A. Lagerstrom and L. N. Edited by THE TO THE OF I

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T H E O R Y O F S I N G U L A R P E R T U R B A T I O N S W I T H A P P L I C A T I O N S T O T H E A S Y M P T O T I C T H E O R Y

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P . A . L a g e r s t r o m a n d L . N . H o w a r d

S a u l K a p l u n d e v e l o p e d his i d e a s about s i n g u l a r p e r ­ t u r b a t i o n s in the c o u r s e of studying a s y m p t o t i c s o l u t i o n s of the N a v i e r - S t o k e s e q u a t i o n s , in p a r t i c u l a r solutions f o r v i s c o u s f l o w at l o w R e y n o l d s n u m b e r s . A s a r e s u l t , a l m o s t a l l his d i s c u s s i o n of a s y m p t o t i c t h e o r y and technique is g i v e n in connection w i t h p r o b l e m s of v i s c o u s f l o w . T h i s I n t r o ­ duction w i l l attempt a s u r v e y of his b a s i c i d e a s . A h i s t o r i c a l account of how he d e v e l o p e d these i d e a s w i l l be g i v e n h e r e . It is h o p e d that s u c h a n a c c o u n t w i l l b e of h e l p f o r u n d e r ­ standing the l o g i c of his i d e a s although, f o r r e a s o n s stated, concepts f r o m F l u i d D y n a m i c s w i l l be u s e d . H o w e v e r , K a p l u n

1

s t h e o r y of s i n g u l a r p e r t u r b a t i o n s c o u l d , in p r i n c i p l e , be p r e s e n t e d without a n y r e f e r e n c e to v i s c o u s f l o w . T h e r e ­ f o r e s o m e s i m p l e m a t h e m a t i c a l m o d e l s w i l l be g i v e n l a t e r

on in this I n t r o d u c t i o n and it w i l l be i n d i c a t e d b r i e f l y how K a p l u n ' s i d e a s c a n be i l l u s t r a t e d w i t h the a i d of t h e s e e q u a t i o n s .

D e v e l o p m e n t of an A s y m p t o t i c T h e o r y f o r F l o w at L o w R e y n o l d s N u m b e r s . In this h i s t o r i c a l a c c o u n t w e s h a l l have to a s s u m e that a r e a d e r is f a m i l i a r w i t h the b a s i c t h e o r y of v i s c o u s f l u i d s . A s r e f e r e n c e f o r this t h e o r y the r e a d e r m a y c o n s u l t s t a n d a r d t e x t b o o k s . W e r e f e r a l s o to V a n D y k e (1964) w h i c h d e a l s s p e c i f i c a l l y w i t h p e r t u r b a t i o n p r o b l e m s a n d a p p l i e s K a p l u n ' s technique to a l a r g e v a r i e t y of p r o b l e m s a n d to L a g e r s t r o m (1964) w h i c h , in the d i s ­ c u s s i o n of v i s c o u s f l o w , m a k e s e x t e n s i v e u s e of i d e a s due to K a p l u n . T h e notation u s e d in this I n t r o d u c t i o n is e x p l a i n e d in the a r t i c l e s r e p r i n t e d h e r e a s C h a p t e r s II and I I I .

The p r o b l e m of d i s c u s s i n g P r a n d t l ' s b o u n d a r y - l a y e r t h e o r y ( f o r f l o w at l a r g e R e y n o l d s n u m b e r s ) in the light of the t h e o r y of a s y m p t o t i c e x p a n s i o n s had o c c u p i e d v a r i o u s r e s e a r c h w o r k e r s at G A L C I T s i n c e b e f o r e 1950. K a p l u n ' s f i r s t c o n t r i b u t i o n to this r e s e a r c h w a s his d o c t o r ' s t h e s i s on the r o l e of the c o o r d i n a t e s y s t e m in b o u n d a r y - l a y e r t h e o r y ; the p u b l i s h e d v e r s i o n of this is r e p r i n t e d h e r e a s C h a p t e r I . V e r y little c o m m e n t is n e e d e d on this c h a p t e r ( i n f a c t , E d i t o r s ' N o t e s w e r e d i s p e n s e d w i t h ) . It is a s e l f - c o n t a i n e d

p a p e r w h i c h s o l v e s the g i v e n p r o b l e m i n d i c a t e d b y the title in an i n g e n i o u s w a y b y going d i r e c t l y to the h e a r t of the m a t t e r . In e s s e n c e the p r o b l e m p o s e d is c o m p l e t e l y s o l v e d , although o b v i o u s l y s e v e r a l g e n e r a l i z a t i o n s of the t h e o r y s u g ­ g e s t t h e m s e l v e s .

A f t e r having c o m p l e t e d his t h e s i s , K a p l u n s t a r t e d his w o r k on the t h e o r y of f l o w at low R e y n o l d s n u m b e r s . T h e p r o b l e m s of f l o w at l a r g e and s m a l l R e y n o l d s n u m b e r s , r e s p e c t i v e l y , e x h i b i t a c e r t a i n s i m i l a r i t y . In the f o r m e r c a s e one m a y obtain an a p p r o x i m a t e s o l u t i o n b y n e g l e c t i n g the v i s c o u s t e r m s in the N a v i e r - S t o k e s e q u a t i o n s . T h i s a p p r o x i m a t i o n ; i s , h o w e v e r , not u n i f o r m l y v a l i d n e a r the b o u n d a r y of a s o l i d . P r a n d t l ' s b o u n d a r y - l a y e r t h e o r y of 1904 w a s an i n g e n i o u s attempt to cope w i t h this difficulty and is in fact a c l a s s i c a l e x a m p l e of one of the i m p o r t a n t m e t h o d s in the t h e o r y of s i n g u l a r p e r t u r b a t i o n s . F o r the c a s e of f l o w at l o w R e y n o l d s n u m b e r s Stokes p r o p o s e d in 1850 that the t r a n s p o r t t e r m s be n e g l e c t e d in the N a v i e r - Stokes e q u a t i o n s . T h e s u c c e s s as w e l l as the difficulties of this t h e o r y a r e w e l l - k n o w n . A p a r t i a l e x p l a n a t i o n of the difficulties w a s found b y O s e e n a r o u n d 1910, w h o pointed out that the Stokes a p p r o x i m a t i o n w a s not u n i f o r m l y v a l i d at infinity. O s e e n i n t r o d u c e d the "extended Stokes e q u a t i o n s

1 1

, now k n o w n as the O s e e n e q u a t i o n s . T h e s e equations e v i d e n t l y w e r e an i m p r o v e m e n t o v e r the Stokes e q u a t i o n s in s o m e

r e s p e c t s ; h o w e v e r a r a t i o n a l d i s c u s s i o n of t h e i r m e a n i n g and t h e i r v a l i d i t y w a s m i s s i n g . A n y w a y in both the c a s e of l a r g e R e y n o l d s n u m b e r s and that of s m a l l R e y n o l d s n u m ­ b e r s a b a s i c fact is the l a c k of u n i f o r m v a l i d i t y of c e r t a i n a p p r o x i m a t i o n s . G u i d e d b y this s i m i l a r i t y K a p l u n s t a r t e d to i n v e s t i g a t e w h e t h e r a unified t h e o r y of s i n g u l a r p e r t u r ­ bations c o u l d be a p p l i e d to both c a s e s . T h e u l t i m a t e r e s u l t of his i n v e s t i g a t i o n s w a s not only the f i r s t s y s t e m a t i c e x ­ p l a n a t i o n of m e a n i n g of the Stokes a n d O s e e n solutions and t h e i r p l a c e in an a s y m p t o t i c e x p a n s i o n of the c o r r e s p o n d i n g N a v i e r - S t o k e s s o l u t i o n s , but a l s o s o m e v e r y s i g n i f i c a n t and d e e p i d e a s about s i n g u l a r p e r t u r b a t i o n p r o b l e m s , p o w e r f u l and g e n e r a l e n o u g h to be a p p l i c a b l e to a w i d e v a r i e t y of pr o b l e m s .

P r e v i o u s l y m a n y a u t h o r s had a p p l i e d the i d e a of l i m i t p r o c e s s e s to t h e o r y of f l o w at l a r g e R e y n o l d s n u m b e r s : a n outer l i m i t , w h i c h y i e l d s the i n v i s c i d solution and a n i n n e r l i m i t w h i c h y i e l d s the b o u n d a r y - l a y e r solution (cf. C h a p t e r I ) . S i m i l a r l y (cf. C h a p t e r s II a n d I I I ) a s R e y n o l d s n u m ­ b e r tends to z e r o one m a y c o n s i d e r t w o p r i n c i p a l l i m i t s .

The outer l i m i t ( O s e e n l i m i t ) m a y be thought of as letting the c h a r a c t e r i s t i c l e n g t h of the b o d y g o to z e r o w h i l e e v e r y ­ thing e l s e is f i x e d . 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 u n ­ c h a n g e d u n d e r this l i m i t . T h e inner l i m i t ( S t o k e s l i m i t ) m a y be c o n v e n i e n t l y thought of as letting v i s c o s i t y tend to infinity; a p p l i e d to the N a v i e r - S t o k e s e q u a t i o n s it y i e l d s the Stokes e q u a t i o n s . K a p l u n i n t r o d u c e d the s y s t e m a t i c use of these two l i m i t s . U n d e r the outer l i m i t the finite b o d y s h r i n k s to a point w h i c h cannot d i s t u r b the f l o w . T h u s it is p l a u s i b l e that the f r e e s t r e a m v e l o c i t y q* at any f i x e d p o i n t tends to the f r e e - s t r e a m v a l u e i . T h e c o n v e r g e n c e is, h o w e v e r , not u n i f o r m n e a r the b o d y s i n c e q* = 0 at the s u r f a c e of the body. O r , to put it s l i g h t l y d i f f e r e n t l y , a v e r y s m a l l b o d y i n t r o d u c e d in a u n i f o r m s t r e a m c a u s e s a p e r t u r b a t i o n w h i c h is v e r y s m a l l e x c e p t at the s u r f a c e of the b o d y . It is this l a c k of u n i f o r m i t y of the outer l i m i t n e a r the b o d y w h i c h s u g g e s t s the i n t r o d u c t i o n of the i n n e r l i m i t d e s c r i b e d a b o v e . A n e q u i v a l e n t d e s c r i p t i o n of the inner l i m i t is the f o l l o w i n g : A s the b o d y s h r i n k s to a point, s a y the o r i g i n , one m e a s u r e s the v e l o c i t y , not a t , a f i x e d point, but at a point w h i c h a p p r o a c h e s the o r i g i n at the s a m e r a t e at w h i c h the b o d y ^ s h r i n k s . U s i n g e i t h e r d e s c r i p t i o n one s e e s that the v a l u e q* = 0 at the b o d y does not

, f

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1

w h e n the inner l i m i t p r o c e s s is a p p l i e d . Note that O s e e n e m p h a s i z e d the l a c k of u n i f o r m i t y of the Stokes e q u a t i o n s at oo; K a p l u n e m p h a s i z e d the l a c k of u n i f o r m i t y of the outer l i m i t n e a r the body. In fact he i n s i s t e d that w h i l e the outer and i n n e r l i m i t s a r e the t w o p r i n c i p a l l i m i t s ( s e e end of S e c t i o n L Z of C h a p t e r I V ) the outer l i m i t has p r i o r i t y , it is " m o r e p r i n c i p a l '

1

. ( S e e end of Section 1.5 of C h a p t e r I V ) . The p h y s i c a l r e a s o n f o r this is that the nature of the p e r t u r ­ b a t i o n p r o b l e m is that a u n i f o r m s t r e a m ( = i = outer l i m i t of q* ) is being p e r t u r b e d b y the p r e s e n c e of a s m a l l b o d y . T h i s is p h y s i c a l l y r e a s o n a b l e , and the m a t h e m a t i c a l technique for c o n s t r u c t i n g the a s y m p t o t i c e x p a n s i o n s c o n s i s t s in f i r s t finding the l e a d i n g t e r m of the outer s o l u t i o n ( i n this c a s e g

Q

The technique of using inner a n d outer l i m i t s w a s , of c o u r s e , not new. The new thing in the r e a s o n i n g d e s c r i b e d a b o v e c o n s i s t e d e s s e n t i a l l y in r e a l i z i n g that this technique c o u l d be a p p l i e d to the p r o b l e m of f l o w at l o w R e y n o l d s n u m ­ b e r s and in finding s u i t a b l e i n n e r and outer l i m i t s a n d f i n a l l y in r e c o g n i z i n g the p r i o r i t y of the outer l i m i t . A c c o r d i n g to the state of the a r t in 195 3-54 the next thing w o u l d be to

c o n s t r u c t the inner and outer e x p a n s i o n s , in p r i n c i p l e o b ­ tainable b y r e p e a t e d a p p l i c a t i o n s of the r e s p e c t i v e l i m i t p r o c e s s e s to the e x a c t solutions and in p r a c t i c e b y s o l v i n g c e r t a i n a p p r o x i m a t e d i f f e r e n t i a l equations w i t h a p p r o p r i a t e b o u n d a r y conditions a n d m a t c h i n g conditions. In b o u n d a r y - l a y e r t h e o r y ( s e e C h a p t e r I ) the s i m p l e s t f o r m of the m a t c h i n g condition states that the t a n g e n t i a l v e l o c i t y c o m ­ ponent at the w a l l g i v e n b y the outer solution has the s a m e v a l u e as that g i v e n b y the inner ( b o u n d a r y - l a y e r ) solution at infinity. A m o r e g e n e r a l f o r m is that the i n n e r l i m i t of the outer l i m i t should a g r e e w i t h the outer l i m i t of the i n n e r l i m i t . A s t i l l m o r e g e n e r a l f o r m is that s o m e p a r t i a l s u m of the outer e x p a n s i o n , e v a l u a t e d for s m a l l v a l u e s of the a r g u m e n t , should a g r e e w i t h s o m e p a r t i a l s u m of the i n n e r e x p a n s i o n , e v a l u a t e d f o r l a r g e v a l u e s of the a r g u m e n t .

C o n s i d e r now the c a s e of t w o - d i m e n s i o n a l f l o w . In this c a s e the f i r s t two f o r m s of the m a t c h i n g condition f a i l c o m p l e t e l y ; the l a s t f o r m m a y be v e r i f i e d a p o s t e r i o r i but hides the nature of matching. T o see that the f i r s t two f o r m s f a i l w e o b s e r v e that the i n n e r l i m i t of q# is z e r o ( a n a n a l o g y w i t h the heat equation m a k e s this quite p l a u s i b l e a p r i o r i and it m a y be m a d e s t i l l m o r e p l a u s i b l e a p o s t e r i o r i ) . Since the outer l i m i t of q* is i , it is o b v i o u s that the i n n e r and outer l i m i t cannot be m a t c h e d b y a n y p r o c e s s . T h i s l e d K a p l u n to a t h o r o u g h rethinking of the p r i n c i p l e s u n d e r l y i n g the techniques of m a t c h i n g .

The b a s i c q u e s t i o n i s : What a r e the e s s e n t i a l conditions for two a s y m p t o t i c a p p r o x i m a t i o n s to m a t c h ? T o a n s w e r this q u e s t i o n K a p l u n c o n s i d e r e d the p a r t i a l l y o r d e r e d set of e q u i ­ v a l e n c e c l a s s e s ( o r d c l a s s e s , o r d f ) of functions f ( R e )

such that o r d R e < o r d f < o r d 1. ( F o r the definition of these t e r m s see the s e c t i o n on I n t e r m e d i a t e L i m i t s in C h a p t e r II or Section 1 of C h a p t e r I V ) . T o e a c h s u c h f ( R e ) t h e r e is a c o r r e s p o n d i n g l i m i t p r o c e s s defined; the o r d e r c l a s s e s a r e a l s o u s e d in d e s c r i b i n g the d o m a i n of v a l i d i t y of an a p p r o x i ­ mation, (cf. C h a p t e r II and Section 1.2 of C h a p t e r I V ) . L e t us c o n s i d e r the t w o - d i m e n s i o n a l p r o b l e m s in p o l a r c o o r d i n a t e s and d i s r e g a r d the d e p e n d e n c e on θ ( w h i c h does not l e a d to n o n - u n i f o r m i t i e s ) . T h e outer v a r i a b l e is then r = U r / v

and the inner v a r i a b l e is r* = r / R e . L e t w be a f l o w quantity, s a y q * , and wj_ and W £ t w o a p p r o x i m a t i o n s . A s s u m e that w\ is u n i f o r m l y v a l i d in the o r d e r d o m a i n Dj_, c o n s i s t i n g of a l l f s u c h that o r d gi < o r d f < o r d 1. T h i s m e a n s that as R e tends to z e r o w - w]_ tends to z e r o

u n i f o r m l y f o r f ^ ( R e ) < r < co w h e r e £\ is a n y function in the o r d e r d o m a i n D . In other w o r d s , g i v e n a n y δ one m a y find an e j§ s u c h that | w - W ] J < δ in the d o m a i n b o u n d e d b y the h o r i z o n t a l l i n e s R e = 6^ 5 and R e = 0 and the c u r v e τ = ^ ( R e ) ( S e e F i g . 1 ) .

F i g . 1 O v e r l a p p i n g D o m a i n s of V a l i d i t y

S i m i l a r l y , w e m a y s p e a k of W 2 b e i n g a u n i f o r m l y v a l i d a p p r o x i m a t i o n in an o r d e r d o m a i n D£ c o n s i s t i n g of a l l f ( R e ) s u c h that o r d R e < f < o r d g£ . T h i s m e a n s that

w - W 2 a p p r o a c h e s z e r o u n i f o r m l y f o r R e < r < f 2 ( R e ) w h e r e o r d £2 is any o r d e r c l a s s in D ^ . C l e a r l y w-^ and W 2 c a n be m a t c h e d only if they have d o m a i n s of v a l i d i t y w h i c h o v e r - lap, that is w h i c h have o r d e r c l a s s e s in c o m m o n . In the p r e s e n t c a s e this m e a n s that o r d g\ < o r d g 2 · The functions fj a n d f 2 m a y then be c h o s e n s u c h that f^ < Î 2 f o r R e > 0 , and t h e r e e x i s t s a functio n f 3 , lyin g i n b e t w e e n f ^ an d f 2 . A m a t c h i n g conditio n i s the n s i m p l y lim £ ( w , - w ^ ) = 0 .

I n c i d e n t a l l y , if r = R e r e p r e s e n t s the s u r f a c e of the b o d y and if and W 2 have o v e r l a p p i n g d o m a i n s as d e s c r i b e d a b o v e , then one has in p r i n c i p l e a u n i f o r m l y v a l i d solution ( t o o r d e r unity). K a p l u n c o n s i d e r e d the p r o b l e m of using

and W 2 to c o n s t r u c t one c o m p o s i t e a p p r o x i m a t i o n ( w h i c h contains both) to be of l e s s f u n d a m e n t a l i m p o r t a n c e , although its p r a c t i c a l i m p o r t a n c e m a y be c o n s i d e r a b l e .

It m a y now be s e e n w h y the i n n e r and outer l i m i t s cannot be e x p e c t e d to o v e r l a p . ( W e s t i l l c o n s i d e r t w o - d i m e n s i o n a l f l o w at l o w R e y n o l d s n u m b e r ) . A p r i o r i one m a y e x p e c t the outer l i m i t to be v a l i d f o r r > A w h e r e A is any constant > 0 and the i n n e r l i m i t to be~~~valid f o r 1 < r * < Β = constant o r R e < r < Β R e . It is then c l e a r that no m a t t e r how s m a l l A is c h o s e n a n d no m a t t e r how l a r g e Β is c h o s e n , t h e r e is no a p r i o r i r e a s o n to e x p e c t o v e r l a p (cf. F i g . 2 ) .

R e

B o d y S u r f a c e —

— r = A

' r = Β R e

Outei

F i g . 2 N o n - o v e r l a p of O u t e r and I n n e r L i m i t s

The E x t e n s i o n T h e o r e m , to be d i s c u s s e d b e l o w , cannot a p r i o r i b e e x p e c t e d to r e m e d y the situation and, of c o u r s e , does not do s o in a c t u a l fact in the p r e s e n t c a s e of t w o - d i m e n s i o n a l f l o w .

The q u e s t i o n then a r i s e s how to find two a p p r o x i m a t i o n s w h i c h b e t w e e n t h e m c o v e r the e n t i r e d o m a i n R e < r < co , i.e. b e h a v e like the w^ a n d w ^ d e s c r i b e d a b o v e . It s e e m s p l a u s i b l e ( b y a n a l o g y w i t h solutions of the O s e e n e q u a t i o n s ) that if f = R e ^ ( 0 <

a <

f

(l-a)T.

a p p l i e d to the N a v i e r - S t o k e s e q u a t i o n s y i e l d s the Stokes equation. T h u s D is the f o r m a l d o m a i n of v a l i d i t y of the Stokes equations (this is j u s t a definition). K a p l u n then m a d e the r e a s o n a b l e H y p o t h e s i s of V a l i d i t y that t h e r e e x i s t s a solution of the Stokes e q u a t i o n s w h o s e a c t u a l d o m a i n of v a l i d i t y c o i n c i d e s w i t h the f o r m a l d o m a i n of v a l i d i t y of the e q u a t i o n s . T h i s s o l u t i o n , w h i c h need not be a l i m i t , w i l l then

o v e r l a p w i t h the outer l i m i t p r o v i d e d the d o m a i n of v a l i d i t y of the outer solution c a n be e x t e n d e d e v e r s o s l i g h t l y , i.e.

to a d o m a i n Di : o r d f^ < o r d f < 1 w h e r e o r d f^ is d e f i n i t e l y s m a l l e r than, though p o s s i b l y v e r y c l o s e to, o r d 1. T h i s E x t e n s i o n T h e o r e m is e a s i l y p r o v e d (the b a s i c i d e a is g i v e n in C h a p t e r II , s e c t i o n on M a t c h i n g ) . A s u i t a b l e s o l u t i o n of the Stokes e q u a t i o n s is the function \T

0

0

0

hides the n a t u r e of m a t c h i n g . It is the v e r y f i r s t d e s c r i p t i o n of u

0

(l-a)T*

(l-a)T

a

. The function e ( R e ) is d e t e r m i n e d b y m a t c h i n g w i t h the outer l i m i t T* . The o v e r l a p d o m a i n is a c t u a l l y v e r y s m a l l ; o r d R e ^ f o r any Ot is outside the o v e r l a p d o m a i n .

One c a n now see the m e a n i n g of the O s e e n e q u a t i o n s . If the outer e x p a n s i o n is i 4 6 g^, then gj ( a s w e l l a s

i + € ) o b e y s the O s e e n e q u a t i o n s . F o r i n s t a n c e , K a p l u n

1

s a n a l y s i s m a d e it obvious that the O s e e n equations a r e v a l i d at s m a l l R e ( b a s e d on r a d i u s of c u r v a t u r e at the n o s e ) f o r f l o w p a s t a p a r a b o l o i d of r e v o l u t i o n , but not f o r f l o w p a s t a p a r a b o l a . In the f o r m e r c a s e the outer l i m i t of q* is i ; in the l a t t e r c a s e it is v i s c o u s f l o w p a s t a s e m i - i n f i n i t e flat plate. It a l s o m a d e it o b v i o u s that the O s e e n equations a r e by nature l i n e a r , i.e. , l i n e a r i z a t i o n a r o u n d f r e e s t r e a m v e l o c i t y , w h e r e a s the Stokes e q u a t i o n s a r e l i n e a r only b y accident. F o r c o m p r e s s i b l e f l o w the Stokes e q u a t i o n s a r e n o n - l i n e a r (cf. L a g e r s t r o m (1964), p. 190 ff. ) . The s t a t e ­ ments j u s t m a d e i l l u s t r a t e s o m e e s s e n t i a l i d e a s of K a p l u n ' s and they w i l l t h e r e f o r e be e l a b o r a t e d . A s o b s e r v e d p r e ­ v i o u s l y , if l i m f is a p p l i e d to the N a v i e r - S t o k e s equations they do not change if o r d f = 1 and tend to the Stokes e q u a ­ tions in the a d j o i n i n g o r d e r d o m a i n o r d R e < o r d f < 1 , K a p l u n ' s b a s i c i d e a is then that a solution of the Stokes equation should m a t c h a s o l u t i o n of the N a v i e r - S t o k e s e q u a ­ tions. T h e Stokes e q u a t i o n s (but not n e c e s s a r i l y the Stokes s o l u t i o n ) a r e thus o b t a i n e d b y a l i m i t p r o c e s s a n d m a y be n o n - l i n e a r (they a c t u a l l y a r e in the c a s e of c o m p r e s s i b l e f l o w ) . The N a v i e r - S t o k e s equations a r e c e r t a i n l y n o n - l i n e a r . H o w e v e r , since they a r e obtained b y l i m i t s p e r t a i n i n g to one c l a s s only ( o r d f = o r d 1) w e m a y a s s u m e that the r e l e v a n t N a v i e r - S t o k e s solution is obtained b y a p p l y i n g lirri£ , w i t h o r d f = o r d 1 , to the e x a c t solution ( w e a s s u m e that o r d f

= o r d R e and o r d f = o r d 1 g i v e the p r i n c i p a l l i m i t s a n d that hence w e a r e only i n t e r e s t e d l i m f f o r o r d R e < o r d f

< o r d 1). U n d e r the outer l i m i t a finite b o d y r e d u c e s to a point and a p a r a b o l o i d of r e v o l u t i o n to a n e e d l e . T h e l i m i t i n g c o n f i g u r a t i o n s cannot influence the flow and hence w e m a y a s s u m e that g^ = outer l i m i t of q* is e q u a l to i w h i c h r e p r e s e n t s the f r e e s t r e a m f l o w . The next t e r m in the outer e x p a n s i o n s , n a m e l y gj then o b e y s the l i n e a r O s e e n equations, i.e. the equations o b t a i n e d b y l i n e a r i z i n g a r o u n d u n d i s t u r b e d f l o w . Nothing in this r e a s o n i n g w o u l d change f o r c o m p r e s s i b l e f l o w ( o m i t t i n g the s u p e r s o n i c c a s e w h e r e the b e h a v i o r of the s h o c k l a y e r s at infinity has to be c o n s i d e r e d ) . T h u s in this c a s e the O s e e n equations a r e l i n e a r a n d the Stokes e q u a t i o n s a r e not. L e t now the b o d y be a p a r a b o l a ; u n d e r the outer l i m i t this b o d y tends to a s e m i - i n f i n i t e flat p l a t e . H e n c e in this c a s e o b e y s e q u a t i o n s o b t a i n e d l i n e a r i z i n g a r o u n d the f l o w f i e l d p a s t a s e m i - i n f i n i t e flat p l a t e . T h e s e equations a r e not the O s e e n e q u a t i o n s ; they a r e b y nature l i n e a r but

cannot be w r i t t e n d o w n e x p l i c i t l y until one d e t e r m i n e s the f l o w f i e l d a r o u n d a s e m i - i n f i n i t e flat p l a t e .

A n o t h e r f e a t u r e of the O s e e n equations is now m a d e c l e a r . A l t h o u g h the O s e e n equations a r e outer equations they have s o l u t i o n s w h i c h a r e u n i f o r m l y v a l i d to o r d e r 6 ( r e l a t i v e to N a v i e r - S t o k e s s o l u t i o n s ) . The r e a s o n is that the O s e e n e q u a t i o n s a r e " r i c h enough" to contain the Stokes equations in the s e n s e that the a p p l y i n g inner l i m i t of the O s e e n e q u a t i o n s y i e l d s the Stokes e q u a t i o n s . A g a i n , this is true only f o r i n c o m p r e s s i b l e f l o w . F o r c o m p r e s s i b l e flow the Stokes equations a r e not contained in the O s e e n equations and one cannot e x p e c t to find a u n i f o r m l y v a l i d O s e e n solution.

The e s s e n c e of the i d e a s and technique d e s c r i b e d a b o v e w e r e f i r s t a n n o u n c e d ( a n d c l e a r l y a t t r i b u t e d to K a p l u n ) in

L a g e r s t r o m a n d C o l e (1955), u n d e r the heading " M e t h o d of K a p l u n

1 1

( S e c t i o n 6.3.2. ) .

W e d e s c r i b e d a b o v e K a p l u n ' s intuitive e x p l a n a t i o n w h y t h e r e e x i s t s a Stokes s olution w h i c h m a t c h e s the outer l i m i t . The c o r r e s p o n d i n g p r o b l e m a r i s e s f o r h i g h e r o r d e r t e r m s . One m a y often v e r i f y b y a d i r e c t c h e c k that t h e r e is an o v e r ­ lap d o m a i n f o r p a r t i a l s u m s of two e x p a n s i o n s ( o r at l e a s t that this a s s u m p t i o n does not l e a d to c o n t r a d i c t i o n s ) . H o w ­ e v e r , it is d e s i r a b l e to have an intuitive e x p l a n a t i o n w h y an o v e r l a p d o m a i n e x i s t s . K a p l u n g a v e this p r o b l e m c o n s i d e r ­ a b l e thought. W e s h a l l r e t u r n to this topic l a t e r on in this Introduction.

M a t h e m a t i c a l M o d e l s . W e i n t e r r u p t the d i s c u s s i o n of the p r o b l e m of f l o w at l o w R e y n o l d s n u m b e r in o r d e r to i n t r o ­ duce s o m e s i m p l e m a t h e m a t i c a l m o d e l s w h i c h m a y b e u s e d to i l l u s t r a t e K a p l u n ' s i d e a s .

The f i r s t m o d e l e q u a t i o n w a s o r i g i n a l l y i n t r o d u c e d b y K. O. F r i e d r i c h s to e x p l a i n P r a n d t l ' s b o u n d a r y - l a y e r t h e o r y ; it has often b e e n e m p l o y e d in the d i s c u s s i o n of s i n g u l a r p e r t u r b a t i o n p r o b l e m s . T h e equation is

e

y

w i t h b o u n d a r y conditions

y(0) = 0 , y ( l ) = 1 ( 2 )

H e r e " p r i m e " denotes d e r i v a t i v e w i t h r e s p e c t to the v a r i a b l e x. W e s e e k a n a s y m p t o t i c solution v a l i d as € J 0 . B y

s t a n d a r d m e t h o d s one finds that t h e r e is a b o u n d a r y l a y e r at χ = 0 a n d that its t h i c k n e s s is of o r d e r 6 . In other w o r d s ,

the a p p r o p r i a t e i n n e r l i m i t is o b t a i n e d b y holding x* = x / e f i x e d , w h e r e a s the outer l i m i t is o b t a i n e d b y holding χ f i x e d . The o r d e r d o m a i n to be c o n s i d e r e d is the

D = { o r d f J o r d e < o r d f < o r d 1 } ( 3 ) W e define the v a r i a b l e xf b y f(6 )x£ = χ ; then l i m f is the l i m i t as 6 | 0 , xf being f i x e d . A p p l y i n g lirri£ ( w i t h ord£

in D ) one obtains t h r e e p o s s i b l e e q u a t i o n s ,

y" = g> ( x ) o r d f = o r d 1 ( 4 )

y

1

= 0 o r d € < o r d f < o r d 1 ( 5 )

6 y » + y" = 0 o r d f = o r d 6 ( 6 )

F r o m E q . ( 5 ) a n d the condition y ( l ) = 1 w e obtain the solution of E q . ( 4 )

h

0

0

0

D

1

x

solution w h i c h is v a l i d in the d o m a i n

D

2

0

l i m

f

0

k

0

i

0

_ X

' ) ( 1 2 )

T h i s solution is v a l i d in s o m e d o m a i n

D

3

3

Since o r d € is in this d o m a i n the condition IQ(0) = 0 m u s t be s a t i s f i e d . T h u s E q . (12) contains only one constant of i n t e g r a t i o n A( e ) . M a t c h i n g w i t h s o m e f s u c h that o r d f is in both and g i v e s us

A ( 0 ) = k

0

T h i s i s , of c o u r s e , the c l a s s i c a l r e s u l t that

i o ( o o ) = h

0

or that

l i m ^ i

0

x

* ) = l i m .

h

o ( x ) (14b) outer

o x 1

i n n e r

o x 1 x

' O b v i o u s l y , the m e t h o d u s e d is m u c h too c l u m s y f o r a c t u a l c o m p u t a t i o n of the p r e s e n t c a s e . H o w e v e r , it d o e s g i v e a d e e p e r e x p l a n a t i o n of the matching ( a l t h o u g h , of c o u r s e , the m a t c h i n g p r i n c i p l e has not bee η p r o v e n w i t h m a t h e m a t i c a l r i g o r ) a n d in tackling m o r e difficult p r o b l e m s the m o r e f u n d a m e n t a l v i e w p o i n t m a y be v e r y helpful. W e note the f o l l o w i n g s i m p l i f i c a t i o n w h i c h is s t i l l in the s p i r i t of the g e n e r a l a p p r o a c h : If a n i n t e r m e d i a t e l i m i t , i.e. a liiri£

w i t h o r d f in is a p p l i e d to E q . ( 4 ) , one obtains E q . ( 5 ) . One m a y s a y that E q . ( 4 ) is " r i c h e n o u g h to y i e l d E q . ( 5 ) "

or s i m p l y that it "contains E q . ( 5 ) " . (cf. the s t a t e m e n t m a d e e a r l i e r that the O s e e n equations contain the Stokes e q u a t i o n s in the i n c o m p r e s s i b l e c a s e ) . S i m i l a r l y E q . ( 6 ) contains E q . ( 5 ) . T h u s it is n a t u r a l to state that the f o r m a l d o m a i n of v a l i d i t y of E q . ( 4 ) is a c t u a l l y the union of the d o m a i n s and D£ and that the f o r m a l d o m a i n of v a l i d i t y of E q . ( 6 ) is the union of and Dy T h u s in this m o d e l e x a m p l e t h e r e is a v e r y l a r g e o v e r l a p d o m a i n , n a m e l y D ^ . A v e r y s i m i l a r situation o c c u r s in the matching of the outer f l o w and b o u n d a r y - l a y e r f l o w in the a s y m p t o t i c t h e o r y of f l o w at l a r g e R e . T h e d i s c u s s i o n g i v e n a b o v e i s , in fact, a p a r a ­ p h r a s e of a l e t t e r b y K a p l u n about b o u n d a r y - l a y e r t h e o r y .

M a n y of the p e c u l i a r i t i e s of the a s y m p t o t i c e x p a n s i o n s of N a v i e r - S t o k e s s o l u t i o n s f o r s m a l l R e y n o l d s n u m b e r s m a y be studied w i t h the a i d of the f o l l o w i n g m a t h e m a t i c a l m o d e l

, 2 r d r

&

d r dr

g = 0 f o r r = ρ

g = 1 for r = oo (15c) The p r o b l e m is to find a n a s y m p t o t i c e x p a n s i o n as ρ \ 0.

H e r e one m a y think of the point r = ρ a s the s u r f a c e of a s m a l l s p h e r e in η - d i m e n s i o n a l s p a c e . The " t e m p e r a t u r e "

g is z e r o at the s u r f a c e of the s p h e r e and unity at infinity.

The t e r m g d g / d r m a y b e thought of as r e p r e s e n t i n g a non­

l i n e a r heat l o s s . T h e d i s c u s s i o n of f l o w at l o w R e y n o l d s n u m b e r g i v e n in this I n t r o d u c t i o n and in C h a p t e r s II and III m a y be e a s i l y t r a n s l a t e d in t e r m s of this m o d e l e x a m p l e . T h e b a s i c p h e n o m e n a a r e the s a m e but the a n a l y t i c a l w o r k is

c o n s i d e r a b l y e a s i e r ; it is e v e n p o s s i b l e to p r o v e r i g o r o u s l y the v a l i d i t y of c e r t a i n r e s u l t s obtained b y a p p l y i n g K a p l u n ' s m e t h o d to this m o d e l ( s e e the b o o k b y J u l i a n D . C o l e ,

" P e r t u r b a t i o n M e t h o d s in A p p l i e d M a t h e m a t i c s " , to be p u b l i s h e d b y B l a i s d e l l P u b l i s h i n g C o m p a n y ) . The outer v a r i a b l e is r , the i n n e r v a r i a b l e is r * = r / p . It is intuiti­

v e l y p l a u s i b l e that, at a f i x e d point, the p e r t u r b a t i o n due to the b o d y d i s a p p e a r s as ρ tends to z e r o ( f o r η > 2). T h u s u n d e r the outer l i m i t g —* 1, but not u n i f o r m l y f o r r n e a r ρ . If l i m f is a p p l i e d to E q . (15a) w i t h o r d ρ < o r d f < o r d 1, the f i r s t t w o t e r m s of the equation r e m a i n and the t h i r d tends to z e r o . T h e r e s u l t i n g equation m a y be c a l l e d the Stokes equation. F o r n = 2 this e q u a t i o n has no solution w h i c h s a t i s f i e s a l l b o u n d a r y conditions ( S t o k e s p a r a d o x ) . H o w ­ e v e r , a c c o r d i n g to K a p l u n ' s r e a s o n i n g d e s c r i b e d e a r l i e r in this I n t r o d u c t i o n this " p a r a d o x " is i r r e l e v a n t . A l l one m a y r e q u i r e is that t h e r e e x i s t a solution of the Stokes equation w h i c h m a t c h e s the outer l i m i t of g. F o r n = 2 such a

solution is - l / l o g 6 · log r # . F o r n = 3 the W h i t e h e a d p a r a d o x o c c u r s , i.e. the f i r s t i t e r a t i o n on the Stokes

solution is l o g a r i t h m i c at infinity. F o r η > 3 the p a r a d o x is s t i l l f u r t h e r d e l a y e d , but w i t h e n o u g h p a t i e n c e a n y b o d y can find a p a r a d o x in a n y d i m e n s i o n . T h e O s e e n equation is obtained b y l i n e a r i z i n g a r o u n d the v a l u e g = 1 , in other w o r d s the n o n - l i n e a r t e r m g d g / d r in E q . (15a) is r e p l a c e d

(15a)

(15b)

by the l i n e a r t e r m d g / d r . In a n a l o g y w i t h the c a s e of i n ­ c o m p r e s s i b l e v i s c o u s f l o w the Stokes e q u a t i o n of the p r e s e n t m o d e l is l i n e a r a n d is a l s o c o n t a i n e d in the O s e e n e q u a t i o n (in the s e n s e d i s c u s s e d in the p r e v i o u s e x a m p l e , i.e. the a p p l i c a t i o n of the i n n e r , or any i n t e r m e d i a t e l i m i t to the O s e e n e q u a t i o n y i e l d s the Stokes e q u a t i o n ) . The l a s t t w o facts a r e a c c i d e n t a l , a s m a y be s e e n b y studying the f o l l o w ­ ing m o d i f i e d v e r s i o n of E q . (15a)

ώ +

+

2

+

g =0 ) ( 1 6

, 2 r d r \ d r /

to

d r dr

(cf. the s t a t e m e n t m a d e e a r l i e r r e g a r d i n g c o m p r e s s i b l e f l o w at l o w R e ) . T h e r e a d e r i n t e r e s t e d in the m a t h e m a t i c a l a s p e c t s of the P a r t I but having no s p e c i a l i n t e r e s t in F l u i d D y n a m i c s is a d v i s e d to t r a n s l a t e the d i s c u s s i o n in C h a p t e r s I I , I I , a n d I V f r o m the N a v i e r - S t o k e s e q u a t i o n s to the m a t h e m a t i c a l m o d e l s d e s c r i b e d a b o v e .

H i g h e r O r d e r A p p r o x i m a t i o n s . F r i n g e R e g i m e . W e r e t u r n to the p r o b l e m of l o w R e y n o l d s n u m b e r f l o w p a s t a t w o - d i m e n s i o n a l finite b o d y ( w h i c h c o r r e s p o n d s to E q . ( 1 5 ) w i t h η = 2). In the f i r s t p a r t of this I n t r o d u c t i o n w e d i s ­

c u s s e d t h e j p r o b l e m of m a t c h i n g the outer l i m i t of the e x a c t solution ( g = i = outer l i m i t of q^ ) w i t h an a p p r o x i m a t i o n ϊΓ w h i c h is v a l i d n e a r the body. One of K a p l u n

1

s b a s i c i d e a s w a s that ξ s h o u l d be m a t c h e d , not w i t h a l i m i t , but w i t h a solution of a l i m i t i n g equation. T h e e s s e n t i a l r e q u i r e ­ ment on the function £Γ is that it s h o u l d have a s u f f i c i e n t l y l a r g e d o m a i n of v a l i d i t y ; this l e d to the i d e a that ϊ Γ s h o u l d be a solution of a n e q u a t i o n w h o s e f o r m a l d o m a i n o i P v a l i d i t y is s u f f i c i e n t l y l a r g e . The function uT is the f i r s t t e r m of an i n t e r m e d i a t e e x p a n s i o n w h e r e the p a r t i a l s u m s a r e d e t e r ­ m i n e d b y t h e i r o r d e r of v a l i d i t y r a t h e r than b y r e p e a t e d a p p l i c a t i o n of a l i m i t p r o c e s s . It s o h a p p e n s that the f i r s t t e r m of the i n t e r m e d i a t e e x p a n s i o n is the s e c o n d t e r m of the i n n e r e x p a n s i o n , etc. K a p l u n c o n s i d e r e d the p o s s i b i l i t y that in g e n e r a l a n y i n t e r m e d i a t e e x p a n s i o n is r e l a t e d in a s i m i l a r w a y to a l i m i t p r o c e s s e x p a n s i o n . Still, the concept of an i n t e r m e d i a t e e x p a n s i o n is m o r e f u n d a m e n t a l than that of an i n n e r e x p a n s i o n . T h u s K a p l u n d e - e m p h a s i z e d the

f u n d a m e n t a l r o l e of l i m i t - p r o c e s s e x p a n s i o n s (without denying t h e i r i m p o r t a n c e a s a m a t h e m a t i c a l t o o l ) . T h e r e a s o n i n g that U

q

F L U I D M E C H A N I C S A N D S I N G U L A R P E R T U R B A T I O N S

r i g o r o u s . H o w e v e r , if one p r o c e e d s to the next a p p r o x i ­ m a t i o n the r e a s o n i n g p r e v i o u s l y a p p l i e d to u~ s h o w s that ϊ Γ + 6 m a y be e x p e c t e d to be v a l i d ( t o o r d e r e ) only in trie d o m a i n o r d R e < o r d f < o r d 6. S i m i l a r l y , the n-th p a r t i a l s u m £Q 6^ vu c a n be e x p e c t e d to be v a l i d ( t o o r d e r e

n

) only f o r o r d R e < o r d f < o r d e

n

. If w e t r y to m a t c h w i t h the outer e x p a n s i o n 6^ g* w e s e e that t h e r e is a

η 3

3

"fringe r e g i m e " o r d 6 < o r d f < o r d 1 w h i c h cannot be b r i d g e d b y the e x t e n s i o n ~ t h e o r e m and w h e r e this o v e r l a p is in doubt. K a p l u n d i s c u s s e s this b r i e f l y in C h a p t e r I I I and in c o n s i d e r a b l e d e t a i l in C h a p t e r I V ( s e e E d i t o r ' s N o t e s to C h a p t e r s II a n d Π Ι ) . In p r i n c i p l e the f r i n g e r e g i m e r e q u i r e s the i n t e r p o l a t i o n of a d d i t i o n a l i n t e r m e d i a t e e x p a n s i o n s ; f o r p r a c t i c a l p u r p o s e s one m a y , h o w e v e r , a s s u m e that the

v a l i d i t y of the outer e x p a n s i o n extends o v e r the f r i n g e r e g i m e . I n t e g r a t e d E f f e c t s a n d S w i t c h b a c k . F o r m of a n

E x p a n s i o n and C o n s i s t e n c y . In a p e r t u r b a t i o n p r o b l e m one is f a c e d w i t h the f o l l o w i n g situation w h i c h in s c h e m a t i c a n d o v e r s i m p l i f i e d f o r m m a y be e x p r e s s e d a s f o l l o w s : T h e e x a c t equation is M = 6 Ν . H e r e M and Ν a r e functions of the independent v a r i a b l e ( o r v a r i b l e s ) a n d the dependent v a r i a b l e s and t h e i r d e r i v a t i v e s . A solution v a l i d to o r d e r unity ( i n s o m e o r d e r d o m a i n m a y be o b t a i n e d b y s o l v i n g the e q u a t i o n M = 0. One then e x p e c t s the m i s t a k e , i . e . the c o r r e c t i o n t e r m , to be of o r d e r 6, and hence the next t e r m in the e x p a n s i o n to be of this o r d e r . T h e c o r r e c t i o n t e r m to the solution of M = 0 m a y be thought of as due to the effect of the "forcing t e r m " € N o v e r the e n t i r e d o m a i n c o n s i d e r e d . T h e w e l l k n o w n p h e n o m e n a of r e s o n a n c e in f o r c e d o s c i l l a t i o n s s h o w that the i n t e g r a t e d effect of a f o r c i n g t e r m of o r d e r 6 m a y be of o r d e r unity p r o v i d e d the length of the time i n t e r v a l c o n s i d e r e d is of o r d e r l/ e . K a p l u n g a v e this p r o b l e m c o n s i d e r a b l e thought. A v e r y s t r i k i n g

i l l u s t r a t i o n of this p r o b l e m w a s g i v e n in P r o u d m a n a n d P e a r s o n ( 5 7 ) w h e r e the a u t h o r s s h o w e d that f o r the c a s e of flow p a s t a s p h e r e at l o w R e , a t e r m R e ^ l o g R e m u s t o c c u r : The t e r m s of o r d e r R e f o r the i n n e r and outer e x p a n s i o n s w e r e c o n s t r u c t e d a n d m a t c h e d ; h o w e v e r , then it w a s found that m a t c h i n g w a s i m p o s s i b l e f o r the t e r m s of o r d e r R e without i n s e r t i n g an i n t e r m e d i a t e t e r m of o r d e r R e ^ l o g R e . K a p l u n u s e d the e x p r e s s i o n " s w i t c h b a c k " to indicate that in t r y i n g to find the t e r m s of a c e r t a i n o r d e r one is f o r c e d to r e c o n s i d e r l o w e r o r d e r t e r m s . A d e t a i l e d

study o f s w i t c h b a c k effect s f o r th e s p e c i a l p r o b l e m o f v i s c o u s f l o w a t l a r g e d i s t a n c e s f r o m a finit e b o d y m a y b e foun d i n C h a n g (1961) ; v a r i o u s othe r e x a m p l e s a r e g i v e n i n V a n D y k e (1964) . S i m i l a r p r o b l e m s a r e d i s c u s s e d b y K a p l u n i n connection w i t h s e p a r a t i o n ( s e e P a r t I I . C h a p t e r I , i n p a r t i c u l a r S e c t i o n 2.5d) . I n a f r a g m e n t o f a d r a f t f o r a n a r t i c l e K a p l u n g i v e s th e f o l l o w i n g e x a m p l e o f s w i t c h b a c k du e to i n t e g r a t e d e f f e c t s . T h e functio n w i s d e f i n e d b y th e e q u a t i o n

ê ⁽

a ? ) • « <"* >

and th e b o u n d a r y condition s

w ( l ) = 0 (17b )

- I I w(a)

€ , a =

e

(17c ) W e c o n s i d e r th e d o m a i n Û function s f( e )

û = { f I f < < a } (18 )

T h e p r o b l e m i s t o fin d a functio n u s u c h tha t f i n / ) i m p l i e s that | w - u | — 0 u n i f o r m l y f o r 1 < r < f ( 6 ) . I f on e a s s u m e s that u ha s th e f o r m u = g ( r ) on e obtain s th e a n s w e r u = 0.

Since h o w e v e r

w = 6 lo g r + ( i - - l ) (19 )

the a n s w e r u = 0 i s f a l s e . On e m u s t a s s u m e u t o hav e th e f o r m u = g ( r ) + 6 h ( r ) . I n othe r w o r d s th e r a n g e o f r i s s o g r e a t tha t th e i n t e g r a t e d effec t o f th e f o r c i n g t e r m 6 i s o f o r d e r unity .

The g e n e r a l c h a r a c t e r o f s w i t c h b a c k i s tha t on e a s s u m e s the e x p a n s i o n t o hav e a c e r t a i n f o r m . T h e n , a s on e p r o c e e d s in th e a c t u a l c o n s t r u c t i o n , e v e r y t h i n g m a y l o o k c o n s i s t e n t u p to a c e r t a i n s t a g e . H o w e v e r , a t th e nex t s t a g e on e find s a n i n c o n s i s t e n c y ; th e p r o b l e m i s r e s o l v e d b y r e v i s i n g th e a s s u m e d f o r m o f th e e x p a n s i o n .

F o u n d a t i o n o f P e r t u r b a t i o n T h e o r y . F o r th e p r a c t i c a l c o n s t r u c t i o n o f e x p a n s i o n s th e i d e a s o f c o n s i s t e n c y , s w i t c h - b a c k , etc . a r e o f g r e a t v a l u e . T h e r e i s th e d a n g e r i n v o l v e d , h o w e v e r , w h i c h i s i l l u s t r a t e d b y th e p r o b l e m j u s t g i v e n . I f

one does not look f o r h i g h e r o r d e r t e r m s then, b y the c o n ­ s i s t e n c y a r g u m e n t , u = 0 s e e m s a p e r f e c t l y v a l i d a p p r o x i ­ m a t i o n to w to o r d e r unity. It is only w h e n one p r o c e e d s to o r d e r t that one finds that the a p p r o x i m a t i o n g i v e n is w r o n g . T h e s w i t c h b a c k is s t r o n g in this c a s e : It is not j u s t that one is f o r c e d to i n s e r t an u n e x p e c t e d t e r m of, say, o r d e r 6 l n 6 ; i n s t e a d one r e a l i z e s that the o r i g i n a l solution to o r d e r unity, w h i c h l o o k e d c o n s i s t e n t , is c o m p l e t e l y w r o n g . T h u s if one p r o c e e d s b y c o n s i s t e n c y a l o n e one n e v e r k n o w s w h a t d a n g e r m a y be l u r k i n g a r o u n d the next c o r n e r . F u r t h e r ­ m o r e , i n the f r a g m e n t j u s t r e f e r r e d to, K a p l u n states that one m a y m o d i f y the p r e v i o u s e x a m p l e ( E q s . (17) ff. ) s o a s to exhibit t r a n s c e n d e n t a l s w i t c h b a c k . B y this is m e a n t that an e x p a n s i o n m a y b e c o n s i s t e n t to a l l o r d e r s 6

n

; h o w e v e r , w h e n one c o n s i d e r s t e r m s of o r d e r e" V

£

one d i s c o v e r s the e x p a n s i o n w a s a c t u a l l y i n c o r r e c t . In other w o r d s , the i n ­ t e g r a t e d effect of a t e r m of o r d e r e" V

e

m a y be of o r d e r €

n

. Thus r e a s o n i n g b y c o n s i s t e n c y alone is f r a u g h t w i t h c e r t a i n d a n g e r s ; f u r t h e r m o r e it is " e s t h e t i c a l l y i m p r o p e r " . It w a s K a p l u n ' s hope that "if f o r c i n g t e r m s w e r e taken into account, p e r t u r b a t i o n t h e o r y w o u l d be c o m p l e t e in s o m e significant s e n s e

1 1

. H o w e v e r , the v a r i o u s f r a g m e n t a r y d r a f t s of a r t i c l e s w h i c h have b e e n f o u n d d o not g i v e a definite i n d i c a ­ tion of w h a t he intended. H i s m o s t i m p o r t a n t , r e l a t i v e l y c o m p l e t e study of the foundations of p e r t u r b a t i o n t h e o r y a r e c o n t a i n e d in C h a p t e r I V e s p e c i a l l y S e c t i o n 1.6. T h i s

r e p r e s e n t s a n o u t g r o w t h of the i d e a m e n t i o n e d at the b e g i n n i n g of this Introduction, that t h e r e s h o u l d e x i s t a solution of the Stokes e q u a t i o n s w h o s e d o m a i n of v a l i d i t y ( r e l a t i v e to the

"exact

11

N a v i e r - S t o k e s s o l u t i o n ) c o i n c i d e s w i t h the f o r m a l d o m a i n of v a l i d i t y . T h e b a s i c i d e a is that if a n e q u a t i o n E

1

is "close" to a n e q u a t i o n Ε than to e v e r y solution of Ε t h e r e s h o u l d be a c o r r e s p o n d i n g c l o s e solution of E

1

. C o n c l u d i n g R e m a r k s . T h e r e a r e m a n y w a y s in w h i c h K a p l u n

1

s r e s e a r c h p r e s e n t e d h e r e in P a r t I m a y be continued.

O b v i o u s l y his technique m a y be a p p l i e d to a v a r i e t y of p r o b l e m s . H e h i m s e l f g i v e s a n i n g e n i o u s e x a m p l e i n the study of lift at l o w R e y n o l d s n u m b e r s ( C h a p t e r V ) . H i s study of the s e p a r a ­ tion p r o b l e m ( P a r t I I of this b o o k ) a l s o contains m a n y e x a m p l e s of v e r y s k i l f u l u s e of p e r t u r b a t i o n t e c h n i q u e s .

A n o t h e r a s p e c t of his w o r k is the i n v e s t i g a t i o n of the s t r u c t u r e of p a r t i a l l y o r d e r e d s p a c e s of o r d e r c l a s s e s g i v e n in C h a p t e r V I w h i c h contains m a n y i n t e r e s t i n g r e s u l t s ; it s h o u l d be p o s s i b l e to put his i d e a s in this c h a p t e r on a m a t h e m a t i c a l l y

r i g o r o u s b a s i s a n d r e l a t e t h e m to m o d e r n a b s t r a c t m a t h e ­ m a t i c s . It s e e m s , h o w e v e r , that a r e a l l y i m p o r t a n t t a s k for f u t u r e r e s e a r c h is to f u r t h e r p u r s u e his intuitive i d e a s about foundations of p e r t u r b a t i o n t h e o r y . M a t h e m a t i c a l r i g o r cannot be e x p e c t e d at a n e a r l y s t a g e , but f u r t h e r c l a r i f i c a t i o n and d e v e l o p m e n t of the b a s i c i d e a s i n C h a p t e r I V ( p o s s i b l y u s i n g the " a r i t h m e t i c " d i s c u s s e d in C h a p t e r V I ) m a y c o n t r i b u t e s i g n i f i c a n t l y to a d e e p e r u n d e r s t a n d i n g

of p e r t u r b a t i o n t h e o r y .

P . A . L .