MOTION
OF
V.
A FINITE
K. Andreev
MASS and
OF
V. V.
FLUID Pukhnachev
UDC 532.529.6
The qualitative p r o p e r t i e s of the nonsteady motion of a finite volume of fluid c o m p l e t e l y bounded by a f r e e s u r f a c e is d i s c u s s e d in this p a p e r . The motion a r i s e s f r o m a specified initial state. The e x t e r n a l body f o r c e s a r e known functions of the c o o r d i n a t e s and the time. The fluid can be viscous o r ideal, p o s s e s s i n g s u r f a c e tension o r devoid of it. 1. S t a t e m e n t of the P r o b l e m . The p r o b l e m of the motion of a finite m a s s of fluid r e d u c e s to a s e a r c h for a r e g i o n ~Qt ~ / P and a solution v(x, t) = (vl(x, t), v2(x, t), v.~(x, t)), and p(x, t) of the N a v i e r - S t o k e s s y s t e m of equations vt ~- v ' v v ~ --VP -~ ~Av ~- g(x, t), div v = 0 (1.1) in this r e g i o n so that on the boundary I"t of the r e g i o n ~2t the boundary conditions v.nr t
= Vn
pnrt -- 2vDnvt
for X ~ Ft; 20Hnrt for x ~ rt,
=
(1.2) (1.3)
and the initial condition a t t = 0 v(x, 0) = re(X), x ~ "(20~ p" a r e satisfied.
(1.4)
The r e g i o n t2 is a s s u m e d to be specified and bounded.
H e r e v is the v e l o c i t y v e c t o r , p is the fluid p r e s s u r e , g is the a c c e l e r a t i o n of the e x t e r n a l body f o r c e s , v-> 0 is the v i s c o s i t y of the fluid, and a_> 0 is the s u r f a c e tension coefficient. The quantities y and a a r e a s s u m e d to be c o n s t a n t s , and the v e c t o r g is a s s u m e d to be a known function of x and t. The fluid density is a s s u m e d to be equal to unity. The d i s p l a c e m e n t v e l o c i t y of the s u r f a c e F t in the direction of the outer n o r m a l is denoted in (1.2) by Vn, and the unit v e c t o r of the o u t e r n o r m a l to Ft is denoted by uFt. If the s u r f a c e Ft is specified by the equation F(x, t) = 0, then V n = - - F t i J v F [ . ,' I n the condition (1.3) D is the s t r a i n r a t e t e n s o r with e l e m e n t s Dik = (0vi/SXk+ ~Vk/aXi)/2 (i, k = 1, 2, 3), and H is double the m e a n c u r v a t u r e of the s u r f a c e Ft. It is a s s u m e d that H > 0 if r t is convex e x t e r n a l to the fluid. Eq. (1.2) Indicates that the s u r f a c e r t bounds the fluid v o l u m e ~t. According to (1.3), e x t e r n a l s u r f a c e f o r c e s on the boundary of the r e g i o n 12t a r e a b s e n t , i.e., the boundary r t is a f r e e s u r f a c e . The v e c t o r field v0 in (1.4) is a s s u m e d to be specified and solenoidah div v 0 = 0, x ~ D.
(1.5)
If y >0, then v 0 still s a t i s f i e s the congruence condition with (1.3): Dnr t --
(nr t
'Dnrt)
nr t =
0, x ~ F0~ t = 0.
(1.6)
The m a i n difficulty with investigating the p r o b l e m (1.1)-(1.4) c o n s i s t s of the n e c e s s i t y o f seeking the r e g i o n ~t- H o w e v e r , the c h a r a c t e r i s t i c s of the condition (1.2) p e r m i t t r a n s f o r m i n g this p r o b l e m into another one in which the r e g i o n in which the solution is defined is specified in advance. This situation is achieved by c o n v e r s i o n to L a g r a n g f a n c o o r d i n a t e s . The t r a j e c t o r y of a p a r t i c l e located a t t i m e t = 0 a t the point $ is specified by the f o r m u l a x = x ( ~ , t),
(1.7)
in which the functions xi( ~, t) (i= 1, 2, 3) a r e d e t e r m i n e d f r o m the Cauchy p r o b l e m dx/dt: v(x, t), x = ~ at t : 0 .
(1.8)
The v a r i a b l e s ~ = (~1, ~2, ~3) a r e called L a g r a n g i a n . If the r e l a t i o n s h i p F(x(~, t), t)-= f(~, t) = 0 defines the f r e e boundary, then it follows f r o m (1.2) and (1.6) that ft = 0 and the equation of a f r e e boundary in L a g r a n g i a n c o o r d i n a t e s is s i m p l y f ( 0 =0. T h e r e f o r e , if one c o n s t r u c t s F t as the shape F0---F with the mapping (1.7), condition (1.2) will be a u t o m a t i c a l l y satisfied. K r a s n o y a r s k and Novosibirsk. T r a n s l a t e d f r o m Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 25-43, M a r c h - A p r i l , 1979. Original a r t i c l e submitted A p r i l 17, 1978.
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0021-8944/79/2002-0144 $07.50
9
Plenum Publishing C o r p o r a t i o n
Let us formulate the p r o b l e m (1.1)-(1.4) in L a g r a n g i a n coordinates in the case v = 0 and G= 0 (an ideal fluid with zero s u r f a c e tension). It is r e q u i r e d to find the vector x(}, t) and the function p(~, t) in the r e g i o n s x [0, T] so that the following equations, the initial and boundary conditions, a r e satisfied: q- V~P = g(x, t), (let M = i;
M*xtt
p ---- 0 for ~ ~ ['; X
---- ~,
xt =
vo(~)
at
(1.9) (1.10)
t =
O,
(1.11)
and divvy0 = 0. Here V~ and div t a r e the gradient and the divergence with r e s p e c t to the v a r i a b l e s (!h, ~2, ~a), M is th6 J a c o b i a n m a t r i x of the~mapping (1.7) for t fixed, and Mik= ~ x i / ~ k (i, k = 1, 2, 3). This a r t i c l e is a r e v i e w of papers which have been devoted to the investigation of the p r o b l e m (1.1)-(1.4) in the e x a c t formulation, as well as of some models of it. The r e s u l t s of investigations conducted during the last 15 y e a r s through the initiative and under the direction of L. V. Ovsyannikov c o m p r i s e the basis of this article. 2. Existence T h e o r e m s . The f i r s t (and still the only) r e s u l t on the solvability of the p r o b l e m (1.9)(1.11) belongs to L. V. Ovsyannikov [1] and is based on the t h e o r y which he developed of the nonlinear Cauchy p r o b l e m on the scale of Banach spaces [2]. Let s be a s i m p l y connected two-dimensional region, v0 b e an i r r o t a t i o n a l t w o - d i m e n s i o n a l v e c t o r field in ~, and g(x, t) = 0. in this c a s e the p r o b l e m (1.9)-(1.11) is equivalent to the following:
fo~
I~1=1, t > o
h~=~h: ~~
\ ~~] ~' ~, ~ ~~-' :{R~
(2.1)
b:l:t
t
; "r '-:~[R "cw'c"~d'~ I~1=~
l
f "c '~ w-cI:d'r i'r
for I~] ~ t, t = 0 w = w0(~), h = h0(~),
(2.2)
where h(s t) is the c o n f o r m a l mapping of the unit c i r c l e of the complex plane ~ onto the r e g i o n ~t of the plane z = x 1 + ix2, and w(~, t) -~ w*(z, t) is the complex potential of the motion. We define the Banach space Bp(p>0) as the set of t r a c e s on the c i r c l e [~ [= 1 of the analytic functions u(~) for which the n o r m
n:--oo
is finite, where u n a r e the coefficients of the L a u r e n t s e r i e s of the function u(~), which is analytic in the r i n g e-P < [~l < e< The set S = U B o is the scale of the Banach spaces. The following t h e o r e m [1] is valid: 0 0 is found such that the solution of the p r o b l e m (2.1) and (2.2) exists and belongs to the space Bp for any P 0 and G= 0 has been investigated in [5, 6], where the condition (1.3) with ~=0 is r e p l a c e d by a m o r e g e n e r a l one, p n - 2 ~ D n = p 0 f x , t)n (P0(X, t) is a specified distribution of the external p r e s s u r e on the fluid surface). Lacking the possibility here of giving a complete presentation of the r e s u l t s of [5, 6], we r e s t r i c t o u r s e l v e s to a reduced formulation of one of them. Let F ~ C2+% v o ~ E~+~(Q) with some a ~ (0, l), g= 0, and P0 = 0, and let the congruence conditions (1.5) and (1.6) be satisfied. We will denote the Holder n o r m Iv0]~+~)= R. For any R > 0 is found a T > 0 such thatthe
145
p r o b l e m (1.1)-(1.4) with v > 0 and ~=0 has a unique solution for t ~ [0, T ] , and v and p belong to c e r t a i n Holder c l a s s e s . In addition, one can find for any T > 0 an R > 0 such that the p r o b l e m (1.1)-(1.4) is uniquely soluble on the i n t e r v a l [0, T]. We note that all the t h e o r e m s mentioned above on the solvability of the p r o b l e m s of the nonsteady motion of a fluid with a f r e e boundary a r e of a local nature in the e x a c t formulation. This r e s t r i c t i o n is a s s o c i a t e d with the e s s e n c e of the m a t t e r . One can d e s c r i b e a situation in which in the p r o c e s s of motion two points of the f r e e s u r f a c e which a r e initially located at a finite distance a p p r o a c h one another with subsequent i m p a c t of one p a r t of the fluid with the other. The m a t h e m a t i c a l nature of the p e c u l i a r i t i e s which a r i s e is c o m p l i c a t e d and has not b e e n i n v e s t i g a t e d up to this time. Also the p r o b l e m of the p o s s i b l e loss of s m o o t h n e s s of the f r e e s u r f a c e as t i m e p r o g r e s s e s h a s not been investigated. Finally, not a single e x a c t r e s u l t of a g e n e r a l nature e x i s t s c o n c e r n i n g the s o l v a b i l i t y of the p r o b l e m (1.1)-(1.4) in the c a s e a ~ 0. 3. F i n i t e - D i m e n s i o n a l Models. Below a r e e n u m e r a t e d e x a m p l e s of solutions of the p r o b l e m (1.1)-(1.4) f o r which the s e a r c h r e d u c e s to the i n t e g r a t i o n of a s y s t e m of o r d i n a r y equations. The r i c h e s t c l a s s of such solutions is allowed in the c a s e v = 0 and a = O. T h e s e involve motions with a linear v e l o c i t y field d i s c o v e r e d by D i r i c h l e t ( s e e [7]) and investigated in detail in [8] (also s e e [9, 10]). The mapping (1.7) is given h e r e by the formula x -~ A ( t ) ~ - x0(t), (3.1) so that M = A . It is n e c e s s a r y for the e x i s t e n c e of such solutions that the v e c t o r of the e x t e r n a l f o r c e s be a linear function of x: g ~ G(t)x ~-go(t). (3.2) By v i r t u e of (1.9)-(1.11) the m a t r i x A and the v e c t o r x0 s a t i s f y the equations A ' " - - GA = q A * - 1 N ;
(3.3)
x0 -- Gx0 = go § q A * - l b
(3.4)
A (0) -----E,
A' (0) = A0;
(3.5)
xo (0) -- 0,
X'o (0) = x~0.
(3.6)
and the initial conditions
H e r e A~ is a n a r b i t r a r y constant m a t r i x with SpA~ = 0, N= diag~nl, n2, n3}, w h e r e nl, n2, n a a r e a r b i t r a r y positive n u m b e r s , b and x~0 a r e a r b i t r a r y constant v e c t o r s , and the notation q= is introduced.
Sp (A-~A') 2 -- SO (A--~GA) Sp (A--~A*--~N)
The f o r m u l a for the p r e s s u r e is o f the f o r m p = q(l ~- 2b.~ -- [ . N ~ ) / 2 ,
w h e r e Z is a constant.
The f r e e s u r f a c e I' is an ellipsoid with the equation l + 2b.~ -- ~.N~ = 0.
We will r e s t r i c t o u r s e l v e s in the following to motions in which g = 0 , b = 0 , and x~0=0. By v i r t u e of (3.2), (3.4), and (3.6), xo(t) =0. CauchyWs p r o b l e m (3.3) and (3.5) is uniquely s o l v a b l e for all t [9]. The s y s t e m (3.3) with G = 0 has eight i n t e g r a l s which e x p r e s s c o n s e r v a t i o n of the m a s s (det A= 1), e n e r g y , c i r c u l a t i o n , and angular m o m e n t u m of a d e f o r m e d fluid ellipsoid [7]. T h e r e e x i s t a n u m b e r of e x a c t solutions of the p r o b l e m (3.1). We will d i s c u s s one of t h e m , which is found in [8]. L e t N = E , and l e t the m a t r i x A~ have nonzero e l e m e n t s 2(a~)22 = 2(a~)33=-(a~)li = - 2 b , and (a~)a2=-(a~)23 = o~. Then the m a t r i x A has nonzero e l e m e n t s all = m, a22 = a33 = k , and a32 = - - a 2 3 n , and =
]/m
\
J0
" " = ~
0m
dr ,
and the function m(t) is found by q u a d r a t u r e f r o m the equation m "~ ~-
4m*
3b~ 4 ~ : (i - - m) l 4- 2m a
with the condition m(0) = 1. The equation of the f r e e boundary r t in E u l e r i a n c o o r d i n a t e s is
146
(3.7)
x~
,
2
"
~- m (z~ § x ~ ) = c ~ _~ l.
T h e i n t e r p r e t a t i o n of the s o l u t i o n i s a s f o l l o w s . A t t h e i n i t i a l i n s t a n t the fluid f i l l e d a s p h e r e F and w a s in a s t a t e o f u n i f o r m r o t a t i o n on w h i c h a n i r r o t a t i o n a l l i n e a r v e l o c i t y f i e l d w a s s u p e r i m p o s e d . L e t us a s s u m e f o r d e f i n i t e n e s s t h a t w ~ 0 and b > 0. T h e n d u r i n g 0 < t < t , t h e s p h e r e F is e l o n g a t e d into a n e l . l i p s o i d of r e v o l u t i o n F t with a x i s x 1 u n t i l i t s s e m i m a j o r a x i s t a k e s o n t h e m a x i m u m v a l u e c m . = c[3(b/r 1]if2 a t t h e i n s t a n t t = t , . A f t e r t h i s t h e e l l i p s o i d s t a r t s to c o n t r a c t , p a s s e s the s h a p e o f the s p h e r e F a t the t i m e t = 2 t . , and t h e n s h r i n k s to t h e p l a n e x l = 0 , m e r g i n g w i t h i t a s t ~ . Now l e t us c o n s i d e r the c a s e w = O . T h e m a t r i x A i s d i a g o n a l , and the m o t i o n i s i r r o t a t i o n a l . If b > 0 , t h e n m > 1 f o r t > 0 and m--* ~ a s t--* ~ . T h u s when b > 0 and w = 0, the e l l i p s o i d F t e x p a n d s in t h e d i r e c t i o n o f t h e x l - a x i s , s h r i n k i n g to t h i s a x i s when t ~ . F r o m the p o i n t of v i e w of t h e s t a b i l i t y o f the m o t i o n t h i s r e s u l t i n d i c a t e s t h a t the i r r o t a t i o n a l m o t i o n s p e c i f i e d b y t h e m a t r i x A f o r w = 0 i s u n s t a b l e with r e s p e c t to a s s m a l l vortical perturbations as desired. T h e q u e s t i o n o f t h e b e h a v i o r o f the s o l u t i o n s of the C a u c h y p r o b l e m (3.3) and (3.5) a s t ~ has n o t y e t b e e n s o l v e d . The h y p o t h e s i s t h a t w h e n A~ r 0 t h i s p r o b l e m h a s no b o u n d e d s o l u t i o n s s e e m s p l a u s i b l e . Now l e t us d i s c u s s t w o - d i m e n s i o n a l m o t i o n s with a l i n e a r v e l o c i t y f i e l d . In t h i s c a s e ~ and x d e n o t e t w o d i m e n s i o n a l v e c t o r s , and A , A~, and N a r e s e c o n d - o r d e r m a t r i c e s . The s o l u t i o n of the p r o b l e m (3.3) and (3.5) h e r e d e s c r i b e s the m o t i o n of a r o t a t i n g d e f o r m e d e l l i p s e . U s i n g the m o t i o n i n t e g r a l s , i t i s p o s s i b l e to i n t e g r a t e t h i s p r o b l e m in q u a d r a t u r e s [11]. It t u r n s o u t t h a t if the i n i t i a l s t a t e is n o t r e s t , the f o l l o w i n g a l t e r n a t i v e o c c u r s . E i t h e r one o f the s e m i a x e s o f the e l l i p s e i n c r e a s e s i n d e f i n i t e l y a s t -~ ~ o r the m o t i o n is u n i f o r m r o t a t i o n o f a fluid c i r c l e a b o u t i t s c e n t e r . We w i l l 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 (3.3) and (3.5) with
a n d N = E ( r is a c i r c l e o f r a d i u s c). In t h i s e a s e t h e s e m i a x e s of the e l l i p s e at(t) and a2(t) a r e r e l a t e d b y t h e equations 1
~C4(02
,2
(a, + a2')
+
-
+
(3.8)
ala2 ---- c2~ al(O) = a2(O) = c,
and t h e a n g u l a r r o t a t i o n a l v e l o c i t y o f t h e e l l i p s e i s e q u a l to 4c2w(al+a2) -2. The c a s e b = 0 c o r r e s p o n d s to r o t a t i o n o f the c i r c l e a s a r i g i d b o d y . I t is e v i d e n t f r o m (3.8) t h a t a n i n i t i a l d e f o r m a t i o n o f t h e v e l o c i t y field a s s m a l l a s d e s i r e d (b ~ 0) d i s r u p t s the i n d i c a t e d s t e a d y m o t i o n . The s u p p l y o f e x a c t s o l u t i o n s of the p r o b l e m (1.1)-(1.4) in the c a s e ~ ~ 0 a n d a s 0 is e x t r e m e l y s p a r s e . The o n l y n o n t r i v i a l e x a m p l e i s the s o l u t i o n w h i c h d e s c r i b e s the r a d i a l m o t i o n b y i n e r t i a of a s p h e r ] : c a l l a y e r [12-15]. T h e t w o - d i m e n s i o n a l a n a l o g u e o f t h i s s o l u t i o n d e s c r i b e s t h e r a d i a l m o t i o n o f a c i r c u l a r riLng [13, 16]. The t w o - d i m e n s i o n a l p r o b l e m o f r o t a t i o n a l l y s y m m e t r i c m o t i o n o f a r o t a t i n g r i n g i s m o r e g e n e r a l . 4. R o t a t i n g R i n g . W e w i l l d i s c u s s t h e t w o - d i m e n s i o n a l p r o b l e m (1.1)-(1.4) with s p e c i a l i n i t i a l d a t a : i s t h e c i r c l e rl0 < r = Ixl 0 (it is a s s u m e d that a 0 - - 0 as T--* oo). This condition guarantees the c o r r e c t n e s s of the p r o b l e m under consideration. The e s t i m a t e (6.11) is valid for the solutions of the p r o b l e m (6.5)-(6.7). This estimate p e r m i t s adopting the quantity N (t) = IIT, [zL,(r)-t- il V$ ~,,.e) as a m e a s u r e of the stability and calling the fundamental solution stable if N(t) is bounded for all t > 0 for any (D0 ~ Wi(~2) and unstable in the opposite case. Finally, such a definition of stability is not uniquely possible. It is proposed in [13] to c h a r a c t e r i z e stability in t e r m s of the behavior as t - - ~ of the component n o r m a l to F t of the p e r t u r b a t i o n v e c t o r of the free boundary X(~, t), ~ ~ F. In the g e n e r a l c a s e it is g i v e n by the equation t
(7.1)
R ~ nrt.X : p S n'M-1Vdt, o
where p =
[V/I(IM,-,VIi) -x,
and nFt is the unit v e c t o r of the outer n o r m a l to the s u r f a c e r at the point x(~, t).
If R(~, t) - ~ as t--oo for some ~ ~ F, then the local departure of the free surface f r o m its tmperturbed state i n c r e a s e s without limit in the space (x). Thus the quantity R(}, t) provides a v e r y delicate c h a r a c t e r i s t i c of the stability of the motion, whereas the function N(t) is its crude integrated c h a r a c t e r i s t i c . It may happen that a c e r t a i n solution is stable in the integrated sense, but the function R i n c r e a s e s without limit at individual points of the boundary as t--* oo. Such a situation actually a r i s e s in p r o b l e m s on the stability of motions with a linear velocity field described by Eqs. (3.1)-(3.6). In the p a r t i c u l a r case g= 0, b = 0, and x~0 = 0,
Op/On = --q(t) lN[[, q(t) = Sp(A'A-')2/Sp(A-~A*-XN). The stability of this class of motion is investigated in m o r e detail below. gral n)
itu(g, t)i'dQ§
.q
The integral identity (nenergy inte-
!t ~ a;'-~ I~',(g, t ) 1 2 d r = i l V~2, ~ o ( ~ ) l ~ ' d ~
-
(7.2}
?
--2~.l'U(g ,
t).igA-~U(~,t)dg~dt+
~
~
JW~(~, t)pdrdt
o c c u r s , which is valid for any solution of the p r o b l e m (6.5)-(6.7), where U(~, t) == A *-~ (t)V~(g, t). It is a l r e a d y possible to e x t r a c t some information f r o m the identity (7.2) about the behavior of ]l~FttlLgD, e.g., for specific motions as t--- ~o without solving the p r o b l e m (6.5)-(6.7). As an example, let us consider the two-dimensional p r o b l e m of the stability of deformed ellipse. Here the fundamental solution is given by Eq. (3.1), w h e r e x0 = 0 and A = diag (al, a2). The functions a~(t) and a2(t ) a r e defined by Eqs. (3.8), in which one sets w = 0. Let us a s s u m e for definiteness that b > 0; then the s e m i m a j o r axis o f the ellipse at(t) is found f r o m the equation d~ billet = "!'(V~+ l)t/2~,
(7.3)
[
whence
al=b242t+O(t-t) as t--*oo. We obtain f r o m the identity (7.2) the estimates 'r
dry,,
-,, }tVr ('~ +~): h
!,o,~ ~,en
0 . One
153
should note that in the case in which the initial function if0 is even with r e s p e c t to ~l and different f r o m a conslant, the e s t i m a t e [lT IIL,r (t-i), t--~-co o c c u r s . The solution which is even in ~l d e s c r i b e s motion with an impenetrable wall ~2 = 0. Calculation of the n o r m a l component of the perturbation v e c t o r a c c o r d i n g to Eq. (7.1) for the case ~0lr = sinn 0, where 0 =arctan(~z/~t), gives
o sin, 0
R'~ (cos20_~_a~sin. O)l/z
' o (t-')] [5'n~-
(7.4)
as t-*~o, Here Yn is some constant, and al(t) is a function defined by Eq. (7.3). It is evident from (7.4) that outside the zones ]0l < e, In -- 0l < e ( e > 0 is specified) R=O(t) as t - - ~ . Thus the free boundary is unstable i n t h i s c a s e . M o r e o v e r , ~llL~(r)-+0 as t - * ~ . If (I)0[r = cos nO , n = l , 2, ..., then as t ~ o o a 1 cos nO
~
,
with constant 6n. It follows f r o m this that for this solution the instability of the free boundary is localized in the r a n g e of angles [0] = O(t -1) and ]Ir-0I = O(Ci). This conclusion is applicable, i n p a r t i c u l a r , to the p r o b l e m of the stability of a deformed eltipse when an impenetrable wall ~2 = 0 is present. As t-~ o0 the fluid approaches the wall, and this stabilizes the free boundary. Another example is the stability of i r r o t a t i o n a l a x i s y m m e t r i c motion with a linear velocity field. The fundamental solution is described in Sect. 3 and is interpreted as the motion of an ellipsoid of revolution. The mapping (3.1) has the f o r m x = diag0n, t / V ~ , t/V:m}~ here. The function m(t) is defined by Eq. (3.7), in which w = 0, and by the condition m(0) = 1. The r e g i o n ~2 is a s p h e r e I} I < c, and r is its boundary. The identity (7.2) leads to the e s t i m a t e s = o(1), as t ~ r162for an oblate ellipsoid (which c o r r e s p o n d s to m ~ 0 as t ~ oo) ; where i = 1~ 2, for a proiate ellipsoid. t--~ ~o with some constant k > 0 [30].
M o r e o v e r , it is possible to point out initial data such that R > k t as
The r e s u l t s presented above indicate the stability for the linear approximation of the fundamental motions considered if one takes the n o r m in L 2 of the values of the potential ~ or its derivatives as the m e a s u r e of stability. However, if one a s s e s s e s the stability f r o m the departure of the free boundary f r o m its unperturbed state, the indicated motions should be r e c o g n i z e d as unstable. Only i r r o t a t i o n a l p e r t u r b a t i o n s have been considered above. If one c o n s e r v e s the irrotationality of the fundamental motion but widens the c l a s s of perturbati~)ns by r e m o v i n g the condition of irrotationality f r o m them, then a solution which is stable with r e s p e c t to i r r o t a t i o n a l perturbations may become unstable. A p p r o p r i a t e examples a r e given in [21]. The exact solutions d i s c u s s e d in Sec. 3, which describe the motions of a rotating ellipsoid and a rotating ellipse, indicate this behavior. In addition to the c a s e s discussed above, the stability of the following i r r o t a t i o n a t motions of an ideal fluid has been investigated in the linear approximation at p r e s e n t : the motion of a s p h e r i c a l layer [12] and the motion of a c i r c u l a r r i n g [13, 16]. Taking c a p i l l a r y f o r c e s into account in the p r o b l e m of s m a l l p e r t u r b a t i o n s leads, as has a l r e a d y been noted, to the c o r r e c t l y formulated p r o b l e m (6.13)-~(6.17). Concerning the effect of capillarity on stability~ the number of specific p r o b l e m s c o n s i d e r e d here is v e r y small. It is known that s u r f a c e tension s u p p r e s s e s the growth of t w o - d i m e n s i o n a l p e r t u r b a t i o n s of the r a d i a l motion of a ring [16] and the motion of a rotating ring [28]. On the other hand, the introduction of s u r f a c e tension r e s u l t s i n t h e exponential growth as t - * ~ of a x i s y m m e t r i c perturbations of the nonsteady motion of a fluid cylinder whose lateral surface is free and whose bases are solid i m p e n e t r a b l e walls [31]. P r o b l e m s of the stability of equilibrium states of a fluid volume a r e not considered in this paper. Sufficient conditions of stability and instability with r e s p e c t to finite perturbations a r e obtained in this p r o b l e m which a r e based on the r e s u l t s of [32], in which an analogue of the Lagrange stability t h e o r e m is established for the motion o f a viscous capillary fluid. An exposition of this group of p r o b l e m s is contained in [19].
154
The p r o b l e m of limiting modes of the motion of a fluid volume of finite mass as t - - ~ is closely related to the stability problem. The examples discussed above of the motions of an ideal fluid and a viscous ring exhibit the d i v e r s i t y of the possibilities which a r i s e here. This p r o b l e m is far f r o m solution in the general case. There a r e only the following partial r e s u l t s [9]. Let us a s s u m e that a c l a s s i c a l solution of the p r o b l e m (1.9)-(1.11) with g = 0 exists for all t > 0 , and that the mapping (1.7) specifies d i f f e o m o r p h i s m of the regions ~2 and ~2t for any t. Let us denote the diameter of the r e g i o n fit as d(t), where d(t) = max{ix -- Yi : x, y ~ Qt}. Letv0(~)~Oandrotv0=Ofor
~O.
Thend(t)--~oast--*~o.
In other words, in the case of the i r r o t a t i o n a l motion by inertia of a finite volume of ideal fluid with v a r i able velocity and zero surface tension the diameter of the volume i n c r e a s e s without limit as time goes by. The proof of this ~esult is based on the s u p e r h a r m o n i c i t y p r o p e r t y of the p r e s s u r e in the i r r o t a t i o n a l motion of the fluid and on the identity [33]
d" t'
~f
!
pdff~ t,
which is valid for a r b i t r a r y motion of an isolated volume of a viscous c a p i l l a r y fluid. 8. Boundary L a y e r s . Let us a s s u m e that the solution of the p r o b l e m (1.1)-(1.4) with ~>0 is known. How does one find its asymptote as v--* 07 It is natural to expect that outside n a r r o w l a y e r s near the free boundary the motion will be close to the motion of an ideal fluid. An abrupt change of the derivatives of the velocities o c c u r s in the boundary l a y e r s which provides for vanishing of tangential s t r e s s e s on the free boundary. A f o r m a l a s y m p t o t e of the solution of this p r o b l e m in the two-dimensional and a x i s y m m e t r i c a l c a s e s is given in [34], and the boundary l a y e r s in problems of the i r r o t a t i o n a l motion of an ellipsoid of revolution and an ellipse with ~= 0 a r e investigated in [35, 36] (see Sec. 3). The only example in which it has proven possible to justify the validity of an a s y m p t o t i c expansion is the p r o b l e m of a rotating ring. The asymptote of the solution is Sought in the f o r m
4 +V
4
+
+
+
r i ~ r : _ ( ( i ) _ :,~Ir / v r- i ( i l ---i - y r , " ) - : . . . ,
+
i=t,
+...,
2.
The notation v0, X, and r i a r e introduced in See. 4. The functions v0(k), r~ k), and )/(k) ( k = 0 , 1, 2, ...) a r e found with the help of the f i r s t L y u s t e r n i - V i s h i c iterative p r o c e s s . When k = 0, we obtain the solution of the p r o b l e m of the motion of a r i n g of ideal fluid. Functions of the b o u n d a r y - l a y e r type ~}k)" (i = 1, 2, k = 1, 2...) a r e d e t e r mined as a r e s u l t of the second iterative p r o c e s s . They compensate discrepancies on condition of the absence of tangential s t r e s s on the free boundaries of the ring. An estimate is given of the e r r o r of the a s y m p t o t i c expansion as v - - 0 which is valid on any finite time interval if ~= 0 and on any interval 0-< t
