c.l. pekeris- stationary spherical vortices in a perfect fluid

8/3/2019 C.L. Pekeris- Stationary Spherical Vortices in a Perfect Fluid

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P r o c . N a t . A c a d . S c i . USAV o l . 6 9 , N o . 9 , p p . 2 4 6 0 - 2 4 6 2 , S e p t e m be r 1 9 7 2

S t a t i o n a r y S p h e r i c a l V o r t i c e s i n a P e r f e c t F l u i d( i n c o m p r e s s i b i l i t y / f l o w / t r a j e c t o r y / d y n a m o t h e o r y / s t a t i o n a r i t y c o n d i t i o n s )

C . L . PEKERISD e p a r t m e n t o f A p p l i e d M a t h e m a t i c s , The Weizmann I n s t i t u t e , R e h o v o t , I s r a e lC o n t r i b u t b y C . L . P e k e r i s , J u n e 2 6 , 1 9 7 2ABSTRACT N e c e s s a r y c o n d i t i o n s a r e d e r i v e d f o r as p h e r i c a l v o r t e x i n a p e r f e c t f l u i d t o b e s t a t i o n a r y i n t h ec a s e wh e n t h e v e l o c i t i e s depend on a s i n g l e s u r f a c e h a r -monic. T h e motion i s i n d e t e r m i n a t e u n l e s s a n a d d i t i o n a lc o n d i t i o n i s i m p o s e d . I n t h e c a s e when t h i s c o n d i t i o n i si n c o m p r e s s i b i l i t y , t h e e q u a t i o n s a r e s o l v e d , y i e l d i n g ac l a s s o f s t a t i o n a r y s p h er i c a l v o r t i c e s .I n t h i s i n v e s t i g a t i o n , I w i s h t o d e t e r m i n e t h e t y p e s o f s t a t i o n -a r y v o r t e x m o t i o n t h a t a r e p o s s i b l e i n a f l u i d t h a t i s b o u n d e db y s p h e r i c a l s u r f a c e s . T h e c o n d i t i o n o f s t a t i o n a r i t y i m p o s e s ar e s t r i c t i o n o n t h e p a t t e r n o f f l o w ( 1 ) , s i n c e t h e s t r e a m l i n e sh a v e t o c o i n c i d e w i t h t h e t r a j e c t o r i e s o f t h e f l u i d p a r t i c l e s .T h e i m m e d i a t e a p p l i c a t i o n I h a v e i n m i n d i s t o t h e d y n a m ot h e o r y o f t h e o r i g i n o f t h e e a r t h ' s m a g n e t i c f i e l d , p r o p o s e d b yL a r m o r ( 2 ) . I n t h e s t a t i o n a r y k i n e m a t i c d y n a m o ( 3 ) , t h em a g n e t i c f i e l d H i s d e t e r m i n e d b y t h e e q u a t i o n

VI H+ V c u r l ( U x H ) = 0 , [ 1 ]w h e r e U i s a g i v e n c o n v e c t i v e f i e l d i n s i d e t h e l i q u i d c o r e o f t h ee a r t h . H e r e , V i s a n o n d i m e n s i o n a l e i g e n v a l u e g i v e n b y

V = 4 T K K b U o , [ 2 ]w h e r e K i s t h e e l e c t r i c a l c o n d u c t i v i t y o f t h e l i q u i d c o r e , bi s t h e r a d i u s o f t h e c o r e , a n d U o t h e v e l o c i t y s c a l e o f t h e c o n -v e c t i o n .

I n d e c i d i n g o n t h e t y p e o f c o n v e c t i v e f i e l d t o c h o o s e , Is h a l l a s s u m e a s a w o r k i n g h y p o t h e s i s t h a t i f a c e r t a i n s t a t i o n -a r y c o n v e c t i v e c e l l i s p o s s i b l e i n t h e c a s e o f a p e r f e c t f l u i d ,s u c h a p a t t e r n o f f l o w i s l i k e l y t o b e e s t a b l i s h e d a l s o i n a r e a lf l u i d , s u b j e c t t o l o c a l m o d i f i c a t i o n s b y d i s s i p a t i v e f o r c e s .NECESSARY CONDITIONS FOR A STATIONARY FLOWI n a s p h e r i c a l s y s t e m o f c o o r d i n a t e s ( r , o , 4 ) w i t h v e l o c i t y c o m -p o n e n t s ( u , v , w ) t h e s t a t i o n a r y m o t i o n h a s t o s a t i s f y t h eE u l e r i a n e q u a t i o n s

6 u V f u w a u ( v 2 + w 2 ) _ Qu + +c r r a 0 r s i n eOf r b r '

o v v d V w d V u V w 2 c o t a nOu-_ + +b r r 6 0 r s i n 0 6 f r r

1 a Qr a s '

, w V 6 w w 6 w w ua r r a 0 r s i n O6 f rwv c o t a nO 1 b Qr r s i n 0 a + '

[ 3 ]

w h e r eQ - ++ ,p [ 6 ]

p d e n o t i n g t h e p r e s s u r e , p t h e d e n s i t y , a n d a t h e p o t e n t i a l o ft h e f o r c e s . I s h a l l r e s t r i c t t h e d i s c u s s i o n t o t h e c a s e w h e r e t h em o t i o n d e p e n d s o n a s i n g l e s p h e r i c a l s u r f a c e h a r m o n i c

nY = Y n m = E ( A n m c o s m O + B n m s i n m O ) P n m ( c o s O ) , [ 7 ]m =Ot h r o u g h ( 4 ) ,

u =U ( r ) Y ,v = V ( r ) ( ) Y / a o ) + [ W ( r ) / s i n o ] ( b Y / b 4 ) ,w = [ V ( r ) / s i n 0 ] ( b Y / l b ) - W ( r ) ( a r / 6 0 ) .

[ 8 ][ 9 ]

[ 1 0 ]H e r e , U ( r ) a n d V ( r ) r e p r e s e n t t h e p o l o i d a l f i e l d , a n d W ( r )t h e t o r o i d a l f i e l d . T h e d i l a t a t i o n i s g i v e n b y

d i v U = X ( r ) Y ,X = t + ( 2 / r ) U - n ( n + 1 ) ( V / r ) ,

[ 1 1 ][ 1 2 ]

t h e d o t d e n o t i n g d i f f e r e n t i a t i o n w i t h r e s p e c t t o r . I n t h e c a s eo f t h e b o d i l y t i d e s , t h e m o t i o n i s p o l o i d a l ( 5 ) i n t h e a b s e n c e o fr o t a t i o n . T h e r o t a t i o n o f t h e e a r t h i n d u c e s a t o r o i d a l c o m -p o n e n t .S u b s t i t u t i o n o f [ 8 ] , [ 9 ] , a n d [ 1 0 ] i n t o [ 3 ] , [ 4 ] , a n d [ 5 ]y i e l d s

-( Q l b r ) = U 6 t Y 2 + ( 1 / r ) (UV - V 2 -W2) f- ( b Q / l B ] = 1 / 2 ( L / b O ) + M Y ( 1 / s i n O ) ( b Y / l 4 o ) ,- O Q / 1 ) = ' / 2 ( b L / b o ) -M s i n O Y ( b Y / 1 G ) ,

w h e r eL = { [ U ( r V + V ) + n ( n + 1 ) W 2 ] Y 2

[ 1 3 ][ 1 4 ][ 1 5 ]

+ ( V 2 + W 2 ) Y } , [ 1 6 ]M = [ U ( r W + W ) - n ( n + 1 ) V W ] ,Y = [ ( b Y / b o ) 2 + ( 1 / s i n 2 O ) ( b Y / 6 4 ) ] 24 ]

[ 1 7 ][ 1 8 ]

I s h a l l a s s u m e , i n t h e f i r s t i n s t a n c e , t h a t Y i n [ 7 ] d e p e n d s on0 . By e q u a t i n g t h e e x p r e s s i o n s f o r ( 6 2 Q / 6 0 6 0 ) d e r i v e d f r o m[ 1 4 ] a n d f r o m [ 1 5 ] , we g e t[ 5 ] M- [ - n ( n + 1 ) Y 2 + f ] = . [ 1 9 ]2 4 6 0


2/4

S t a t i o n a r y S p h e r i c a l V o r t i c e s 2 4 6 1S i m i l a r l y , b y c r o s s d i f f e r e n t i a t i n g [ 1 3 ] a n d [ 1 4 ] , w e g e tN Y ( 6 Y / a O ) + P ( b Y / ' a )

- M ( 1 / s i n O ) Y ( I b Y / 4 k ) = 0 , [ 2 0 ]w h e r e

N = ( d / d r ) [ U 2 - U ( r V + V ) - n ( n + 1 ) W 2 ] , [ 2 1 ]P= [ ( 1 / r ) ( U V - V 2 - W 2 ) - VV - WW]. [ 2 2 ]

T h e e x p r e s s i o n s o f ( b 2 Q / & r b 4 ) y i e l d , b y [ 1 3 ] a n d [ 1 5 ] ,N Y ( b Y / Y 6 ) + P ( a Y / a k ) + M s i n G Y ( b Y / 6 O ) = 0 . [ 2 3 ]

C o n d i t i o n s [ 1 9 ] , [ 2 0 ] , a n d [ 2 3 ] a r e n e c e s s a r y f o r a p e r f e c tf l u i d i n o r d e r t h a t t h e s p h e r i c a l v o r t e x r e p r e s e n t e d b y [ 8 ] ,[ 9 ] , a n d [ 1 0 ] s h a l l b e s t a t i o n a r y . S i n c e t h e e x p r e s s i o n i nb r a c k e t s i n E q . [ 1 9 ] d o e s n o t v a n i s h f o r a n y s p h e r i c a l s u r f a c eh a r m o n i c Y , w e m u s t h a v eM = U(rW + W ) - n ( n + 1)VW = . [ 2 4 ]

I t f o l l o w s f r o m [ 2 0 ] a n d [ 2 3 ] t h a tNY2 + 2PY = ( r ) . [ 2 5 ]

C o n d i t i o n [ 2 5 ] t a k e s o n a s p e c i a l f o r m i n t h e c a s e n = 1 :Y = Y 1 = c o s o + A s i n ~ c o s ( o - 4 o ) y [ 2 6 ]w h e n t h e f o l l o w i n g r e l a t i o n h o l d s :

Y 1 2 + y i= 1 + A 2 [ 2 7 ]F o r t h e c a s e n = 1 , [ 2 5 ] t h e r e f o r e r e q u i r e s t h a t

N1 - 2 P 1 = 0 , [ 2 8 ]2 P ( 1 + A 2 ) = f ( r ) . [ 2 9 ]

F o r n > 1 , c o n d i t i o n [ 2 5 ] r e q u i r e s , i n a d d i t i o n t o t h e v a n i s h i n go f f ( r ) , t h a tN = 0 ,P = 0 .

[ 3 0 ][ 3 1 ]

E q . [ 3 0 ] i n t e g r a t e s i n t oU(rV+ V ) + n ( n + 1 ) W 2 - U2 = 0 , [ 3 2 ]

w h i l e [ 3 1 ] c a n b e p u t i n t h e f o r mUV-V(rV+ V)- W(rW7+ W ) =0. [ 3 3 ]

T h e t h r e e E q s . [ 2 4 ] , [ 3 2 ] , a n d [ 3 3 ] d o n o t s u f f i c e t o d e t e r -m i n e t h e f u n c t i o n s U , V , a n d W , b e c a u s e t h e y a r e n o t i n -d e p e n d e n t :

U [ 3 3 ] + V [ 3 2 ] + W [ 2 4 ] = 0 . [ 3 4 ]T h e r e f o r e , w e c a n t a k e t w o o f t h e m , s a y [ 2 4 ] a n d [ 3 2 ] , a s t h en e c e s s a r y c o n d i t i o n s f o r t h e e x i s t e n c e o f a s t e a d y m o t i o n o ft h e t y p e r e p r e s e n t e d b y E q s . [ 8 ] , [ 9 ] , a n d [ 1 0 1 .E l i m i n a t i n g V f r o m [ 1 2 ] a n d [ 2 4 ] , w e g e t

rW =[ X / n ( n + 1 ) ] r 2 U e x p [ f r ( X / U ) d r J , [ 3 5 ]w h i c h i s v a l i d a l s o f o r t h e c a s e n = 1 . H e r e , X i s a n a r b i t r a r yc o n s t a n t . S u b s t i t u t i n g f r o m [ 1 2 ] a n d [ 3 5 ] i n t o [ 3 2 ] , w e a r r i v ea t( d 2 / d r 2 ) ( r 2 U ) - n ( n + 1 ) U + X 2 r 2 U G 2

=( d / d r ) ( r 2 X ) , [ 3 6 ]

w h e r eG[(r)e x p [ f ( X / U ) d r ]

I n t h e c a s e n = 1 , E q . [ 2 8 ] c a n b e p u t i n t h e f o r m( d / d r ) l n [ U 2 - U ( r V + V ) - 2 W 2 ] = ( 2 V / r U )

= ( d / d r ) l n ( r W ) .H e n c e ,

U 2 - U ( r V + V ) - 2W2 = ArW.By [ 1 2 ] a n d [ 3 5 ] , E q . [ 3 9 ] y i e l d s( d 2 / d r 2 ) ( r 2 U ) - 2U + X 2 r 2 U G 2 = ( d / d r ) ( r 2 X j

[ 3 7 ]

[ 3 8 ][ 3 9 ]

+ K r 2 G . [ 4 0 ]L e t ( 3 )

r 2 U ( r ) = n ( n + 1 ) S ( r ) ,t h e n

V = ( S / r ) - [ r 2 X / n ( n + 1 ) ] ,W ( X / r ) S G .

E q s . [ 3 6 ] a n d [ 4 0 ] t h e n t a k e o n t h e f o r mS + { X 2 G 2 - [ n ( n + 1 ) / r 2 ] } S = [ 1 / n ( n + 1 ) ]

X ( d / d r ) ( r 2 X ) , n> 1 ,S + [ X 2 G 2 - ( 2 / r 2 ) ] S = 1 / 2 ( d / d r ) ( r 2 X )

+ K ' r 2 G , n = 1 .CASE OF AN INCOMPRESSIBLE FLUID

I n t h e c a s e o f i n c o m p r e s s i b i l i t y , we h a v eX=0, G=1,

V = ( S / r ) , W = ( X / r ) S ,S n + { X 2 - [ n ( n + 1 ) / r 2 ] } S n = O n > 1 ,

3 , + [ X 2 - ( 2 / r 2 ) ] S 1 = K ' r 2 , n = 1 .The s o l u t i o n o f [ 4 8 ] i sS n =A ( X r ) ' / 1 J n + 1 / 2 ( X r ) + B ( X r ) ' / ' J - n - / 2 ( X r ) I

[ 4 1 ][ 4 2 ][ 4 3 ]

[ 4 4 ][ 4 5 ]

[ 4 6 ][ 4 7 ][ 4 8 ]

[ 4 9 ]

n > 1 , [ 5 0 ]w h i l e t h e s o l u t i o n o f [ 4 9 ] i s

S i =A ( X r ) ' / 2 J 3 / 2 ( X r ) + B ( X r ) l / 2 J - 3 , 2 ( X r ) + C r 2 . [ 5 1 ]To s u m m a r i z e , i n t h e c a s e o f an i n c o m p r e s s i b l e i d e a l f l u i d

t h e r e e x i s t s a c l a s s o f s t a t i o n a r y s p h e r i c a l v o r t i c e s d e f i n e d b yu = [ n ( n + 1 ) / r 2 ] S ( r ) Y , [ 5 2 ]

v = ( S / r ) ( b Y / 1 O ) + ( T / r s i n O ) ( b Y / b 4 ) , [ 5 3 ]w = ( S / r s i n O ) ( b Y / ? 0 ) - ( T / r ) ( b Y / 6 O ) , [ 5 4 ]

w h e r e Y i s a s u r f a c e s p h e r i c a l h a r m o n i c d e f i n e d i n [ 7 ] ,T ( r ) = X S ( r ) , [ 5 5 ]

a n d S ( r ) i s g i v e n b y E q s . [ 5 0 ] a n d [ 5 1 ] .

P r o c . N a t . A c a d . S c i . USA 6 9 ( 1 9 7 2 )


3/4

2 4 6 2 A p p l i e d M a t h e m a t i c s : P e k e r i sTHEHICKS SPHERICAL VORTEX

T h e s p e c i a l c a s e o f a n i n c o m p r e s s i b l e i d e a l f l u i d w h e r e t h em o t i o n p o s s e s s e s a x i a l s y m m e t r y w a s t r e a t e d b y H i c k s ( 6 ) .I n t h i s c a s e o f a x i a l s y m m e t r y ( 7 ) , M ( r ) m u s t v a n i s h b y [ 1 5 ] ,a n d w i t h i t E q . [ 2 3 ] i s s a t i s f i e d . T h e r e m a i n i n g c o n d i t i o n [ 2 0 ]t a k e s t h e f o r m( o r / s ) I [ N - 2 n ( n + 1 ) P ] Y

- 2P c o t a n = ( b Y / 6 0 ) 0 . [ 5 6 ]C o n d i t i o n [ 5 6 ] r e q u i r e s t h a t [ 2 8 ] b e s a t i s f i e d i n c a s e n = 1 ,a n d t h a t [ 3 0 ] a n d [ 3 1 ] b e s a t i s f i e d f o r n > 1 .

SOLUTION FOR THEPRESSUREI n o r d e r t o s o l v e f o r t h e p r e s s u r e , w e n o t e t h a t , b y v i r t u e o f[ 2 4 ] , a s o l u t i o n o f [ 1 4 ] a n d [ 1 5 ] i sQ = - 1 / 2 L + C = C - 1 / 2 [ U 2 y 2 + ( V 2 + W 2 ) Y ]

= - 1 / 2 ( u 2 + v 2 + W 2 ) + C , [ 5 7 ]w h e r e u s e h a s b e e n m a d e o f [ 3 2 ] . T h e s o l u t i o n [ 5 7 ] s a t i s f i e s[ 1 3 ] b y v i r t u e o f [ 3 2 ] . S u b s t i t u t i n g f r o m [ 6 ] , w e g e t a B e r -n o u l l i e q u a t i o n

( p / p ) + 1 = - ' / 2 ( u 2 + v 2 + W 2 ) + C . [ 5 8 ]T h e c o n s t a n t C i n [ 5 8 ] i s a n a b s o l u t e c o n s t a n t , a n d d o e s n o td i f f e r f r o m o n e s t r e a m l i n e t o a n o t h e r ( 8 ) .A c h a r a c t e r i s t i c f e a t u r e o f o u r s p h e r i c a l v o r t i c e s , d e f i n e db y e q u a t i o n s [ 5 2 ] - [ 5 5 ] , i s t h a t t h e i r v o r t e x l i n e s c o i n c i d e w i t ht h e s t r e a m l i n e s :

V xU =XU. [ 5 9 ]I t f o l l o w s f r o m t h e h y d r o d y n a m i c e q u a t i o n t h a t( V X U ) X U = - g r a d [ ( p / p ) + Q + 1 / 2 ( U 2 + V 2 + W 2 ) ] = 0 ,[ 6 0 ]h e n c e r e l a t i o n [ 5 8 ] .T a k i n g t h e c u r l o f [ 5 9 ] we g e t

V X ( V X U ) = g r a d ( V . U ) - V 2 U = X 2 U , [ 6 1 ]V 2 U + X 2 U = 0 [ 6 2 ]o f w h i c h [ 5 2 ] - [ 5 5 ] a r e s o l u t i o n s .

T h i s r e s e a r c h w a s s u p p o r t e d b y t h e O f f i c e o f N a v a l R e s e a r c h( U S A ) u n d e r C o n t r a c t N 0 0 0 1 4 - 6 6 - C - 0 0 8 0 .1 . L a m b , H . ( 1 9 3 2 ) i n H y d r o d y n a m i c s ( C a m b r i d g e U n i v e r s i t yP r e s s , C a m b r i d g e ) , p . 2 4 3 .2 . L a r m o r , J . ( 1 9 1 9 ) R e p . B r i t . A s s . , 1 5 9 - 1 6 0 ; ( 1 9 1 9 ) E l e c .R e v . ( L o n d o n ) 8 5 , 4 1 2 .3 . B u l l a r d , E . C . & G e l l m a n , H . ( 1 9 5 4 ) P h i l . T r a n s . R o y . S o c .L o n d o n 2 4 7 , 2 1 3 - 2 7 8 .4 . A l t e r m a n , Z . , J a r o s c h , H . & P e k e r i s , C . L . ( 1 9 5 9 ) P r o c . R o y .S o c . S e r . A 2 5 2 , 8 0 - 9 5 .5 . P e k e r i s , C . L . & A c c a d , Y . ( 1 9 7 2 ) P h i l . T r a n s . R o y . S o c .L o n d o n ( i n p r e s s ) .6 . H i c k s , W. M. ( 1 8 9 9 ) P h i l . T r a n s . R o y . S o c . L o n d o n S e r . A1 9 2 , 3 3 - 1 0 0 .7 . P e k e r i s , C . L . ( 1 9 5 3 ) P r o c . N a t . A c a d . S c i . USA 3 9 , 4 4 3 -4 5 1 .8 . L a m b , H . ( 1 9 3 2 ) i n H y d r o d y n a m i c s ( C a m b r i d g e U n i v e r s i t yP r e s s , C a m b r i d g e ) , p . 2 1 .

P r o c . N a t . A c a d . S c i . USA 6 9 ( 1 9 7 2 )


4/4

P r o c . N a t . A c a d . S c i . USA 6 9 ( 1 9 7 2 )C o r r e c t i o n . I n t h e a r t i c l e " S t a t i o n a r y S p h e r i c a l V o r t i c e s i n aP e r f e c t F l u i d , " b y C . L . P e k e r i s , w h i c h a p p e a r e d i n t h eS e p t e m b e r 1 9 7 2 i s s u e o f t h e P r o c . N a t . A c a d . S c i . USA 6 9 ,2 4 6 0 - 2 4 6 2 , o n p . 2 4 6 0 , E q . [ 1 8 ] s h o u l d r e a d : P = [ ( b Y /W ) 2 + ( 1 / s i n 2 0 ) ( ( 3 y / a < ) 2 ] . I n t h e c a s e o f a s p h e r i c a l h a r -m o n i c n o f o r d e r 1 , we h a v e , i n p l a c e o f [ 5 9 1 ,V X U =XU -X 2 r C 1 b Y l e + X 2 r C b y Is i n 0 o p 0) n = 1 ,w h e r e C s t h e c o n s t a n t a p p e a r i n g i n E q . [ 5 1 1 . The B e r n o u l l ie q u a t i o n t a k e s on t h e f o r m( P I P ) + = - / 2 ( u 2 + v2 + W 2 ) + X 2 C S I

X ( 1 + A2-Y2), n = 1 ,A d e n o t i n g t h e c o n s t a n t i n E q . [ 2 6 ] .

C o r r e c t i o n : S h i m o m u r a e t a l . 3 8 4 9E q . [ 6 2 ] , p . 2 4 6 2 , r e a d s

X l r C b Y , b y ,v2 +x2U = AR l e -XrC-s i n 0 o a 0 n = 1 .

C o r r e c t i o n . I n t h e a r t i c l e " R e a c t i o n s I n v o l v e d i n B i o -l u m i n e s c e n c e S y s t e m s o f L i m p e t ( L a t i a n e r i t o i d e s ) a n dL u m i n o u s B a c t e r i a , " b y S h i m o m u r a , O . , J o h n s o n , F . H . &K o h a m a , Y . , w h i c h a p p e a r e d i n t h e A u g u s t 1 9 7 2 i s s u e o f t h eP r o c . N a t . A c a d . S c i . USA 6 9 , 2 0 8 6 - 2 0 8 9 , i n t h e A b s t r a c t( p . 2 0 8 6 , l i n e 1 3 ) , " 0 . 1 7 + 0 . 1 p h o t o n s " s h o u l d r e a d : " 0 . 1 7 0 . 0 1 p h o t o n s . " On p a g e 2 0 8 8 , r i g h t - h a n d c o l u m n , l i n e 9f r o m t o p , " 0 . 1 5 4 i 0 . 1 e i n s t e i n / m o l " s h o u l d r e a d : " 0 . 1 5 44 0 . 0 1 e i n s t e i n / m o l " ; a n d , s a m e p a g e a n d c o l u m n , l i n e 1 3f r o m t o p , " 0 . 1 7 0 . 1 " s h o u l d r e a d : " 0 . 1 7 4 0 . 0 1 . "

c.l. pekeris- stationary spherical vortices in a perfect fluid

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