r e g i o n su n d e r t h e c u r v e s a re n o t t h e s a m e f o r th e s a m e v a l u e o f E 2 ; t h a t d e p e n d s o n w h e t h e r ~ < e ta , o r
F i n a l ly , th e c a s e o f R a y l e i g h - T a y l o r i n s t a b i l i t y is a ls o i n v e s t i g a t e d b y u s i n g t h e S t u r m f u n c t i o n st o e x a m i n e t h e n a t u r e o f t h e r o o t s o f t h e r e s u lt in g e q u a t i o n a n d t o o b t a i n t h e c o n d i t i o n s f o rs t a b i l i ty i n t h e c a s e o f s t r a t i fi e d f l u i d s w i t h t h e i n f l u e n c e o f a n o r m a l e l e c t r i c fi el d .
AcknowledgementsW e w o u l d l i k e t o t h a n k o n e o f t h e r e f er e es fo r h is u s e f u l c o m m e n t s , s u g g e s t i o n s , a n d i n t e re s t in
t h i s w o r k .
AppendixT h e S t u r m i a n f u n c t i o n s f o r E q . (6 .7 ) a r e:
f ( c ) = 21 c5 -- ,~2 c3 -- ,~.3C2 -~- ,~4c --~ "~-5,f l ( c ) --- 5 '~ 1 c 4 - - 3 ~ 2 c 2 - - 2 2 a c + ) ~4 ,
( A .1 )( A .2 )
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A.A. Mohamed et al./Journal of Computational and Applied Mathematics 60 (1995) 331-346 345f 2 ( c ) = z 2 2 c 3 + ~ 2 3 C 2 _ _ ~ 2 , , C - - 2 5 , ( A .3 )
1 { 42 2 23 + 3 0 2 1 2 3 2 ` _ 2 5 2 1 2 2 2 5 } c{ 1 2 2 3 _ 4 5 2 1 2 3 2 _ 4 0 2 , 2 2 2 , , } c 2 + 2--~223 ( c) =1+ ~ 2 2 { 7 5 2 , 2 2 2 5 - - 4 2 2 2 2 ,,} , ( A . 4 )
f 4 ( c ) = [{ { 4 8 2 ~ - 1 8 0 2 1 2 2 2 32 - 1 6 0 2 1 2 2 2 4 + 1 5 0 2 1 2 2 2 5 - 8 2 3 2 , , } { 1 2 2 3 - 4 5 2 1 2 3 2 - 4 0 2 1 2 2 2 , , }- - { 2 0 2 2 3 2 3 - 2 4 0 2 1 2 2 2 3 2 4 - 1 35 2 1 2 3 --I- 1 0 0 2 1 2 2 2 5 } { 8 2 2 2 3 - 4 - 6 0 2 1 2 3 2 4 - 5 0 2 1 2 2 2 5 }} C+ { 5 25 {1223 - 4 5 2 1 2 3 2 - 4021222 4 } 2 __ { 2 0 2 3 2 3 _ _ 2402122 2 3 2 4 . - - 1 3 5 2 1 2 3 3 - Jr- 1002122225}x { 7 5 2 1 2 2 2 5 - - 4 2 2 2 4 } } ] / 5 { 1 2 2 3 - 4 5 2 , 2 2 - 4 0 2 1 2 2 2 , } 2 , ( A .5 )
f s ( c ) = [ { { S 2 2 2 3 - - I - - 6 0 2 , 2 3 2 , , - 5 0 2 1 2 2 2 5 }( {4 8 2 2 4 - 1 8 0 2 , 2 2 2 2 - 1 6 0 2 , 2 2 2 , , + 1 5 0 2 , 2 2 2 5- 8 2 3 2 , , } { 1 2 2 3 - 4 5 2 12 32 - 4 0 2 1 2 2 2, , } - { 2 0 2 3 - 2 4 0 2 1 2 2 2 3 2 , , - 1 3 5 2 , 2 3 + 1 0 0 2 ,2 2 2 2 5 }x { 8 2 2 2 3 + 6 0 2 1 2 3 2 4 - - 5 0 2 1 2 2 2 s } ) ( 5 2 5 { 1 2 2 3 - 4 5 2 1 2 2 - 4 0 2 1 2 2 2 4 } 2- { 2 0 2 2 2 2 3 - 2 4 0 2 , 2 2 2 3 2 , , - 1 3 5 2 , 2 ~ + 1 0 0 2 , 2 ~ 2 5 } { 7 5 2 , 2 2 2 5 - 4 2 2 2 2 , , } )- { 12 23 - 45 2 1 2 2 - - 4 0 2 , 2 2 2 , , } ( 5 2 5 { 1 2 2 3 - - 4 5 2 1 2 2 - - 4 0 2 1 2 : 2 , , }2- { 2 0 2 3 2 3 - 2 4 0 2 , 2 2 2 3 2 , , - 1 3 5 2 1 2 ~ + 1 0 02 1 22 2 25 } { 7 5 2 1 2 2 2 5 - 4 2 ~ 2 , , } ) 2 }- { { 7 5 2 , 2 2 2 5 - 4 22 2 2 4 } ( { 4 8 2 4 - 1 8 0 2 1 2 2 2 2 - 1 6 0 2 , 222 2 , , + 1 5 0 2 , 222 2 5 - 8 2 3 2 4.} { 12 2 3 - 4 5 2 1 2 2 - 4 02 1 22 2 `* } - { 2 0 2 3 2 3 - 2 4 0 2 , 2 2 2 3 2 ` * - 1 3 5 2 , 2 3 + 1 0 0 2 1 2 2 2 5 }
~2X { 82 2 22 3 Jr- 6 0 2 , 2 3 2 4 - - 5 0 2 1 2 2 2 5 } ) 2 } ] / 4 2 2 [ { 4 8 2 ~ - - 1 8 0 2 1 2 2 A 3 - - 1 6 0 2 , 2 2 2 4+ ~ 5 0 2 1 2 ~ 2 5 - 8 2 3 2 ` * } {1 2 2 3 - 4 5 2 , 2 3 2 - 4 0 2, ,~ 2 2 `* } - { 2 0 2 3 2 3 - 2 4 0 2 1 2 2 2 3 2 , ,- - 1 3 5 2 , 2 3 + 1 0 0 2 1 2 2 2 5 } { 82 2 22 3 " 4 - 6 0 2 1 2 3 2 , , - 5 0 2 1 2 2 2 5 } ] 2 . ( A .6 )
R e f e r e n c e s[ 1 ] W . H . H . B a n k s , P . G . D r a z i n a n d M . B . Z a t u r s k a , O n t h e n o r m a l mo d e s o f p a r a ll e l fl o w o f i n v i s c id s tr a ti fi e d f lu i d , J .Fluid Mech. 75 (1976) 149 -171 .[ 2 ] S . B a r n a r d a n d J . M . C h i ld , Higher Algebra (Macmi l l an and Co . , New York , 1952) .[ -3] R. Betchov and W.O. Criminale, J r . , Stability of Parallel Flow (Academ ic Pres s , New York , 1967) .[4 ] H. B lock an d J .P . Ke l ly , E lec t ro - rheo logy , J. Phys. D: Appl. Phys. 21 (1988) 1661-1677.[ 5 ] S . C h a n d r a s e k h e r , Hydrodynamic and Hydromagnetic Stability (Oxford Univers i ty Press , Oxford, 1961).[6 ] P .G . Dra z in an d L .N. How ard , H ydr od yna m ic s t ab il ity o f pa ra l le l f low o f inv i sc id f lu id s ,Adv. Appl. Mech. 9 (1966)
1 - 8 9 .[7 ] M .F. E l -Sayed , E le c t roh ydro dyn am ic s t ab il ity o f two superposed f lu id s , M as te r o f Sc ience Thes i s , Ain Sham sUnivers i ty , Cairo , Egypt , 1985.
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346 A.A. Moh am ed et al. /Jou rnal o f Computational and Applied Mathematics 60 (1995) 331 346[ 8 ] A . P . G a s t a n d C . F . Z u k o s k i , E l e c t r o r h e o l o g i c a l fl u id s a s c o l lo i d a l s u sp e n s io n s , Adv. Colloid Interface Sci. 30 (1989)153 -202 .[ 9 ] S . G o l d s t e i n , O n t h e s t a b i l i t y o f s u p e r p o s e d s t r e a m s o f d if f er e nt d e n s it ie s , Proc. Roy. Soc. London A 132 (1931 )524 -548 .
[ 1 0 ] T . C . H a l s e y , E l e c t r o r h e o l o g i c a l fl u id s , Science 258 (1992) 761 765 .[ 1 1 ] T . C . H a l s e y , J .E . M a r t i n a n d D . A d o l f , R h e o l o g y o f e l e c t r o r h e o l o g i c a l f lu i d s, Phys. Rev. Lett. 68 (1992 ) 1519 -1522 .[ 1 2 ] L . D . L a n d a u a n d E . M . L if s h it z, Electrodynamics of Continuous Me dia ( P e r g a m o n P r e s s , O x f o r d , 1 9 6 0 ) .[ 13 ] J .R . Me lc he r , Field Coupled Surface Waves ( M I T P r e s s , C a m b r i d g e , 1 9 63 ).[ 14 ] J .R . Me lc he r , Continuum Electromechanics ( M I T P r e s s , C a m b r i d g e , 1 9 81 ).[ 1 5 ] K . O . M i k a e l i a n , T i m e e v o l u t i o n o f d e n s i ty p e r t u r b a t i o n s i n a c c e l e r a t i n g s t ra t i f ie d f lu i ds , Phys. Rev. A 28 (1983)1637 1646 .[ 1 6 ] A . A . M o h a m e d a n d N . K . N a y y a r , E l e c t r o h y d r o d y n a m i c R a y l e i g h - T a y l o r i n s ta b i li t y o f a p l a n e c i r c u la r in te rf ac e ,Nuovo Cimento B 16 (1973) 285-292.[ 1 7] A . A . M o h a m e d a n d E . F. E l s h e h a w e y , N o n l i n e a r e l e c t r o h y d r o d y n a m i c R a y l e i g h - T a y l o r i ns ta b il it y, P a r t I : a p e r -p e n d i c u l a r f ie l d in t h e a b s e n c e o f s u r f a c e c h a rg e s , J. Fluid Mech. 129 (1983) 473 -494 .[ 1 8] A . A . M o h a m e d a n d E . F. E l s h e h a w e y, N o n l i n e a r e l e c t r o h y d r o d y n a m i c R a y l e i g h - T a y l o r i n s ta b i li ty , P a r t I I :a p e r p e n d i c u l a r f i e l d p r o d u c i n g s u r f a c e c h a r g e s , Phys. Fluids 26 (1983) 1724 1731 .[ 1 9 ] A . A . M o h a m e d , E . F . E l s h e h a w e y a n d M . F . E I - S a y e d , E l e c t r o h y d r o d y n a m i c s ta b i li ty o f t w o s u p e r p o s e d v i s c o u sf lu ids, J. Colloid Interface Sci. 169 (1995 ) 65 -78 .1 -2 0] P . A . M o n k e w i t z a n d P . H u e r r e , I n f l u e n c e o f th e v e l o c i t y r a t io o n t h e s p a t i a l i n s t a b i l it y o f m i x i n g l a y e r s , Phys.Fluids 25 (1982) 1137-1143 .[ 2 1 ] M . S . P l e s s e t a n d D . -Y . H s i e h , G e n e r a l a n a l y s i s o f t h e s t a b i l i ty o f s u p e r p o s e d f l u id s , Phys. Fluids 7 (1964 ) 1099 -1108 .[ 2 2 ] M . S . P l e s s e t a n d A . P r o s p e r e t t i , G e n e r a l a n a l y s i s o f t h e s t a b i l i ty o f s u p e r p o s e d f l u id s ( R e p l y t o c o m m e n t s ) , Phys.Fluids 25 (1985) 911 912 .[ 2 3 ] K . R . R a j a g o p a l a n d A . S . W i n e m a n , F l o w o f e l e c tr o - r h e o lo g i c a l m a t e r i a ls , Acta Mech. 91 (1992) 57 75.[ 24 ] A .J . Rose n tha l a nd R.S . L ind e n , In s t a b i l i t i e s i n a s t r a t i f i e d f lu id ha v in g one c r it ic a l l e ve l , Pa r t I : re su l ts , J. Atmosph.Sci. 40 (1983) 509 -520 .1 -2 5] B . K . S h i v a m o g g i , A g e n e r a l iz e d t h e o r y o f t h e s t a b i l i t y o f s u p e r p o s e d f l u id s i n h y d r o m a g n e t i c s , Astrophys. Space Sci.84 (1982) 477-484.
1 -26] J .F . S on to w s k i , B .S . Se ide l a nd W .F. A me s , O n the s t a b i l i ty o f t he f low o f a s t r a t i f i e d ga s ov e r a li qu id , Quart. Appl.M a th . 27 (1969) 335-348.[ 27] K . S t e w a r t son , Ma rg ina l ly s t a b l e i nv i sc id f l ow w i th c r i t i c a l l a ye r s , IMA J , Appl . Math . 27 (1981) 133 175.[ 28] R . T a o , E le c t ri c - fi e ld - induc ed ph a se t r a ns i t i o n in e l e c t ro rhe o log ic a l f l u id s , Phys. Rev. E 47 (1993 ) 423 426 .[ 2 9 ] G . I . T a y l o r , E f f e ct o f v a r i a t i o n o f d e n s i t y o n t h e s t a b i l i t y o f t h e s u p e r p o s e d s t r e a m o f fl u id , Proc. Roy. Soc. LondonA 132 (1931 ) 499 -523 .1 -3 0] W . R . T o o r , S t r u c t u r e f o r m a t i o n i n e l e c t r o r h e o l o g i c a l fl u id s , J. Colloid Interface Sci. 156 (1993 ) 335 349 .[ 3 1 ] E . T . W h i t t a k e r a n d G . N . W a t s o n , Modern Analys is ( C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e , 1 9 5 2 ) .[ 3 2 ] U . Z i m m e r m a n n , E l e c t r i c f i e l d - m e d i a t e d f u s i o n a n d r e l a t i o n e le c tr ic a l p h e n o m e n a b i o c h i m , Biophys. Acta 69 4(1982) 227-277.