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P e r g amon A p p l . M a t h . L e t t . Vol . 9 , No , 4 , pp . 109 -113 , 1996C o p y r i g h t @ 1 9 9 6 E l s e v i e r S c i e n c e L t dP r i n t e d i n G r e a t B r i t a i n . A l l r i g h t s r e s e r v e d0 8 9 3 - 9 6 5 9 / 9 6 $ 1 5 . 00 + 0 . 0 0S0893-9659(96)00061-4
A p p l i c a t i o n o f A d o m i a n ' s A p p r o x i m a t i o n st o t h e N a v i e r - S t o k e s E q u a t i o n si n C y l i n d r i c a l C o o r d i n a t e sK . HALDARPhys i c s and Appl i ed Ma thema t i c s Uni t , Ind i an St a t i s t i ca l Ins t i t u t e203 Bar rackpore Trunk Road , Ca l cut t a 700 035 , Ind i a
(Received June 1994; accepted August 1995)A b s t r a e t - - A d o m i a n h a s a p p l i ed t h e d e c o m p o s i t io n m e t h o d t o t h e N a v i e r -S t o k e s e q u a t i o n s i n t h eC a r t e s i a n c o o r d i n a t e s y s t em . T h e p r e s e n t a n a l y si s gi v es t h e a p p l i c a t io n o f t h e d e c o m p o s i t io n m e t h o dt o t h e N a v i e r - S t o k e s e q u a t i o n s .K e y w o r d s - - N a v i e r - S t o k e s e q u a ti o n s, D e c o m p o s it io n m e t h o d , A d o m i a n 's p o l yn o m i al s , I n fi n it es e r i e s s o l u t i o n , R a p i d l y c o n v e r g e n t .
1 . I N T R O D U C T I O NT h e b a s i c d y n a m i c a l e q u a t i o n s o f f l ui d m e c h a n i c s a r e r e p r e s e n t e d b y t h e N a v i e r - S t o k e s e q u a t i o n s .T h e s e e q u a t i o n s a r e n o n l i n e a r p a r t i a l d i f f e r e n t ia l e q u a t i o n s w h i c h g o v e r n t h e f lo w fi e ld o f a i ra r o u n d a i r c r a f t , i n r a m j e t , b l o o d c i r c u l a t i o n i n t h e c a r d i o v a s c u l a r s y s t e m i n t h e h u m a n b o d y ,a n d i n m a n y o t h e r d i s c i p l i n e s .
S i n c e t h e N a v i e r - S t o k e s e q u a t i o n s a r e n o n l i n e a r i n c h a r a c t e r , i t i s n o t p o s s i b l e t o s o l v e th e s ee q u a t i o n s a n a l y t i c a l l y . I n o r d e r t o s o l v e t h e s e e q u a t i o n s w e n e e d s o m e s i m p l i f ic a t io n s , s u c h a sl i n e a r i z a t i o n o r a s s u m p t i o n s o f ' w e a k ' n o n l i n e a r i ty , s m a l l f l u c t u a t io n s , e t c . O t h e r w i s e , w e u s em a n y t r a d i t i o n a l n u m e r i c a l t e c h n i q u e s w h i c h r e s u l t i n m a s s i v e c o m p u t a t i o n s .
O u r o b j e c t i v e i s t o f i n d o u t c o n t i n u o u s a n a l y t i c s o l u ti o n s w i t h o u t m a s s i v e o u t p r i n t s a n d r e s t r ic -t i v e a s s u m p t i o n s w h i c h c h a n g e t h e p h y s i c a l p r o b l e m i n t o a m a t h e m a t i c a l l y t r a c t a b l e p r o b l e m .T h e s o l u t i o n o f t h e r e d u c e d p r o b l e m c a n b e o b t a i n e d a n a l y t i c a l ly , b u t t h i s s o l u t i o n i s n o t c o n -s i s t e n t w i t h t h e s o l u t i o n o f t h e o r i g i n a l p r o b l e m . R e c e n t l y , a p o w e r f u l m e t h o d w h i c h i s c a l le dD e c o m p o s i t i o n M e t h o d o l o g y d e v e lo p e d b y A d o m i a n [1 -3 ] c a n p r o v i d e a n a ly t i c a p p r o x i m a t i o n s t oa w i d e c l a s s o f n o n l i n e a r o r d i n a r y d i f f e r e n t ia l e q u a t io n s , s y s t e m s o f d if f e r en t i a l e q u a t i o n s , p a r t i a ld i f f e r en t i a l e q u a t i o n s , a n d s y s t e m s o f p a r t i a l d i f f e re n t i a l e q u a t i o n s , a n d t h i s m e t h o d d e m a n d s t ob e p a r a l l e l t o a n y m o d e r n s u p e r c o m p u t e r .
A d o m i a n [3] h a s a p p l i e d t h e d e c o m p o s i t i o n m e t h o d t o t h e N a v i e r - S t o k e s e q u a t i o n s i n t h eC a r t e s i a n c o o r d i n a t e s y s t e m . T h e p r e s e n t a n a ly s i s g i ve s t h e a p p l i c a t i o n o f t h e d e c o m p o s i t i o nm e t h o d t o t h e N a v i e r -S t o k e s e q u a ti o n s i n c y l i nd r i ca l c o o r d in a t e s b y m e a n s o f w h i c h t h e s t e a d yt w o - d i m e n s i o n a l i r r o t a t i o n a l f l u i d f lo w p r o b l e m s i n t u b e s o f n o n u n i f o r m c i r c u l a r c ro s s - s e c t io n sc a n b e s t u d i e d .
2 . A N A L Y S I SC o n s i d e r s t e a d y , t w o - d i m e n s i o n a l m o t i o n o f a v is c o u s fl u id in a t u b e . T h e e q u a t i o n s o f m o t i o n
w h i c h g o v e r n t h e f lo w fi el d in t h e t u b e a r e t h e N a v i e r - S t o k e s e q u a t i o n s i n c y l i n d r i c a l c o o r d in a t e s~Ty pe s e t by A A., IS -TEX
109
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i i 0 K . H A L D A R
a n d t h e y a r e g i v en b y_ _ _ _ ( 0 2 u 1 0 u c 9 2 u ' ~O u O u 1 O p + u + - + (1 )~ - 5- ;z + v ~ = p O z \ O r ~ r ~ " 5 -j z2 '
O V + v ~ _r r = l o p ( 0 2 v 1 0 v 0 2 v v 2 )U -~ z p O r + v \ 0 r 2 + - + - - (2 )- - - r ~ r O z 2 - r2 '
a s s u m i n g t h a t t h e r o t a t i o n a l f lu i d m o t i o n i s n e g l e c te d . T h e c o n t i n u i t y e q u a t i o n i s1 c 9 ( r v ) + Ou; . O r ~ = 0 . (3 )
H e r e ( u , v ) a r e t h e c o m p o n e n t s o f f l u id v e l o c i ty i n th e a x i a l c o o r d i n a t e x a n d r a d i a l c o o r d i n a t e r ,r e s p e c t i v e l y , p i s t h e f l u id p r e s s u r e , p is t h e f l u i d d e n s i t y a n d u i s t h e k i n e m a t i c v i s c o s i t y o f f l u id .
I f w e i n t r o d u c e t h e s t r e a m f u n c t i o n d e f i n e d b y1 0 1 0 u - v = - - - ( 4 )r O r ' r O z '
t h e c o n t i n u i t y e q u a t i o n i s i d e n t i c a l ly s a ti s fi e d . I f w e e l i m i n a t e p b e t w e e n ( 1 ) a n d ( 2 ), a n d t h e nu s e t h e r e l a t i o n s ( 4 ) , w e h a v e th e d y n a m i c a l e q u a t i o n o f m o t i o n i n t e r m s o f a s
1 0 ( L , ) 2 0 L 9 = vL 2~ #, (5 )7 o (r , z----7- - r-~ - f f zo02 o02w h e r e L = ~ r - ( l / r ) . ~ r + ~)-~"
0 2 o0~C A S E 1 . L e t L 1 - - ~ - ( l / r ) o . T h e n t h e o p e r a t o r L b e c o m e s L = L 1 + ~ a n d e q u a t i o n (5 )t a k e s t h e f o r m 02L ~ = u - l g 040z 2 ~Z2 (L 1 ), (6)w h e r e N i s t h e n o n l i n ea r t e r m d e n o t e d b y
N = - 1 . 0 ( L , ) _ 2.0 __ _ __ . L . (7 )r 0(r, z ) r 2 O zO p e r a t i n g w i t h t h e a p p r o p r i a t e i n v e r se L ~ 2 o n ( 6 ) , w e h a v e [ 1,2 ]
0 2 ( 0 4 ~~ p( z, r ) = ~ P o (Z , r ) + u - I L ~ 2 ( N ) - 2L~-20-~z2 L I ) - L~ 2 \ Oz 4 ] , ( s )w h e r e ~P0 i s t h e s o l u t i o n o f th e h o m o g e n e o u s e q u a t i o n L 2 g ) 0 - - 0 s u b j e c t t o c e r t a i n b o u n d a r yc o n d i t i o n s .
I n o r d e r t o d e t e r m i n e t h e i n v e rs e o p e r a t o r o f L 1 , w e c o n s id e r t h e e q u a t i o n [3]L 1 = 0 . ( 9 )
o02I f we de f in e L~r --- ~ an d L~ = ~-;,o0 t he n th e op e ra to r L1 b eco m es L1 = Lr~ - ( 1 / r ) L r a n de q u a t i o n ( 9 ) t a k e s t h e f o r m 1Lrr~# - - " L ~ = 0. (10)rS o l v i n g f o r t h e l i n e a r t e r m s L r r W a n d L r F , w e h a v e
L r r = _ 1. L ~ , ( 11 )rL ~ = r L ~ r . (12)
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N a v i e r- S t o k e s q u a t io n s 1 lO p e r a t i n g w i t h L r ~ o n ( 1 1 ) a n d L ~ 1 o n ( 1 2 ) , w e c a n w r i t e
= 1 + L ~ -~ ( 1 L ~ ) , ( 1 3 ) = 2 + L ~ I ( r L ~ ) , (14)
w h e r e 1 a n d 2 a re t h e s o lu t io n s o f t h e h o m o g e n e o u s e q u a t io n s L r r = 0 a n d L ~ = 0 . T h ei n v e r se o p e r a t o r s L ~ 1 a n d L ~ 1 a r e d e f i n e d b y
L ~# = f f [.] dr dr,J J (15),
L~ 1 = / [ . ] dr.JA d d i n g ( 1 3 ) a n d ( 1 4 ) , a n d t h e n d i v i d i n g b y 2 , w e h a v e
]= o + -~ L - ~ L r + L ; I ( r L r ~ ) ,w h e r e 0 = ( 1 / 2 ) ( 1 + 2 ). T h e n1 /~ 1 = ~ [ L ~ J ( l ' L r ) ~ - L ~ - l ( rL r r ) ] o ,
I f t h e q u a n t i t y w i t h i n t h e b r a c k e t i s d e n o t e d b y S , t h e n w e c a n w r i te1 1 = ~ S o ,
2 = ~ S 2 0 ,
1~ n + ~ = 2 n + l S n + 1 o ,
a n d ( 1 ) : ~ C n = ~ " S n O -~ - ~ L - rJ L r Q- L < l ( r L r r o ,n : O r ', ,: O n : O
s o t h a t t h e i n v e r s e L { 1 ---- (Lr r - (1 / r )Lr ) -1 h a s b e e n i d e n t i f i e d a s
n=O
N o w w e c a n r e t u r n t o e q u a t i o n ( 8) a n d w r i t e t h e p a r a m e t r i z e d f o r m o f i t a s [ 2][ 0 2 ( 0 4 ~ ' ~ I~p(z,r) = o(z , r ) + A v - t L T 2 ( N ~ p ) - 2LT2b--~z2 Lt t~) - L 12 \-b-~z4 ] .
( 1 6 )
(17)
(18)
( 1 9 )
( 2 0 )
( 2 1 )
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112 K. HALDARL e t a n d t h e n o n l i n e a r t e r m N b e d e c o m p o s e d i n t o th e f o l lo w i ng f o rm s :
o o = (22)n = 0
o og = E A ndn ' ( 2 3 /n-----O
w h e r e A n a r e s pe c i a l po l ynom i a l s o f A do m i a n [ 1 ,2 ] t o be d i s c u s s e d l at e r . T he p a r a m e t e r Ai n t r od u c e d he r e is no t a pe r t u r b a t i on pa r a m e t e r , a nd i t i s u s e d f o r g r ou p i ng t he t e r m s on l y .T he n e qu a t i on ( 2 1 ) t a k e s t he f o r m
E A n C n = ~ b o + A u - I L ~ 2 A n g , ~ - 2 n 7 ~ L I E A n C n - L 1 2 ~ z 4 E A n C n n = O n = O n = O / n = O
(24)
E qu a t i ng t he t e r m s o f l ik e pow e r o f A , w e ha ve02 coao 1 = v - l L 1 2 A o - 2L1-2 ~ ( L i e 0 ) - L 1 2 COz4 ,CO2 cO412 = v - t L t 2 A 1 - 2 L t 2 ~ ( L 1 1) - L ~ 2 cOz4 ,
cO~ cO4n n + 1 = v - I L ~ 2 A n - 2 L 1 2 -ff~z2 ( L i e n ) - L ~ 2 cOz4
( 2 5 )
T h e p o l y n o m i a l s A 0 , A 1 , . . . , A n a r e A dom i a n ' s po l yno m i a l s ( s ee [2 ]) . T h e y a r e de f i ne d i n s u c ha w a y t h a t e a c h A n d e p e n d s o n l y o n 0 , 1 , . . . , n . T h u s A 0 = A o ( o ), A1 = A 1 ( 0 , 1 ) ,A2 --- A2( 0, 1 , 2) , e t c . From (6) and (23) we have
o oE A ' ~An 1 c O( L , ~b) 2 0 ~ bn = 0 = r " cO (r,z ) - r ' 2 " ~ z ' L " ( 26 )S u b s t i t u t i n g ( 2 2) i n t o ( 2 6 ) a nd t he n e qu a t i n g l ik e pow e r s o f A f r om bo t h s i de s o f ( 2 6) , w e ob t a i n
A o = 1 . 0 ( L e o , G o ) _ 2 . a G0 . L 0 ,r a( r , z) r 2 OzO ( L ~ , ~ o ) 2 0 o . L e t + . . . .A 1 = " O ( r , z ) - r~ " Oz cO(r , z ) r 2cO (L 2, o ) 2 0 0 . L ~ + . . . .A2 = " c O( r , ) - r'~ " cOz cO(r, z) r 2
[ 1 c O ( L I , I ) 2 c O 1 . L 1 ] ,+ " c O( r , z ) - r - - ~ ' cOz
0 1 . L 0 ]O z J020z "L0]
( 2 7 )( 2 s )
( 2 9 )
I f ~ 0 i s onc e ob t a i ne d , t he n w e c a n ob t a i n ~Pl i n t e r m s o f 0 . S i m i la r ly , 2 c a n be ob t a i ne d i no ot e rm s o f ~P t, e t c . So , a l l com po ne nt s of ~p a re ca lcu la ble a nd tp = ~ '~n=o Cn .02C A S E 2 . W h e n L = L r r - ( 1 / r ) L r + L z z , w h e r e L z z = ~ ' , x , e qu a t i on ( 5 ) be c om e s
L 2 ~ = v - i N C . (30)
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Navier-Stokes EquationsO p e r a t i n g w i t h t h e i n v e r s e o p e r a t o r L - 2 o n ( 3 0 ) , w e g e t
113
~b ---- u - I L - 2 ( N ) . ( 3 1 )T h e L - 2 i n v o l v e d i n ( 3 1) c a n b e o b t a i n e d i n t h e s a m e w a y a s L ~ 2 , a n d i t is g iv e n b y
1 [ L r J ( 1 L r L z z ) + L ; l { r ( L r r L z z ) } + L ; 1 ( ! L r L ~ ) ] nL - 2 = E 3---~ - - - (32 )
T h e p a r a m e t r i z e d f o r m o f e q u a t i o n ( 31 ) is~b = u - I A L - 2 ( N ) . ( 33 )
I f w e s u b s t i t u t e ( 2 2 ) a n d ( 2 3 ) i n t o ( 3 3 ) , a n d t h e n e q u a t e l ik e p o w e r s o f A o n b o t h s i d e s o f ( 3 3 ) ,w e o b t a i n
~)1 ~ u-IL-2Ao,~ 2 = u - I L - 2 A 1 ,~3 = ~-IL-2A2,
~P~+I = u-IL-2A~,
w h e r e A o , A 1 . . . . , A n a r e A d o m i a n ' s s p e c ia l p o l y n o m i a l s [2] a n d c a n b e o b t a i n e d f r o m ( 2 7 ) - ( 2 9 ) ,e t c . T h e u s e o f t h e o p e r a t o r L i n th i s a n a l y s i s in c r e a s e s t h e s p e e d o f c o n v e r g e n c e , b u t t h e i n v e r seo f L a n d c o n s e q u e n t i n t e g r a t i o n s a r e m o r e d i f fi c u lt [4 ].
T h e c o n v e r g e n c e o f t h e s e ri es s o lu t io n o b t a i n e d b y t h e d e c o m p o s i t i o n m e t h o d h a s b e e n e s t a b -l i s h e d [ 2 , 5 - 7] a n d i t i s a l s o o b s e r v e d i n [ 4,5 ] t h a t a r a p i d s t a b i l i z a t i o n t o a n a c c e p t a b l e a c c u r a c yis e v i d e n t w h e n n u m e r i c a l c o m p u t a t i o n o f t h e a n a l y t i c a p p r o x i m a t i o n is c a r r i e d o u t .
R E F E R E N C E S1. G. Ado mian, Non l inear Stochastic Operator Equat ions , Aca dem ic Press, (1986).2 . G. Ado mian, Non l inear Stochast ic Syste ms Theory and Appl icat ions to Physics, Kluwer Academic, (1989).3 . G. Adom ian, Appl icat ion of the decom posi t ion method to the N avier -Stokes equat ions , Jour. Math. Anal .A p p l . 119, 340-360 (1986) .4 . G. Adom ian, Nonl inear t ransp or t in moving f lu ids , Appl . Math. Let t . 6 (5), 35-38 (1993).5 . G. A dom ian and R . Rach, A nalyt ic solu t ion of nonl inear boundary-va lue problems to several dimensions,Jour . M ath . A na l . A pp l . 174, 118-137 (1993).6 . Y. Cherruau l t , Convergence of Ado mian 's method , Kybernetes 18 (2), 31-39 (1989).7 . Y. Ch erruau l t , Some new resu l ts for convergence of Adom ian 's method applied to integral equat ions , Mathl~
Comput . Model l ing 16 (2), 85-93 (1992).
