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Existence of smooth, stationary interfaces for Marangoni-type flow
NonlinearAnalysis, Theory, Methods&Applications, Vol. 27, No. 11, pp. 1329-1350, 1996
Copyright© 1996ElsevierScienceLtd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00+ 0.00
Pergamon
0362-546X(95)00118-2 EXISTENCE OF SMOOTH, STATIONARY INTERFACES FOR MARANGONI-TYPE FLOW F. ABERGEL and C. DUPAIX Laboratoire d'Analyse Num6rique, Universit6 Paris Sud B,~timent 425, 91405 Orsay, France
(Received December 1993; received in revised form January 1995; received for publication 1 June 1995) Key words and phrases: Navier-Stokes equations, Marangoni effect, free surface, weighted spaces, existence and uniqueness, thermohydraulics, Marangoni effect.
1. I N T R O D U C T I O N
In this paper, we consider the flow of a viscous liquid, when its free surface is subject to a nonuniform thermal flux. This type of flow, known as Marangoni type flow, see e.g. [1-3], is characterized by both a convection phenomenon within the fluid and an additional tangential stress acting at the interface. Concerning the convection flow, we shall make the classical Boussinesq approximation for variation of the density, so that the really interesting terms in the modeling equations will be provided by the surface law. Our main result concerns the existence and uniqueness of a stationary interface, in the neighbourhood of the capillary solution, i.e. one that is obtained when the flux is zero and the fluid is at rest. This result is obtained via an adequate implicit function theorem. Unlike similar works for the Navier-Stokes systems [4, 5], the main technical difficulties lie in the presence of corners at the boundary of the domain occupied by the fluid, and use is made of the machinery developed in [6, 7]. We also need to study Sturm-Liouville type problems in weighted spaces. This work generalizes the results obtained by Lagunova [8] to the case that the gravity constraint is taken into account. Moreover, we want to mention [9], where a numerical study is performed, and [10] where a bifurcation problem is studied. The paper is organized as follows: Section 2 contains the main geometrical and functional hypotheses and notations; in Section 3, the physical problem is described, and its mathematical formulation given. Section 4 is devoted to the proof of our main result: after presenting the variations of domains suitable for the problem, we study in Section 4.3 Sturm-Liouville problems in weighted spaces, and finally prove in Section 4.4 the implicit function theorem that we need. Essentially, we show that, when a certain parameter (the Marangoni number times the Froude number) does not belong to a finite set of real numbers, then, for small enough values of the thermal flux, there exists a unique stationary interface in weighted H61der spaces. Despite and unlike the results of Lagunova [8], a consequence of taking into account the gravity is that, when the Marangoni number times the Froude number is in this finite set, the implicit function theorem does not apply any more, and bifurcation phenomena may possibly appear. 1329
1330
F. ABERGEL and C. D U P A I X 2. F U N C T I O N A L
SETTING
Let O be an open bounded set of ~2 whose boundary Of~ is a piecewise smooth Jordan curve, and denote by 8(O) the set of points where the tangent vector to 0 0 is discontinuous. We define the following functional spaces.
2.1. The space C m'~ of c~-HOlderian functions of order m Let m be a nonnegative integer and o~ e (0, 1) a real number, e ' ( O ) will stand for the space of m-times continuously differentiable functions in f2. We then define the space of m-times continuously differentiable functions in O whose mth derivatives satisfy a H61der condition with exponent c~. We denote this space by cm'~(O) and endow it with the norm
lule-.o(,)=
E
Sup lOku(x)[ +
I/¢l
E
Sup
IDku(x) - Dku(y)]
Ikl=mx,yef~ x¢y
[X-Y] ~
2.2. The space C~ of weighted H6lderian functions of order [l] and exponent I - [1] For x ~ ~, we define [x] as the integer part of x. For y e O we set
d(y) :=
Inf
z • ~(fl)
Ily-
zll,
where I[" I[ stands for the Euclidian norm. Let l be a positive noninteger number, and s a noninteger real number. For s e (0, l], we define the space C~ (0, $(0)) for which the following weighted norm is finite s
+
~]
Sup
I
Min(dt-~(x), dt-S(y))
[k] = [l] x,y ~ fl x#y
IO%(x~ - Oku(y)[~ ['X -- y ~ ]
")'
and for s < 0, we equip Cts(O, 8(0)) with the norm
lule'.
E
Sup
Id(x)lel-'Oku(x)l
O
=
Sup
-b iklE[l ] X,y e fl x ~y
Min(dt-~(x), dt-~(y)) IDku(x) - Dku(y) I~ [-X -- ) [ l - ~ ]
")"
In the same way, and for all noninteger s and all positive noninteger / satisfying s < 1, we define the space C~(I, 8(I)), where I is a segment. 8(1) is then composed of the endpoints of I. 2.3. Remarks and notation A quite complete study of weighted H61derian spaces can be found in [11]. We now give some properties of weighted H61derian spaces which will be useful later. Let I e R ÷* and s e [R be two given noninteger numbers. (i) Let k be a positive integer satisfying k_< [1]. We suppose that s_< I + k. If u e e~+~(O, $(0)) then Dku e C~_k(O, $(0)).
Stationary interfaces for Marangoni-type flow
1331
(ii) We suppose that s _ 1. If u e C~(o, $(f~)) then if U is a primitive of u, it satisfies U ~ col+l(f~, 8(~)). '~s+l (iii) Let be given (11, 12) ~ (JR+*)z and (s I , s2) e •2 satisfying s I _< l 1, s 2 _< 12 on one hand, and 11 12 l 1 < l2, S1 < S2 on the other hand. Then if u ~ Csl(D, $(~)) and v e Cs2(~, $(~)), we have (u, v) ~ C~1(~, $(D)). Moreover, there exists a positive constant C, independent of u and v such that the following estimate holds
Notation: • In the following, when we consider the space C~, we will always assume that l > Max(0, s). • lul® will stand for Sup [u(x)[. x~fl
3. S T A T E M E N T
OF THE PROBLEM
3.1. The physical problem
We consider a container C, partially filled with a viscous liquid. The boundary o f the container OC is composed o f two half-lines (x = - 1 , y > 0) and (x -- + 1, y > 0) which are connected at the points ( - 1 , 0) and (+1, 0) by a smooth curve OC-, which is supposed to lie in the half-plane (y < 0). The gravity g, is directed along the vector - j ( 0 , - 1 ) . We denote by f2 the part of C containing the fluid, f2 is the open bounded set o f [~2 defined as: £) := {M(x, y) ~ C Iy < f(x)] and r. := [M(x, y) e C I - 1 < x < + 1 and y = f(x)} is the interface between the fluid and the atmosphere. Let F := all\Y, and $(f~) := F A Y.. Let I := ( - 1 , +1) and $(I) := ([-1}; {+11). We suppose that the angles of contact between F and E are both equal to fl ~ (0, ~r) with the convention that fl = 0 if t~ = - j at x = - 1 , where tr. is the unit tangent vector to E. The flow is supposed to be stationary. The interface is subject to a nonuniform thermal flux 5. Thus, the temperature gradients induce, on one hand, a superficial stress which generates a Marangoni flow and, on the other hand, a volume force inside of the fluid, which generates a convection flow. Several authors have considered the physical aspects of this problem, for example, [1-3]. 3.2. The mathematical formulation We first make the two following assumptions. (A1) We assume that the Boussinesq approximation holds, namely, on one hand, the external force satisfies pc(1 - a ( T - Tc))g and on the other hand, the mass density p is constant and equal to Pc in the volume of the fluid. Where Pc = p(Tc), Tc is the temperature of the boundary of the container (i.e. the rigid part of the boundary). Tc is supposed to be a constant, a is a given positive constant. (A2) The surface tension coefficient y is given as a nonincreasing affine function o f the temperature.
1332
F. ABERGEL and C. D U P A I X
The boundary conditions are as follows: (BC1) the fluid satisfies a no-slip condition at the boundary of the container; (BC2) the temperature at the boundary of the container is constant and equal to Tc; (BC3) the interface is subject to a nonuniform thermal flux 9; (BC4) the interface is in thermal and dynamical equilibrium. We then write the conservation laws of mass, momentum and energy and obtain, in a dimensionless form, the following system of partial differential equations for the unknowns ( u , 0, a , f , C ) ,
V. u = 0
in ~
(1)
(u" V)u - Div a(u) - :t0j = 0
in ~2
(2)
1 u" V0 - - - A 0
in ~2
(3)
onF
(4)
onZ
(5)
on E
(6)
one
(7)
Pr Re
u=0
and
00
--=-e~
On
(sD
iT(u) • n . t -
= 0
0=0 and
u'n=0
Ma* 00 = 0 Ot l f _ M a . ~ ( O ) ( \~ )
(7(u) • n . n + P. + F r
f:: x - C = O
1 iT(u)" n. n dx = 0
(9)
-I
f
d x d y = V,
(10)
12
where • u, 0, a are, respectively, the velocity field, the temperature and the stress tensor with 2
o(u)
= - p I d + Ree
e(u),
where e(u) = ½(Vu + Vu t) and p + y/Fr is the pressure, • C is a constant which has to be determined, • A is a contant equal to 1 if we consider an evaporation problem and - 1 if we consider a condensation problem, • ~ = ~(x) is the nonuniform thermal flux, • e is a small parameter, • Pa is the atmospheric pressure,
Stationary interfaces for Marangoni-typeflow
1333
a(0) = 0 + 00 is the surface tension, where 00 is a constant, • Pr is the Prandtl number (cinematic viscosity over diffusion coefficient), • Re is the Reynolds number (characteristic velocity V* of the flow times characteristic length L* over the cinematic viscosity), • Fr is the Froude number (square of the characteristic velocity over characteristic length times gravity field), • Ma* = (dy/dO)(1/pL*V*2), where d y / d 8 is the derivative o f the surface tension with respect to the temperature O, and p is the mass density. Note that Ma* is related to the usual Marangoni number Ma, with Ma* = - M a / P r Re 2, • V is the volume occupied by the fluid. •
Remark 1. Equations (6) and (7) result from the local decomposition o f the vector equation a(u)n + (pa + l f ) n - Cn + Ma*(~(O)Hn - Va(O)t) = O, which expresses the dynamical equilibrium of the interface. (a(u) + Pa + (1/Fr)f)n is the superficial density of force describing the action of the fluid on the interface. (o40)Hn) is, following the classical interpretation of [12], the first variation o f the surface energy, and it measures the forces to apply to obtain a deformation at the interface. Finally, ( - V a ( 0 ) t ) is induced by the variation of the surface tension and characterizes the Marangoni type flows. 4. WELL-POSEDNESS IN THE NEIGHBOURHOOD OF THE CAPILLARY SOLUTION
4.1. The main result Our purpose is to prove the following result. THEOREM 4.1. If F belongs to C t+2 then: (a) There exist a finite sequence o f real numbers, 0 < / ~ 1 ( "'" < /~K < + QOand real numbers I7"> 0, eo > 0 and s o E (0, 1] such that if Ma*FrO o # )t i for all i = 1, ..., K then
vlel < e0 v s • (0, So) v V > 17" ¥ 9 • C;+ltI s-ix , 8(1)) satisfying the condition 5(M (i)) = 0 if fl = r~/2 with 8(/) = (-Ji= 1,2 [M i I, there exists a unique solution (u, 0, a , f , C) o f (S,) satisfying: n•,
tet+2tf2 ~ , ,~(f~)))2
O • et+ztf2 s , ,8(t~)) 1+1 (~'~, 8(~'~))) 4 a • (@s_l
f • @s+x(/, 1+3 8(1)) C•[R.
1334
F. ABERGEL and C. DUPAIX
(b) Moreover, there exists a real number d . > lr/2 such that if the angle o f contact/? satisfies 0 < fl < f l . , then there exists a positive number sl > 1 such that the conclusion of (a) remains true for every s e (1, sl) satisfying s ___ / + 2. The rest o f this section is devoted to the p r o o f of theorem 4.1. 4.2. A particular solution o f the problem Let us suppose e = 0 and V = V0, V0 given, respectively, in (5) and (10). In this case, (u, O, or) = (0, 0, constant) satisfy equations (1) to (6) and thanks to (9), cr = 0. Moreover, integrating (7) with respect to x between - 1 and +1, we deduce that Co := C = Pa + V° - V 2Fr
Ma*Oocos(d)
so that (7) and (8) can be written as
fax) rr
\x/1 + fx(x)~/x
- - f ~ =
+cos(B),
vo
V_
2Fr
(11)
41 + fx(+_l) where V_ is the volume o f the fluid enclosed between the line (y = 0) and the curve 8C_. However, - 1 < cos(]~) < +1, 1/Fr > 0 and Ma*Oo < 0 and, therefore, thanks to the results of [13, 14] it is well-known that there exists a unique solution ~ e C~(I) of (11). This solution is the so-called capillary solution, and determines the interface of the fluid at rest.
Remark 2. If ¢ is the solution o f (11) for V = Vo then ¢ + V~/2 is the solution of (11) for V = V0 + V1 . Thus, we will choose Vlarge enough, so that ¢(x) > ~ > 0 for all x ~ [ - 1 , +1], where g is a positive constant. We will denote by 17 the smallest volume such that the condition above can be satisfied. We now state a first proposition. PROVOSlrlON 4.2. Suppose B e (0, ~r) given. Let V = V0 , where V0 e ~+* is given and satisfies Vo > 17. Then there exists a unique solution (Uo, 00, ao, ¢, Co) of problem (So) for (e = 0). This solution satisfies (Uo, 0o, ao) = (0, 0, 0)
I
Co = Pa + -V~F-~-:- M a * ~ cos(fl) e C (1) is the unique solution o f (11).
Proof. It remains to prove the uniqueness o f the equation (3) by 0 and integrating over ~o, the open deduce, using (4), that 0 = 0 in f~0. Moreover, taking u, integrating over ~o, and using equations (4)-(6), we
solution of problem (So). Multiplying bounded set o f Rz associated to ~, we the scalar product o f equation (2) with deduce thanks to Korn's inequality that
u --- 0 in ~ 0 .
Finally, we deduce from (9) that a(u) -= 0, which completes the p r o o f of proposition 4.2.
Stationary interfaces for Marangoni-type flow
1335
4.3. Preliminary result We establish in this section a result that will be useful in the sequel. We denote by g the capillary solution of equation (11) and we set Xl = - 1 , x2 = + 1. We introduce the two linear operators ~ and 03 defined as £p(x) . - x/1 + ~xZ(X)p(x) - Ma*OoH(p(x)),
Fr
03p(x) ..=
(p
)
~+ ~
p (x),
with
Pxx Hip) = 1 + ~
~
~
-2
-2
gxgxx {gxZ_x(1 - 3gx) gxCxxx (l + ~)2 p~ + \ ~ ¥ ~ + <1 + ~ ) V p"
These operators arise naturally in the linearization of equation (11). We are now able to deduce the following proposition. PROPOSITION 4.3. Let 1 > 0 and s > 0 be two real numbers satisfying s < l + 2. There exists a finite sequence of real numbers 0 < )-i < "'" < '~k < +oo -(ol+lj(l $(I)), such that ifMa*FrOo ~ )ti for all i : 1. . . . . K t h e n , for all (a~, a2) ~ [RE and all h ~ .-.s-~-, there exists a unique function p ~ .col+3tr ~ + ~ . , $(I)) solution of
I £p(x) = h(x) 03p(xi)
ai
vx ~ I with i = 1, 2.
Proof.
Existence of a solution. We first assume that h e @"(I) for some nonnegative integer n. Letting R(x)-
1
1 + gZx(X)' 1 XR r(x'=exp(~f_,R(t'd r(x)
p(x) =
R3/2(x),
1 Rxx q(x) = ~ r(x) --~ (x), r(x) h(x) = -h(x) Ma*OoR(x) ' 1
Ma* FrOo
O,
1336
F. A B E R G E L
a n d C. D U P A I X
the problem above can be written as a Sturm-Liouville problem
I (r(x)p,(x))x + (/up(x) - q(x))p = h(x)
(Bp(xi) = ai
(i = 1,2);
thus, using for example the results of Churchill [15, pp. 260-264] or Ince [16, Chapter IX], we deduce the existence of a unique function p ~ C"+Z(I), provided/a is not a characteristic number (P~)i~ N of the Sturm-Liouville operator, namely a n u m b e r / t for which the homogeneous system
l (r(x)px(X))x + (UP(X) - q(x))p = 0 63p(x~) = 0 (i = 1, 2) n
possesses a nontrivial solution. This yields the condition
Ma*FrOo ~
2i for all i e N with
2i = 1/bli. However, on one hand Ma*Oo and Fr are positive real numbers. Since, on the other hand, no more than a finite number of the 2i are positive numbers [15, theorem 4, p. 267], we deduce the existence of the finite sequence mentioned above. We now come back to the weighted H61derian spaces. If s > 1, which implies [s] - 1 _> 0, we have then (01+1l'1 "~s-lX-, $(I)) C C[sl-l(I), so that the existence of a solution follows. IfO < s < 1, the previous inclusion does not hold; we use the density of ek(I) in ot+l x J S - l (I, 8(I)) for all k _> [l] + 1, to prove the existence.
Regularity of the solution.
Letting
p(x)= w(x) exp ( - ~ l x R x d-1
(t) d
t)
,
K
then, w satisfies the following ordinary differential equation
(Sva0
~ Wxx(X) + (p(x)w(x)= H(x) (,. (Wx + Bi w)(xi) Ai
¥xeI withi=
1,2,
where, using the previous notations, we have set
H(x) : =
h(x)
~o(x) .-
p R3/E(x)
Bi:=
Ai :=
exp(~ f ~ -,
Rx (t) dt),
3 Rxx(X) 3 R~(x) 4 R(----~+ 16 RZ(x) '
3 Rx
-;-ff(xi)
i = 1,2,
z4/~
ai exp
~ - (t) dt -1
i = 1,2.
Stationary interfacesfor Marangoni-typeflow
1337
W e first c o n s i d e r the similar p r o b l e m with c o n s t a n t coefficient, n a m e l y
I (Sc~t)
z,~(x) + b z ( x ) = H(x)
vx • I
(z,, + Biz)(xi) = Ai
with i = 1, 2.
r°t+3(I, 8 ( I ) ) for H • ¢°l+1(I, "~-1 L e t us p r o v e t h a t z • "~s+l
z(x) = ~,zo(x) + VZl(X) + ~ 1
8(I)).
-1 (Zo(X - t) - Zl(X - t))H(t)dt,
if b _< O, ~1 = i @ otherwise, Zo(X) = e ~ the s o l u t i o n o f the linear system w h e r e o~1 = ~
Otl(),e-al
I
Otl(y e ~
with
A2 = 2 -1
--
F o r x • I, z can be w r i t t e n as
a n d z~(x) = e -~x, y a n d v b e i n g
v e ~ ) + Bl(Te -~1 + v e ~) = A 1
- v e -'~)
+ B2(Y
e ~ + v e - ~ ) = A2 - ,42,
ear(l-t) + (1 - ~ )
1+
e-"l(1-t)l g ( t ) dt.
F o r n o t a t i o n a l convenience, we write z in the f o r m
z(x) = q/(x) + where
i
x -1
ep(x, OH(t)dt,
~u(x) := yZo(X) + vzl(x), 1
4,(x, t) : = ~ 1 (Zo(X - t) - zl(x - t)). T h u s , using the fact t h a t
v p • N,
02p Ox2p oh(x, x) = O,
02p+l ax2p+l ¢(x, x) = ,~P, we d e d u c e t h a t v p • N with 2p + 1 _< [1] + 3,
02P 02P ok I x 02p Ox~ z ( x ) = 0 - ~ q / ( x ) + k f f ~ aE(p-1)-k ox~H(x) + -1 0 - ~ 4~(x' t)H(t) dt,
02p+l z(x ) - 0X 2p+102p+1~(x) + Ox2p--~
Ok H(x) + I x 0x2p+102p+I ~ O/12P_1_k oxk 4~(x, t)H(t) dt, ke~ 2 -1
w h e r e 3( 1 = {0,2 . . . . . 2p - 2] a n d 3( 2 = [1, 3 . . . . . 2p - 1].
1338
F. A B E R G E L and C. D U P A I X
Thus, using the results o f remark 2.3, we deduce that H e e~_+~(I,$(I)) implies z~, e e,t+_1(I, $(I)). Then, if s > 1, we have Izlet+-,t~,w))= Iz[= + Iz~[® + Izx~let+_l(z,zm) and H e e~_+l(I, 8(1)) implies that IHI® < +oo, so that Iz(x)l <-I~(x)l +
¢(x, t)H(t) -1
-< [~u[~ + 2[4,1®[Hl® (
-poO,
and thus, [z[® < +0o. In the same way, we deduce that [z~(x)[ < +oo. Thus, for s > 1, $(I)). On the other hand, if 0 < s < 1, we have Is] = 0, and thus Z e c°t+3(I, ~'Js+l [zle~t~](/.s(/)) = [zl.o +
[Zxl~ + Sup
~,y~1
IzAx) - zAy)l + Iz~les_l(,, w)) Ix-yl ~
x~y
However, for 0 < s < 1,
i and H e e tS--1 + l ( I , $(I)) implies Sup
x 1 -1 dl-S(t) dt < +oo,
[dl-*(x)H(x)[ < +o% so that
XEI
-1
IH(t)l dt
=
<
IH(t)l dt
-1 dl-*(t)
SuPldl_S(t)H(t) [ ( 1 - - 1d t < t~l
,1 -1 d l - S ( t )
+oo.
Thus, we deduce that [ z ( x ) [ - [~'(x)[ +
Ifx 4~(x,t)H(t)dt r -1
-< I~,1~ + I~1®
f'
In(t)[ dt
-1
<+oo
and, in the same way, we deduce that [z.[= < +oo. Finally, we have [z,(x) -
Zx(Y)l
Ix - yr ~
= <
[~u~(x) - ~,,(y) + 151 (O/Ox)~(x, t)H(t)dt Ix - yl s IqJ. (x) - q/x (y)[ + Ix - yl s +
[J~-1 ((O/Ox)4~(x, t)
[ff ( a / O x ) o ( y , t ) H ( t ) dtl fx - yl s
-
- JY--1(a/Ox)¢(y, t)H(t)dt[
(O/Ox)4~(y, t))H(t) tit[
Ix - yl s
S t a t i o n a r y interfaces for M a r a n g o n i - t y p e flow
-< 2'-sl~xxl,~ +
x
I
IH(t)l dt Sup Sup I(O/Ox)4~(x, t) - (O/Ox)4a(y, t)l
-1
Ix
1339
X,yel x#y
IX -- Y[+
tel
Y[ ~ yx n ( t )
< +oo.
Thus, H e co++t t+3 8(I)), which completes the p r o o f for the ~,F$_ l (I, 8(I)) implies that z is in Cs+l(I, regularity o f z solution of (Scs,). We now come back to the problem (Sw0. Let T be the Green's function associated to problem (So+t) and H that associated to problem (S+a0. We have for all x e / , x
z(x) =
I
T(x, t)H(t) dt,
-1
w(x) =
H(x, OH(t) dt. -1
We then write w in the form
w(x) =
S,
(rI(x, t) - T(x, t))H(t) dt + z(x).
-1
Thus, we want to prove that F(x) :=
i
x (H(X, t) - T(x, t))H(t) dt ~ tat+3tt .~+, ,., g(I)). -1
Using the properties of the Green's functions T and H, see for example [17, p. 305], and in particular that (T - H ) ( . , t) is C' everywhere and is a C z function except possibly for x = t, we deduce, proceeding as in the case of constant coefficient, that H ~ C~_+] (I, 8(1)) implies that tot+3tt 8(I)), and the p r o o f of proposition 4.3 is completed. 4.4. P r o o f o f theorem 4.1. To prove theorem 4.1, namely the well-posedness of the problem (S,) for small enough values of the parameter e, we use an implicit function theorem. Thus, we consider the problem (S,) on a domain f ~ obtained as a perturbation of the domain D O determined by g. We transform the problem ($8) on Dn to a problem on the fixed domain f~o, and we then apply an implicit function theorem to a mapping defined on a space of suitable deformations. 4.4.1. T h e s p a c e ~pt,s o f admissibleperturbations. Let t~ e PP+*. In the following, D o will stand for the open bounded set of ~2 determined by the capillary solution ff of (11). For s e ~ ÷* and _ l + 2 and (p, r/) e ~,-.s+1 t,ot+3 (I, 8(i)))2, we define the transformation 1 ~ R +* satisfying s < Mo ~ Zo ~ M~ = M o + J ( P n o + r/to), where n o and to are, respectively, the outward unit normal and unit tangent vector to Yo at M o . Z~ will stand for the set of the points M~ so obtained. (p, r/) cannot, o f course, be chosen in an arbitrary way.
1340
F. ABERGEL and C. DUPAIX
Indeed, since the container has vertical rigid walls, M o M ~ and j must satisfy a collinearity condition at x = __1, namely rl = Pgx
at x = +1.
Moreover, for simplicity, we choose tangential deformations r/such that rl(x) = p(x)gx(x)
V x ~ [.
This choice amounts to considering deformations in the j direction only. This kind o f deformation is allowed because of the existence of a representation of Eo in term of a function g o f x . For a more general case, namely for a parametric representation of Zo, the choice of the deformations is a little bit more complicated but remains possible. Indeed, we would then choose deformations in parametric forms, as graphs over Zo. Then we denote by (P~'s(I) the space of admissible perturbations. 4.4.2. P e r t u r b a t i o n s o f t h e o p e n b o u n d e d set D o. For p ~ (Pl's(I), we set
X~ -- {M~ I M~ -- Mo + Op(no + gx to) qMo 6 Eo] =
(x~,y~)
Y~
g(Xo) + ~p(Xo)X/1 + g~(xo)
¥xo ~ ( - 1 , 1) .
We then associate to p a global transformation Z = X(P) mapping f~o onto f ~ and defined as
M o ~ X(Mo),
where X(Mo) = (O(Xo,Y O ) ( y : )
/ x° + (1 - ~O(Xo,Yo) )
)
Yo . . . . \ ~ - ~ o ) tgtXo~ + Op(Xo)~/1 + ~2(Xo))
'
where tp is a smooth function such that
~0(x, y) = 0
if h 1 < y _< ~(x)
0_< ~p(x,y)_< 1 ~0(x, y) = 1
ifhz<_y
<
hi
if y < h 2 ,
where h I > h z are two positive constants. For ~ small enough, we can define the inverse transformation 2~-1, which maps ~n onto ~ o , as X -~ = X-I(P): ~
~ ~o
Mn ~ X-I(Mo),
Stationary interfaces for Marangoni-type flow
where
1341
~9(gi~,Ys)(;:)
)~-I(Mts) =
+ (1 - ~o(x~,yD)
y~g(x~)
.
g(x~) + 6p(x~),,/1 + g~(x.) Thus for 6 small enough, say 6 < 6 o, the transformations X and X -1 a r e as smooth as the function p. Moreover, if p e ~Bt'~(0, 1/6o), the ball of
X(ff) e
t+31 ( D o , $ ( D o ) ) ) 2 I,/co XFS+
1+3 (D~, 8(D~))) z. z - l ( p ) e ,,co t2"-~s+l
Definition. Let us suppose that 0 < 6 < 60. Then D6 will be called a perturbation of the open bounded set Do if there exists a function p e (~t's(0, 1/6o) such that Do = Z-I(D,). 4.4.3. Transformation of the problem on Dn to a problem on the fixed domain Do. Let D~ be a perturbation of D o. Thus, we want to find (us, 0n, a~,fn, Cn) solution of (S,) on Ds. However, D~ is a perturbation of D O, so that this problem is equivalent to finding (u, 0, a , f , C) solution of " ~'.u
in flo
=0
(u" (7)u - D ] v o ( u ) - ~.ej = 0 u" ~0
u=O
1
---7x0= Pr Re and
O0 ----~5 Oh
0
and
o n Fo
u.h
=0
o n Xo
Ma* ~ = 0
a(u)- ~. h + P. +
fx(-+ 1)
-
41 + f ) ( _ t )
I
in ~ o
0 = 0
a ( u ) • ft. { -
i
in ~ o
t u(u) .fl.[ldx
-
+_cos(B) 0
-1
d e t ( o Sdx )dy
= V,
on Zo
Ma*~(O)\
ix
_
o
on Zo
1342
F. A B E R G E L and C. D U P A I X
where • ~ and ~-1, respectively, stand for the Jacobian matrix associated to X(P) and X-l(p), and where we set: •
U =
U60)~,
• 0 = O~oz, •
0"=
0"~0)~,
• f =Aox, •
9=
5~ox,
• E(Mo) = n(X(Mo) ) = n(M~), • t(Mo) = t ( z ( g o ) )
= t(M~).
4.4.4. Using the implicit function theorem. We define the function 5: acting from ×
(~, +oo)
× ,r°t÷3(I, ~s+l
8(1)) x ~ x (es1+2 (~o, 8(~o))) 2
into e~+](I, 8(1)) x IR × ~ × (e~+2(~ o, 8(~o))) 2 × R2 as
~(e, v, p,w) - c l
(7(11)
• ~ . ~ dx
I: o
: (e, V, p, C, w) ~*
t det(~) dx dy - V 0 rio n--w
fx(-+l)
:~ cos(B)
41 + f)(___l) where ,
~(e, V, p, w) = a(u)" fi" h + Pa + -Frrf - Ma*ot(O) and (u, 0, a) is the solution of the system f7 • u = 0
in ~0
( w " fT)u - Dlv tr(u) - 2 0 j = 0
in D O
w"
fTO
u=O
p 1 R e AO = 0
in f~o
0=0
onFo
and
00 O--~=-e9
and
u'E=O
one o
00 a(u)- E- t - Ma* ~-~ = 0
on Z o.
,
Stationary interfaces for Marangoni-type flow
1343
If we set So := (t = 0, V = Vo , p = 0, C = C o , w = 0), it is clear that ~(So) = 0. The m a i n result in the course o f proving t h e o r e m 4.1, is stated in the following proposition. PROPOSITION 4.4. Let us suppose s > 0 and l > 0 given and satisfying s _< ! + 2. There exists a finite sequence of real n u m b e r s 0 < J[1 < "'" < ~K < -~-OO
such that if Ma*FrOo ;~ Ai for all i = 1, ..., K then (a) there exists an open n e i g h b o u r h o o d Wo o f So in [R x (l?, +~o) x e s1+3 + t ( / , $(I)) × R x (C S/+2 (D o, $(Do))) z such that ~Y is a ~ t function on Wo; (b) Dto, c, w)lY(So), the Fr~chet derivative o f 5: with respect to (p, C , w ) at So, is an i s o m o r p h i s m f r o m sc°t+31r+l~., v $(I)) x rR × (es/+2 ([2o, ~(~'~0)))2 onto C sI+1 - z ( I, $(I)) × [R × [R x 1+2 (~'~0, 8(~')0))) 2 X IN2. ((~s
T h e o r e m 4.1 is a straightforward consequence o f p r o p o s i t i o n 4.4. In order to p r o v e p r o p o s i t i o n 4.4, we need several auxiliary results. Therefore, we introduce the following functional spaces a'(ol+l (~')0, ~(~'~0))) 4/[R X 1= ( C s1+2 (~"~0, 8(~')0))) 2 X t'ol+Ect'~ "~s ~,'c'O' ~(~"~0)) X I,~'~s-1
Y
:=
e~i~ (~o, S(~o)) x (e~_2(~o, S(~o)))2 x e~_df~o, $(f~o)) x (e~+2(Fo, $(Fo))) 2 x e~+2(Fo, $(Fo)) x e~+2(T_,o,$(T-,o)) × e~+-~ (~o,
S(~o)) ×
e~+_~(~o,
S(~o)),
and for S = (e, V, p, C, w) e [R x (l?, +oo) x r°t+3(l, ,~,+1 $(I)) x • x the linear o p e r a t o r A = A(S) such that
(C,/ + 2 (Do, $(D0))) 2 we
A(S): X ~ Y (u, o, a) ~- A(S)(u, O, ~), where ~7 " U
(w" ~7)u w.~O
a(u)
Dlv
-
~0j
1
- ~ o
Pr Re
Ulro A(S)(u, 0, a) =
Olro U • n[~ o
o(u) • ~. i -
M.*
oo~
at/l%
define
1344
F. ABERGEL and C. DUPAIX
Then, we consider the following problem: Given F = (fi)/s= 1 ~ Y, find(u, 0, a) • X such that A(So)(U, 0, a) = F. This p r o b l e m can be f o r m u l a t e d as: Given F = (f/)/8 1 • Y, find (u, 0, a) • X such that - - -
Pr Re
A 0 = f3
in f~o
(12)
0 = f5
on Fo
(13)
on Zo
(14)
O0
On
f7
and V • u = fl
in Do
(15)
in D O
(16)
u = f4
o n F0
(17)
u" n = f6
o n Zo
(18)
on Eo.
(19)
- D i v a ( u ) = 2 0 j + f2
O0
a ( u ) " n - t -- Ma* ff~ + fs
Let f] be an open b o u n d e d set o f ~2. Let F be a part of 0 ~ with endpoints $(f]) = [,-)i {M(i)} • Let us suppose that there exist n e i g h b o u r h o o d s o f M (i) and functions gi such that, in the local coordinates system (M a), n(M 0, such that if s e (0, C~o) then, for all (f3,f5 ,f7) • C~-2(Do, $([2o)) × ~+2(Fo, 8(I"0)) × Cst+~(Zo, $(Eo)) satisfying the compatibility condition Ofs/Onz = f7 at points o f $(f~o), there exists a unique solution 0 • e~+2(f2o, 8(f~o)) o f equations (12)-(14). (b) M o r e o v e r , there exists a real n u m b e r c~1 satisfying olI > Max(t~ o, z) 2 such that, if cot+3 with s • (1, cq) and s < 1 + 2, then the conclusion o f (a) remains true. Fo • ~ + 2 and Z o • "~+1 rot+2 and E o e x'~a+l rot+3 with a e (0, 1). There exists a real n u m b e r ~oo > 0, LEMMA 4.6. (a) Let F o e "-'a+l such that if s • (0,~0) then, if ( f l , f z - 20j, f 4 , f 6 , f s + Ma*(a/Ot)O)•YI, there exists a unique solution (u, a) • (Cs1+2(Do, $(Do))) 2 × [(ol+lt ~'~s-l~ D o, $(~o)))4/[R o f equations (15)-(19). (b) M o r e o v e r , there exists an angle fl, such that, if fl < fl. then there exists a real n u m b e r -oq - such that, if Fo • -~t+2 s and )"~0 • rot+3 ~'s+l with s • (1, ~1) and s < 1 + 2, then the conclusion o f (a) remains true.
Stationary interfaces for Marangoni-type flow
1345
Y1 is the subspace of Y o f functions (gl, g2, g4, g6, g8) E e~+~(~o, 8(~0)) × (cls_2(~o, 8(~')0))) 2 )< CI+2tE ~ o, $(Fo)))2 × co/+2iv "+ ~-0, $(Zo)) × co/+1 "-'s-l~t E o, $(Zo)) satisfying the compatibility conditions: •
• g6(Mi) = g4" nc(Mi), • i f B ~ 7r/2,
l[
gl(Mi) - sin(B)
+)]
+
- n~ • 0~r g4 + cos(B)
l[
+
g6 - g+" ~
(
n~
+
(Mi)
0)]
+ cos(~-) t~. ~ r g+ + sin(B) Reg8 - 0T~ g6 + g4 "~t-£~n~:
(Mi),
• if B = 7r/2,
(-+t~.) ~ r
g4 (M3 =
(Reg8 - ~+g 6
+ g4" ~
+ )nz: (Mi),
where 8(f2) = U2= l [Mi} and where tv is chosen so that tv = - n ~ for fl = z#2. The p r o o f of lemma 4.6 is given in [7], and that of lemma 4.5 follows step by step that of lemma 4.6. From these two lemmas, we deduce the following corollary. COROLLARY 4.7. The assumptions on Fo and Zo are the same as in lemmas 4.5 and 4.6. (a) There exists s o with s o ~ (0, 1), such that, for all s e (0, So) and all F e Y satisfying the compatibility conditions of lemma 4.5 and 4.6, there exists a unique solution (u, 0, tr) e X of equations (12)-(19). (b) Moreover, i f f l < f l . , there exists sl > 1 such that for all s ~ (1, Sl) satisfying s _< l + 2 and all F e Y satisfying the compatibility conditions of lemmas 4.5 and 4.6, the conclusion of (a) remains true. Therefore, we deduce from the result above that A(So) is an isomorphism from X onto the subspace of Y of functions satisfying the compatibility conditions of lemmas 4.5 and 4.6. We now give another useful result. LEMMA 4.8. Let 6o > 0 be as in Section 4.4.2. For 0 < 5 < 50, S+ ~ A(S~) is a E 1 function from IR × (17, +oo) × (Bl.s(0, 1/5o) × [R × (C~+2(~ o, 8(~o))) z into L ( X , Y ) , space of linear continuous operators from X into Y.
Proof. The operator A is a linear function of (e, V, C, w). Thus, we have to prove that, for all fixed (e, V, C, w) in ~ × (I7, +oo) x R x (e~+z(f~o, 8(~o))) 2, the mapping .A(') = A(e, V , . , C, w): (Bt'~(O, 1/~o) --. L ( X , Y)
p ~ ~(p), is C ~.
1346
F. ABERGELand C. DUPAIX
This mapping can be written as p ~. z - l ( p )
~. d ( p ) .
Thus, we first consider the mapping p ~ x-l(p). Let (P.)n ~ N be a sequence of 63t's(0, 1/~o) converging in Cs+l t+3 (I, 8(1)) to p belonging to (Bt's(0, 1/~0). We have (x-l(P")
- x-I(P))(x'Y)
=
(
6~.(x,y)(p
o
- p.)(x,y)
)
'
with ~ . bounded in es÷l(I, t+3 $(I)) independently of n. Thus, there exists d > 0 such that [X-I(P.) - X-l(p)let++l(o6,s(n,)) <- d i p .
-
Plel;~(,,s(n),
and thus 63t's(0, 1/6o) --' ~-~,+l~r°l+3(f~,8(QD))2 p ~" x - ' ( p ) ,
is a Lipschitz continuous function. Moreover, and in the same way, it is easily seen that x - l ( p . ) - X - ~ ( p ) = L ( p . - p) + ]p. - ple~+](z,z(O)~(pn - p), where L(.) is a linear operator and where ~(p. - p) goes to 0 in (Cs+I t+3 (f2~, 8(f~)))2 as p tends to Pn in ¢ol+3/r 8(i)). Using for example the results of [5], we infer that x"S+1 \~) 63t,s(0, 1/~0 ) ~ (es+l t+3 (~')8 ' 8(~"~8)))2 p ~ Z-I(p) is ~1, We now consider the mapping 1+31 (~'~, ~(~'~,)))2 ..._}L ( X , (es+
Y)
x - l ( p ) ~ ~Zl(p). -~ depends on X - ~ ( p ) through its Jacobian matrix ~q-~ in a polynomial way. Then, the fact that the function g - l ( p ) ~ ~-1 is smooth from ~--~+ltr°t+3(f~6,$(f~)))2 into (Cs_ lt+2 (f~, $(~)))4 makes the proof of lemma 4.8 complete. We can now prove proposition 4.4. The mapping 5: can be written
S L
(Us, Os, as) L
where (Us, Os,as) is solution of A(S)(u, 0, a) = 0.
5:(S)(ns, Os, as),
Stationary interfaces for Marangoni-type flow
1347
We set Qs := (Us, Os, as). Thus, we first want to prove that there exists an open neighbour[ r $(I)) × ~ × (es1+2 ( f ~ , g(f~)))z such that the mapping hood Wo of SOin ~ × (V, +oo) x gol+3 ,~,+1~-, ~1 is of class C 1 on Wo. Let ~ := R × (17, +oo) × 6~1.s(0, 1/C~o) × R × ( e ,~+z( f ~ , 8(~)))2. Let Qso be the solution of A(So) • Qso = 0. Let 17 ~ R +*. For S ~ 'It such that IS - Solar < r/, let Qs be the solution of A ( S ) • Qs = 0. Then A(So) .
[QSo - Osl =
( A ( S ) - A ( S o ) ) . Os
so that, using lemma 4.8,
A(So)" [Qso - Qs]
=
DA(So)(S - So)" [Qs - Oso] + DA(So)(S +
Is
Sol~¢(s
-
-
-
So)" Qso
So),
where DA(So) is the derivative o f A at point So and ~(S - So) tends to zero in Y when S tends to So in '1/. Thus, since A(So) ~ L ( X , Y), S ,-. A ( S ) is a C 1 function, and (Qso - Qs) remains bounded in X , we deduce that, when S tends to So in '11, then A(So). [Qso - Qs] goes to zero in
L(X, Y). Thus, since A-I(So) remains bounded in L ( Y , X ) thanks to corollary 4.7, we deduce that Qs - Qso = IS - Solat~(S - So). Thus, Qs = Qs - A-I(So)(DA(So)( S - So)" Qso) + IS - S0[~t((S - So), where ((S - So) tends to zero in )~ when S tends to So in qL, so that Dffx(So) = 0 and there exists an open neighbourhood I41oo f So in ~ in which S ~ Qs is C 1. Using the expression of ~F(S) • Qs, it is now easily seen that the mapping Qs ~ ~F(S) • Qs is ~ 1 Taking into account the fact that (u o, 00, ao) = (0, 0, 0), we can see that the derivative of 5: at So evaluated at (p, C, ~¢), denoted by D(p. c,,o $(So)(P, C, #), has the following expression IT
.
1
~2
"no no + ~rr(P~f[ + gx + Dwf(Cv))
- Ma*(T(~-~
,, ,,41 + ~,~/~
-Oo(DpH(g)'p+
r" n o n o dx
I
o
~o
l
l p41 + ~ a x -1 V--~'
DwH(~¢)))-C
1348
F. ABERGEL and C. D U P A I X
where we have set v = Dpv(p) + D~v(@), T = Dp T(p) + Dw T(@), z = Dpr(p) + D.r(@), and where the derivative o f the m e a n curvature H with respect to p at ~, denoted by D p H ( g ) . p, is given by D p H ( g ) . p = _ 1 Pxx + ~2 + (1 + ~ ) 2 P x
\-O+-gx2-~
+ (1 + g ~ ) 2 ] P "
(v, T, r) are the solution o f the linearized p r o b l e m V •v = 0
in ~o
-Div(z) - 2Tj = 0
in f~o
1 - -
-
Pr Re
v=0
(Plso)
A T =
and
in ~o
0
T=0
onFo
v • no = 0
on E o
0T = 0 Üno
on E o OT
r-
no.
to -
Ma*~
Oto
= 0
on E o .
Thus, we are n o w going to prove that Do,,c,w)5(So), defined above, is an isomorphism f r o m e~++~(I, 8(1)) × [P x (e~+2(f~o, 8(~'~0))) 2 o n t o C~_+](I, 8(I)) × ~ x ~ × (C~ + 2(~ o, 8(~'~0))) 2 X [~2. Therefore, we n o w show that, under the assumption that, Ma*FrOo ~ 2 / f o r all i = 1 . . . . . K, tr 8(I)) x R × Ct+2/~ there exists a unique solution (p, C, w) e cot+3 --.s+lv, s ~ o, 8(~o)) 2 satisfying L l ( z , T ) + Dp£(p) + Lz(w) - C = hi
vx~I
"t" n o n o dx ~-. C 1 I
f
o
F,o
f -11
p~/1
v-w=h
÷
= c2 2
together with the condition
(px + P 1exexx + ~,zxJ +
( D w f ( W ) ) = a+l
at x = +1,
Stationary interfaces for Marangoni-type flow
1349
where (v, T, r) satisfies (Plso), and where we have set r)
=
.no " n 0
-
Ma
*T( gx
'x
--
,
1 -z Ma*~DpH(p), Dp£(p) = -Fr px/1 + gx + 1
L2(w) = ~rrDw f ( w ) + Ma*OoDwH(w). First,we remark that for all ( h i , C1, C2, h2, al, a_l) E C~_+~(I, 8(1)) x [R × R × (~+2(~'~0, 8(~'~0)))2 X [R2,
there exists a unique solution (v, T, r) satisfying the system (Plso) (v, T, r) = (0, 0, constant). Moreover, the condition Jg0 r" n o • n o dx = C1 yields z = (C1/Mes(Eo))Id, where Id stands for the identity matrix of [R4. Thus, v = 0 together with equation v - w = h 2 , gives w = - h 2 , SO that in order to prove proposition 4.4, we only need to study
Dp£(p) - C = h
f
l px~- + -1
Ip
x÷
vx ~ I
(20)
= C2
gxgxx \ l +gx/
(21)
,,
for/=
(22)
1,2
with x~ = - 1 Xz = +1.
However, assuming that p is known, then the equation (20) gives p equation (21) gives C as the implicit solution of
i
1 (Dp£)-l(h
C)(x)qI
+ gx2(X) d x =
=
(Dp£)-l(h
+
C), so that
C2.
-1
Thus, it only remains to prove, for all function h ~ "c°t+t/I ~ - 1 ~ , 8(I)) and all (al, a2) ~ ~2, the col+3tr 8(1)) satisfying existence of a unique function p ~ ,~s+~-,
x + P l ~ +g~" "~'x t i )"= a i
fori=
1,2
with Xl = - 1 X2 = + 1 but this is precisely the result given by the proposition 4.3. This ends the p r o o f of proposition 4.4.
1350
F. ABERGEL and C. DUPAIX REFERENCES
1. COWLEY S. J. & DAVIS S. H., Viscous thermocapillary convection at high Marangoni number, J. Fluid. Mech. 135, 175-188 (1983). 2. MANNEVILLE P., Structures Dissipatives, Chaos et Turbulence. Collection Al~a Saclay. 3. SCHWABE D., Surface-Tension Driven Flow in Crystal Growth Melts, Crystals Growth, Properties and Applications, Vol. 11. Springer, Berlin (1988). 4. ABERGEL F., A geometric approach to the study of stationary free surface flows for viscous liquids, Proc. R. Soc. Edinb. (1993). 5. ABERGEL F. & BONA J. L., A mathematical theory for viscous, free surface flows over a perturbed plane, Archs ration. Mech. Analysis 118, 71-93 (1992). 6. SOLONNIKOV V. A., Solvability of a problem on a plane motion of a heavy viscous incompressible capillary liquid partially filling a container, Math. USSR Izves. 14(1), 193-221 (1980). 7. SOLONNIKOV V. A., On the Stokes equations in domains with nonsmooth boundaries and on viscous incompressible flow with a free surface, in Nonlinear Partial Differential Equations and Applications. Coll~ge de France Seminar III, Paris (1980). 8. LAGUNOVA M.V., Solvability of the plane problem of thermocapiUary convection, J. Soy. Math. 45(1), 1130-1140 (1989). 9. BOUILHAC I., Ecoulement Marangoni-Rayleigh induit par un flux thermique sur la surface libre, Th~se de l'Universit6 Paris 11 (1991). 10. ALLAIN G., R61e de la tension superficielle dans la convection de B6nard, Math. Modelling numer. Analysis 24(2), 153-175 (1990). 11. MAZ'YA V. G. & PLAMENEVSKII B. A., Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points, Trans. Am. math. Soc. 123, 89-108 (1984). 12. LANDAU L. & LIFCHITZ E., Physique Th~orique. M~canique des Fluides. MIR, Moscow (1971). 13. FINN R., Equilibrium Capillary Surfaces, Grundlehren der Mathematischen Wissenschaften, Vol. 284. Springer, Berlin. 14. GERHARDT C., Boundary value problems for surfaces of prescribed mean curvature, J. Math. pures appL 58, 75-109 (1979). 15. CHURCHILL R. V., Operational Mathematics. McGraw-Hill, New York. 16. INCE E. L., Ordinary Differential Equations. Dover Publications, Inc., New York. 17. CODDINGTON E. A. & LEVINSON N., Theory of Ordinary Differential Equations. McGraw-Hill, New York.
Copyright© 1996ElsevierScienceLtd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00+ 0.00
Pergamon
0362-546X(95)00118-2 EXISTENCE OF SMOOTH, STATIONARY INTERFACES FOR MARANGONI-TYPE FLOW F. ABERGEL and C. DUPAIX Laboratoire d'Analyse Num6rique, Universit6 Paris Sud B,~timent 425, 91405 Orsay, France
(Received December 1993; received in revised form January 1995; received for publication 1 June 1995) Key words and phrases: Navier-Stokes equations, Marangoni effect, free surface, weighted spaces, existence and uniqueness, thermohydraulics, Marangoni effect.
1. I N T R O D U C T I O N
In this paper, we consider the flow of a viscous liquid, when its free surface is subject to a nonuniform thermal flux. This type of flow, known as Marangoni type flow, see e.g. [1-3], is characterized by both a convection phenomenon within the fluid and an additional tangential stress acting at the interface. Concerning the convection flow, we shall make the classical Boussinesq approximation for variation of the density, so that the really interesting terms in the modeling equations will be provided by the surface law. Our main result concerns the existence and uniqueness of a stationary interface, in the neighbourhood of the capillary solution, i.e. one that is obtained when the flux is zero and the fluid is at rest. This result is obtained via an adequate implicit function theorem. Unlike similar works for the Navier-Stokes systems [4, 5], the main technical difficulties lie in the presence of corners at the boundary of the domain occupied by the fluid, and use is made of the machinery developed in [6, 7]. We also need to study Sturm-Liouville type problems in weighted spaces. This work generalizes the results obtained by Lagunova [8] to the case that the gravity constraint is taken into account. Moreover, we want to mention [9], where a numerical study is performed, and [10] where a bifurcation problem is studied. The paper is organized as follows: Section 2 contains the main geometrical and functional hypotheses and notations; in Section 3, the physical problem is described, and its mathematical formulation given. Section 4 is devoted to the proof of our main result: after presenting the variations of domains suitable for the problem, we study in Section 4.3 Sturm-Liouville problems in weighted spaces, and finally prove in Section 4.4 the implicit function theorem that we need. Essentially, we show that, when a certain parameter (the Marangoni number times the Froude number) does not belong to a finite set of real numbers, then, for small enough values of the thermal flux, there exists a unique stationary interface in weighted H61der spaces. Despite and unlike the results of Lagunova [8], a consequence of taking into account the gravity is that, when the Marangoni number times the Froude number is in this finite set, the implicit function theorem does not apply any more, and bifurcation phenomena may possibly appear. 1329
1330
F. ABERGEL and C. D U P A I X 2. F U N C T I O N A L
SETTING
Let O be an open bounded set of ~2 whose boundary Of~ is a piecewise smooth Jordan curve, and denote by 8(O) the set of points where the tangent vector to 0 0 is discontinuous. We define the following functional spaces.
2.1. The space C m'~ of c~-HOlderian functions of order m Let m be a nonnegative integer and o~ e (0, 1) a real number, e ' ( O ) will stand for the space of m-times continuously differentiable functions in f2. We then define the space of m-times continuously differentiable functions in O whose mth derivatives satisfy a H61der condition with exponent c~. We denote this space by cm'~(O) and endow it with the norm
lule-.o(,)=
E
Sup lOku(x)[ +
I/¢l
E
Sup
IDku(x) - Dku(y)]
Ikl=mx,yef~ x¢y
[X-Y] ~
2.2. The space C~ of weighted H6lderian functions of order [l] and exponent I - [1] For x ~ ~, we define [x] as the integer part of x. For y e O we set
d(y) :=
Inf
z • ~(fl)
Ily-
zll,
where I[" I[ stands for the Euclidian norm. Let l be a positive noninteger number, and s a noninteger real number. For s e (0, l], we define the space C~ (0, $(0)) for which the following weighted norm is finite s
+
~]
Sup
I
Min(dt-~(x), dt-S(y))
[k] = [l] x,y ~ fl x#y
IO%(x~ - Oku(y)[~ ['X -- y ~ ]
")'
and for s < 0, we equip Cts(O, 8(0)) with the norm
lule'.
E
Sup
Id(x)lel-'Oku(x)l
O
=
Sup
-b iklE[l ] X,y e fl x ~y
Min(dt-~(x), dt-~(y)) IDku(x) - Dku(y) I~ [-X -- ) [ l - ~ ]
")"
In the same way, and for all noninteger s and all positive noninteger / satisfying s < 1, we define the space C~(I, 8(I)), where I is a segment. 8(1) is then composed of the endpoints of I. 2.3. Remarks and notation A quite complete study of weighted H61derian spaces can be found in [11]. We now give some properties of weighted H61derian spaces which will be useful later. Let I e R ÷* and s e [R be two given noninteger numbers. (i) Let k be a positive integer satisfying k_< [1]. We suppose that s_< I + k. If u e e~+~(O, $(0)) then Dku e C~_k(O, $(0)).
Stationary interfaces for Marangoni-type flow
1331
(ii) We suppose that s _ 1. If u e C~(o, $(f~)) then if U is a primitive of u, it satisfies U ~ col+l(f~, 8(~)). '~s+l (iii) Let be given (11, 12) ~ (JR+*)z and (s I , s2) e •2 satisfying s I _< l 1, s 2 _< 12 on one hand, and 11 12 l 1 < l2, S1 < S2 on the other hand. Then if u ~ Csl(D, $(~)) and v e Cs2(~, $(~)), we have (u, v) ~ C~1(~, $(D)). Moreover, there exists a positive constant C, independent of u and v such that the following estimate holds
Notation: • In the following, when we consider the space C~, we will always assume that l > Max(0, s). • lul® will stand for Sup [u(x)[. x~fl
3. S T A T E M E N T
OF THE PROBLEM
3.1. The physical problem
We consider a container C, partially filled with a viscous liquid. The boundary o f the container OC is composed o f two half-lines (x = - 1 , y > 0) and (x -- + 1, y > 0) which are connected at the points ( - 1 , 0) and (+1, 0) by a smooth curve OC-, which is supposed to lie in the half-plane (y < 0). The gravity g, is directed along the vector - j ( 0 , - 1 ) . We denote by f2 the part of C containing the fluid, f2 is the open bounded set o f [~2 defined as: £) := {M(x, y) ~ C Iy < f(x)] and r. := [M(x, y) e C I - 1 < x < + 1 and y = f(x)} is the interface between the fluid and the atmosphere. Let F := all\Y, and $(f~) := F A Y.. Let I := ( - 1 , +1) and $(I) := ([-1}; {+11). We suppose that the angles of contact between F and E are both equal to fl ~ (0, ~r) with the convention that fl = 0 if t~ = - j at x = - 1 , where tr. is the unit tangent vector to E. The flow is supposed to be stationary. The interface is subject to a nonuniform thermal flux 5. Thus, the temperature gradients induce, on one hand, a superficial stress which generates a Marangoni flow and, on the other hand, a volume force inside of the fluid, which generates a convection flow. Several authors have considered the physical aspects of this problem, for example, [1-3]. 3.2. The mathematical formulation We first make the two following assumptions. (A1) We assume that the Boussinesq approximation holds, namely, on one hand, the external force satisfies pc(1 - a ( T - Tc))g and on the other hand, the mass density p is constant and equal to Pc in the volume of the fluid. Where Pc = p(Tc), Tc is the temperature of the boundary of the container (i.e. the rigid part of the boundary). Tc is supposed to be a constant, a is a given positive constant. (A2) The surface tension coefficient y is given as a nonincreasing affine function o f the temperature.
1332
F. ABERGEL and C. D U P A I X
The boundary conditions are as follows: (BC1) the fluid satisfies a no-slip condition at the boundary of the container; (BC2) the temperature at the boundary of the container is constant and equal to Tc; (BC3) the interface is subject to a nonuniform thermal flux 9; (BC4) the interface is in thermal and dynamical equilibrium. We then write the conservation laws of mass, momentum and energy and obtain, in a dimensionless form, the following system of partial differential equations for the unknowns ( u , 0, a , f , C ) ,
V. u = 0
in ~
(1)
(u" V)u - Div a(u) - :t0j = 0
in ~2
(2)
1 u" V0 - - - A 0
in ~2
(3)
onF
(4)
onZ
(5)
on E
(6)
one
(7)
Pr Re
u=0
and
00
--=-e~
On
(sD
iT(u) • n . t -
= 0
0=0 and
u'n=0
Ma* 00 = 0 Ot l f _ M a . ~ ( O ) ( \~ )
(7(u) • n . n + P. + F r
f:: x - C = O
1 iT(u)" n. n dx = 0
(9)
-I
f
d x d y = V,
(10)
12
where • u, 0, a are, respectively, the velocity field, the temperature and the stress tensor with 2
o(u)
= - p I d + Ree
e(u),
where e(u) = ½(Vu + Vu t) and p + y/Fr is the pressure, • C is a constant which has to be determined, • A is a contant equal to 1 if we consider an evaporation problem and - 1 if we consider a condensation problem, • ~ = ~(x) is the nonuniform thermal flux, • e is a small parameter, • Pa is the atmospheric pressure,
Stationary interfaces for Marangoni-typeflow
1333
a(0) = 0 + 00 is the surface tension, where 00 is a constant, • Pr is the Prandtl number (cinematic viscosity over diffusion coefficient), • Re is the Reynolds number (characteristic velocity V* of the flow times characteristic length L* over the cinematic viscosity), • Fr is the Froude number (square of the characteristic velocity over characteristic length times gravity field), • Ma* = (dy/dO)(1/pL*V*2), where d y / d 8 is the derivative o f the surface tension with respect to the temperature O, and p is the mass density. Note that Ma* is related to the usual Marangoni number Ma, with Ma* = - M a / P r Re 2, • V is the volume occupied by the fluid. •
Remark 1. Equations (6) and (7) result from the local decomposition o f the vector equation a(u)n + (pa + l f ) n - Cn + Ma*(~(O)Hn - Va(O)t) = O, which expresses the dynamical equilibrium of the interface. (a(u) + Pa + (1/Fr)f)n is the superficial density of force describing the action of the fluid on the interface. (o40)Hn) is, following the classical interpretation of [12], the first variation o f the surface energy, and it measures the forces to apply to obtain a deformation at the interface. Finally, ( - V a ( 0 ) t ) is induced by the variation of the surface tension and characterizes the Marangoni type flows. 4. WELL-POSEDNESS IN THE NEIGHBOURHOOD OF THE CAPILLARY SOLUTION
4.1. The main result Our purpose is to prove the following result. THEOREM 4.1. If F belongs to C t+2 then: (a) There exist a finite sequence o f real numbers, 0 < / ~ 1 ( "'" < /~K < + QOand real numbers I7"> 0, eo > 0 and s o E (0, 1] such that if Ma*FrO o # )t i for all i = 1, ..., K then
vlel < e0 v s • (0, So) v V > 17" ¥ 9 • C;+ltI s-ix , 8(1)) satisfying the condition 5(M (i)) = 0 if fl = r~/2 with 8(/) = (-Ji= 1,2 [M i I, there exists a unique solution (u, 0, a , f , C) o f (S,) satisfying: n•,
tet+2tf2 ~ , ,~(f~)))2
O • et+ztf2 s , ,8(t~)) 1+1 (~'~, 8(~'~))) 4 a • (@s_l
f • @s+x(/, 1+3 8(1)) C•[R.
1334
F. ABERGEL and C. DUPAIX
(b) Moreover, there exists a real number d . > lr/2 such that if the angle o f contact/? satisfies 0 < fl < f l . , then there exists a positive number sl > 1 such that the conclusion of (a) remains true for every s e (1, sl) satisfying s ___ / + 2. The rest o f this section is devoted to the p r o o f of theorem 4.1. 4.2. A particular solution o f the problem Let us suppose e = 0 and V = V0, V0 given, respectively, in (5) and (10). In this case, (u, O, or) = (0, 0, constant) satisfy equations (1) to (6) and thanks to (9), cr = 0. Moreover, integrating (7) with respect to x between - 1 and +1, we deduce that Co := C = Pa + V° - V 2Fr
Ma*Oocos(d)
so that (7) and (8) can be written as
fax) rr
\x/1 + fx(x)~/x
- - f ~ =
+cos(B),
vo
V_
2Fr
(11)
41 + fx(+_l) where V_ is the volume o f the fluid enclosed between the line (y = 0) and the curve 8C_. However, - 1 < cos(]~) < +1, 1/Fr > 0 and Ma*Oo < 0 and, therefore, thanks to the results of [13, 14] it is well-known that there exists a unique solution ~ e C~(I) of (11). This solution is the so-called capillary solution, and determines the interface of the fluid at rest.
Remark 2. If ¢ is the solution o f (11) for V = Vo then ¢ + V~/2 is the solution of (11) for V = V0 + V1 . Thus, we will choose Vlarge enough, so that ¢(x) > ~ > 0 for all x ~ [ - 1 , +1], where g is a positive constant. We will denote by 17 the smallest volume such that the condition above can be satisfied. We now state a first proposition. PROVOSlrlON 4.2. Suppose B e (0, ~r) given. Let V = V0 , where V0 e ~+* is given and satisfies Vo > 17. Then there exists a unique solution (Uo, 00, ao, ¢, Co) of problem (So) for (e = 0). This solution satisfies (Uo, 0o, ao) = (0, 0, 0)
I
Co = Pa + -V~F-~-:- M a * ~ cos(fl) e C (1) is the unique solution o f (11).
Proof. It remains to prove the uniqueness o f the equation (3) by 0 and integrating over ~o, the open deduce, using (4), that 0 = 0 in f~0. Moreover, taking u, integrating over ~o, and using equations (4)-(6), we
solution of problem (So). Multiplying bounded set o f Rz associated to ~, we the scalar product o f equation (2) with deduce thanks to Korn's inequality that
u --- 0 in ~ 0 .
Finally, we deduce from (9) that a(u) -= 0, which completes the p r o o f of proposition 4.2.
Stationary interfaces for Marangoni-type flow
1335
4.3. Preliminary result We establish in this section a result that will be useful in the sequel. We denote by g the capillary solution of equation (11) and we set Xl = - 1 , x2 = + 1. We introduce the two linear operators ~ and 03 defined as £p(x) . - x/1 + ~xZ(X)p(x) - Ma*OoH(p(x)),
Fr
03p(x) ..=
(p
)
~+ ~
p (x),
with
Pxx Hip) = 1 + ~
~
~
-2
-2
gxgxx {gxZ_x(1 - 3gx) gxCxxx (l + ~)2 p~ + \ ~ ¥ ~ + <1 + ~ ) V p"
These operators arise naturally in the linearization of equation (11). We are now able to deduce the following proposition. PROPOSITION 4.3. Let 1 > 0 and s > 0 be two real numbers satisfying s < l + 2. There exists a finite sequence of real numbers 0 < )-i < "'" < '~k < +oo -(ol+lj(l $(I)), such that ifMa*FrOo ~ )ti for all i : 1. . . . . K t h e n , for all (a~, a2) ~ [RE and all h ~ .-.s-~-, there exists a unique function p ~ .col+3tr ~ + ~ . , $(I)) solution of
I £p(x) = h(x) 03p(xi)
ai
vx ~ I with i = 1, 2.
Proof.
Existence of a solution. We first assume that h e @"(I) for some nonnegative integer n. Letting R(x)-
1
1 + gZx(X)' 1 XR r(x'=exp(~f_,R(t'd r(x)
p(x) =
R3/2(x),
1 Rxx q(x) = ~ r(x) --~ (x), r(x) h(x) = -h(x) Ma*OoR(x) ' 1
Ma* FrOo
O,
1336
F. A B E R G E L
a n d C. D U P A I X
the problem above can be written as a Sturm-Liouville problem
I (r(x)p,(x))x + (/up(x) - q(x))p = h(x)
(Bp(xi) = ai
(i = 1,2);
thus, using for example the results of Churchill [15, pp. 260-264] or Ince [16, Chapter IX], we deduce the existence of a unique function p ~ C"+Z(I), provided/a is not a characteristic number (P~)i~ N of the Sturm-Liouville operator, namely a n u m b e r / t for which the homogeneous system
l (r(x)px(X))x + (UP(X) - q(x))p = 0 63p(x~) = 0 (i = 1, 2) n
possesses a nontrivial solution. This yields the condition
Ma*FrOo ~
2i for all i e N with
2i = 1/bli. However, on one hand Ma*Oo and Fr are positive real numbers. Since, on the other hand, no more than a finite number of the 2i are positive numbers [15, theorem 4, p. 267], we deduce the existence of the finite sequence mentioned above. We now come back to the weighted H61derian spaces. If s > 1, which implies [s] - 1 _> 0, we have then (01+1l'1 "~s-lX-, $(I)) C C[sl-l(I), so that the existence of a solution follows. IfO < s < 1, the previous inclusion does not hold; we use the density of ek(I) in ot+l x J S - l (I, 8(I)) for all k _> [l] + 1, to prove the existence.
Regularity of the solution.
Letting
p(x)= w(x) exp ( - ~ l x R x d-1
(t) d
t)
,
K
then, w satisfies the following ordinary differential equation
(Sva0
~ Wxx(X) + (p(x)w(x)= H(x) (,. (Wx + Bi w)(xi) Ai
¥xeI withi=
1,2,
where, using the previous notations, we have set
H(x) : =
h(x)
~o(x) .-
p R3/E(x)
Bi:=
Ai :=
exp(~ f ~ -,
Rx (t) dt),
3 Rxx(X) 3 R~(x) 4 R(----~+ 16 RZ(x) '
3 Rx
-;-ff(xi)
i = 1,2,
z4/~
ai exp
~ - (t) dt -1
i = 1,2.
Stationary interfacesfor Marangoni-typeflow
1337
W e first c o n s i d e r the similar p r o b l e m with c o n s t a n t coefficient, n a m e l y
I (Sc~t)
z,~(x) + b z ( x ) = H(x)
vx • I
(z,, + Biz)(xi) = Ai
with i = 1, 2.
r°t+3(I, 8 ( I ) ) for H • ¢°l+1(I, "~-1 L e t us p r o v e t h a t z • "~s+l
z(x) = ~,zo(x) + VZl(X) + ~ 1
8(I)).
-1 (Zo(X - t) - Zl(X - t))H(t)dt,
if b _< O, ~1 = i @ otherwise, Zo(X) = e ~ the s o l u t i o n o f the linear system w h e r e o~1 = ~
Otl(),e-al
I
Otl(y e ~
with
A2 = 2 -1
--
F o r x • I, z can be w r i t t e n as
a n d z~(x) = e -~x, y a n d v b e i n g
v e ~ ) + Bl(Te -~1 + v e ~) = A 1
- v e -'~)
+ B2(Y
e ~ + v e - ~ ) = A2 - ,42,
ear(l-t) + (1 - ~ )
1+
e-"l(1-t)l g ( t ) dt.
F o r n o t a t i o n a l convenience, we write z in the f o r m
z(x) = q/(x) + where
i
x -1
ep(x, OH(t)dt,
~u(x) := yZo(X) + vzl(x), 1
4,(x, t) : = ~ 1 (Zo(X - t) - zl(x - t)). T h u s , using the fact t h a t
v p • N,
02p Ox2p oh(x, x) = O,
02p+l ax2p+l ¢(x, x) = ,~P, we d e d u c e t h a t v p • N with 2p + 1 _< [1] + 3,
02P 02P ok I x 02p Ox~ z ( x ) = 0 - ~ q / ( x ) + k f f ~ aE(p-1)-k ox~H(x) + -1 0 - ~ 4~(x' t)H(t) dt,
02p+l z(x ) - 0X 2p+102p+1~(x) + Ox2p--~
Ok H(x) + I x 0x2p+102p+I ~ O/12P_1_k oxk 4~(x, t)H(t) dt, ke~ 2 -1
w h e r e 3( 1 = {0,2 . . . . . 2p - 2] a n d 3( 2 = [1, 3 . . . . . 2p - 1].
1338
F. A B E R G E L and C. D U P A I X
Thus, using the results o f remark 2.3, we deduce that H e e~_+~(I,$(I)) implies z~, e e,t+_1(I, $(I)). Then, if s > 1, we have Izlet+-,t~,w))= Iz[= + Iz~[® + Izx~let+_l(z,zm) and H e e~_+l(I, 8(1)) implies that IHI® < +oo, so that Iz(x)l <-I~(x)l +
¢(x, t)H(t) -1
-< [~u[~ + 2[4,1®[Hl® (
-poO,
and thus, [z[® < +0o. In the same way, we deduce that [z~(x)[ < +oo. Thus, for s > 1, $(I)). On the other hand, if 0 < s < 1, we have Is] = 0, and thus Z e c°t+3(I, ~'Js+l [zle~t~](/.s(/)) = [zl.o +
[Zxl~ + Sup
~,y~1
IzAx) - zAy)l + Iz~les_l(,, w)) Ix-yl ~
x~y
However, for 0 < s < 1,
i and H e e tS--1 + l ( I , $(I)) implies Sup
x 1 -1 dl-S(t) dt < +oo,
[dl-*(x)H(x)[ < +o% so that
XEI
-1
IH(t)l dt
=
<
IH(t)l dt
-1 dl-*(t)
SuPldl_S(t)H(t) [ ( 1 - - 1d t < t~l
,1 -1 d l - S ( t )
+oo.
Thus, we deduce that [ z ( x ) [ - [~'(x)[ +
Ifx 4~(x,t)H(t)dt r -1
-< I~,1~ + I~1®
f'
In(t)[ dt
-1
<+oo
and, in the same way, we deduce that [z.[= < +oo. Finally, we have [z,(x) -
Zx(Y)l
Ix - yr ~
= <
[~u~(x) - ~,,(y) + 151 (O/Ox)~(x, t)H(t)dt Ix - yl s IqJ. (x) - q/x (y)[ + Ix - yl s +
[J~-1 ((O/Ox)4~(x, t)
[ff ( a / O x ) o ( y , t ) H ( t ) dtl fx - yl s
-
- JY--1(a/Ox)¢(y, t)H(t)dt[
(O/Ox)4~(y, t))H(t) tit[
Ix - yl s
S t a t i o n a r y interfaces for M a r a n g o n i - t y p e flow
-< 2'-sl~xxl,~ +
x
I
IH(t)l dt Sup Sup I(O/Ox)4~(x, t) - (O/Ox)4a(y, t)l
-1
Ix
1339
X,yel x#y
IX -- Y[+
tel
Y[ ~ yx n ( t )
< +oo.
Thus, H e co++t t+3 8(I)), which completes the p r o o f for the ~,F$_ l (I, 8(I)) implies that z is in Cs+l(I, regularity o f z solution of (Scs,). We now come back to the problem (Sw0. Let T be the Green's function associated to problem (So+t) and H that associated to problem (S+a0. We have for all x e / , x
z(x) =
I
T(x, t)H(t) dt,
-1
w(x) =
H(x, OH(t) dt. -1
We then write w in the form
w(x) =
S,
(rI(x, t) - T(x, t))H(t) dt + z(x).
-1
Thus, we want to prove that F(x) :=
i
x (H(X, t) - T(x, t))H(t) dt ~ tat+3tt .~+, ,., g(I)). -1
Using the properties of the Green's functions T and H, see for example [17, p. 305], and in particular that (T - H ) ( . , t) is C' everywhere and is a C z function except possibly for x = t, we deduce, proceeding as in the case of constant coefficient, that H ~ C~_+] (I, 8(1)) implies that tot+3tt 8(I)), and the p r o o f of proposition 4.3 is completed. 4.4. P r o o f o f theorem 4.1. To prove theorem 4.1, namely the well-posedness of the problem (S,) for small enough values of the parameter e, we use an implicit function theorem. Thus, we consider the problem (S,) on a domain f ~ obtained as a perturbation of the domain D O determined by g. We transform the problem ($8) on Dn to a problem on the fixed domain f~o, and we then apply an implicit function theorem to a mapping defined on a space of suitable deformations. 4.4.1. T h e s p a c e ~pt,s o f admissibleperturbations. Let t~ e PP+*. In the following, D o will stand for the open bounded set of ~2 determined by the capillary solution ff of (11). For s e ~ ÷* and _ l + 2 and (p, r/) e ~,-.s+1 t,ot+3 (I, 8(i)))2, we define the transformation 1 ~ R +* satisfying s < Mo ~ Zo ~ M~ = M o + J ( P n o + r/to), where n o and to are, respectively, the outward unit normal and unit tangent vector to Yo at M o . Z~ will stand for the set of the points M~ so obtained. (p, r/) cannot, o f course, be chosen in an arbitrary way.
1340
F. ABERGEL and C. DUPAIX
Indeed, since the container has vertical rigid walls, M o M ~ and j must satisfy a collinearity condition at x = __1, namely rl = Pgx
at x = +1.
Moreover, for simplicity, we choose tangential deformations r/such that rl(x) = p(x)gx(x)
V x ~ [.
This choice amounts to considering deformations in the j direction only. This kind o f deformation is allowed because of the existence of a representation of Eo in term of a function g o f x . For a more general case, namely for a parametric representation of Zo, the choice of the deformations is a little bit more complicated but remains possible. Indeed, we would then choose deformations in parametric forms, as graphs over Zo. Then we denote by (P~'s(I) the space of admissible perturbations. 4.4.2. P e r t u r b a t i o n s o f t h e o p e n b o u n d e d set D o. For p ~ (Pl's(I), we set
X~ -- {M~ I M~ -- Mo + Op(no + gx to) qMo 6 Eo] =
(x~,y~)
Y~
g(Xo) + ~p(Xo)X/1 + g~(xo)
¥xo ~ ( - 1 , 1) .
We then associate to p a global transformation Z = X(P) mapping f~o onto f ~ and defined as
M o ~ X(Mo),
where X(Mo) = (O(Xo,Y O ) ( y : )
/ x° + (1 - ~O(Xo,Yo) )
)
Yo . . . . \ ~ - ~ o ) tgtXo~ + Op(Xo)~/1 + ~2(Xo))
'
where tp is a smooth function such that
~0(x, y) = 0
if h 1 < y _< ~(x)
0_< ~p(x,y)_< 1 ~0(x, y) = 1
ifhz<_y
<
hi
if y < h 2 ,
where h I > h z are two positive constants. For ~ small enough, we can define the inverse transformation 2~-1, which maps ~n onto ~ o , as X -~ = X-I(P): ~
~ ~o
Mn ~ X-I(Mo),
Stationary interfaces for Marangoni-type flow
where
1341
~9(gi~,Ys)(;:)
)~-I(Mts) =
+ (1 - ~o(x~,yD)
y~g(x~)
.
g(x~) + 6p(x~),,/1 + g~(x.) Thus for 6 small enough, say 6 < 6 o, the transformations X and X -1 a r e as smooth as the function p. Moreover, if p e ~Bt'~(0, 1/6o), the ball of
X(ff) e
t+31 ( D o , $ ( D o ) ) ) 2 I,/co XFS+
1+3 (D~, 8(D~))) z. z - l ( p ) e ,,co t2"-~s+l
Definition. Let us suppose that 0 < 6 < 60. Then D6 will be called a perturbation of the open bounded set Do if there exists a function p e (~t's(0, 1/6o) such that Do = Z-I(D,). 4.4.3. Transformation of the problem on Dn to a problem on the fixed domain Do. Let D~ be a perturbation of D o. Thus, we want to find (us, 0n, a~,fn, Cn) solution of (S,) on Ds. However, D~ is a perturbation of D O, so that this problem is equivalent to finding (u, 0, a , f , C) solution of " ~'.u
in flo
=0
(u" (7)u - D ] v o ( u ) - ~.ej = 0 u" ~0
u=O
1
---7x0= Pr Re and
O0 ----~5 Oh
0
and
o n Fo
u.h
=0
o n Xo
Ma* ~ = 0
a(u)- ~. h + P. +
fx(-+ 1)
-
41 + f ) ( _ t )
I
in ~ o
0 = 0
a ( u ) • ft. { -
i
in ~ o
t u(u) .fl.[ldx
-
+_cos(B) 0
-1
d e t ( o Sdx )dy
= V,
on Zo
Ma*~(O)\
ix
_
o
on Zo
1342
F. A B E R G E L and C. D U P A I X
where • ~ and ~-1, respectively, stand for the Jacobian matrix associated to X(P) and X-l(p), and where we set: •
U =
U60)~,
• 0 = O~oz, •
0"=
0"~0)~,
• f =Aox, •
9=
5~ox,
• E(Mo) = n(X(Mo) ) = n(M~), • t(Mo) = t ( z ( g o ) )
= t(M~).
4.4.4. Using the implicit function theorem. We define the function 5: acting from ×
(~, +oo)
× ,r°t÷3(I, ~s+l
8(1)) x ~ x (es1+2 (~o, 8(~o))) 2
into e~+](I, 8(1)) x IR × ~ × (e~+2(~ o, 8(~o))) 2 × R2 as
~(e, v, p,w) - c l
(7(11)
• ~ . ~ dx
I: o
: (e, V, p, C, w) ~*
t det(~) dx dy - V 0 rio n--w
fx(-+l)
:~ cos(B)
41 + f)(___l) where ,
~(e, V, p, w) = a(u)" fi" h + Pa + -Frrf - Ma*ot(O) and (u, 0, a) is the solution of the system f7 • u = 0
in ~0
( w " fT)u - Dlv tr(u) - 2 0 j = 0
in D O
w"
fTO
u=O
p 1 R e AO = 0
in f~o
0=0
onFo
and
00 O--~=-e9
and
u'E=O
one o
00 a(u)- E- t - Ma* ~-~ = 0
on Z o.
,
Stationary interfaces for Marangoni-type flow
1343
If we set So := (t = 0, V = Vo , p = 0, C = C o , w = 0), it is clear that ~(So) = 0. The m a i n result in the course o f proving t h e o r e m 4.1, is stated in the following proposition. PROPOSITION 4.4. Let us suppose s > 0 and l > 0 given and satisfying s _< ! + 2. There exists a finite sequence of real n u m b e r s 0 < J[1 < "'" < ~K < -~-OO
such that if Ma*FrOo ;~ Ai for all i = 1, ..., K then (a) there exists an open n e i g h b o u r h o o d Wo o f So in [R x (l?, +~o) x e s1+3 + t ( / , $(I)) × R x (C S/+2 (D o, $(Do))) z such that ~Y is a ~ t function on Wo; (b) Dto, c, w)lY(So), the Fr~chet derivative o f 5: with respect to (p, C , w ) at So, is an i s o m o r p h i s m f r o m sc°t+31r+l~., v $(I)) x rR × (es/+2 ([2o, ~(~'~0)))2 onto C sI+1 - z ( I, $(I)) × [R × [R x 1+2 (~'~0, 8(~')0))) 2 X IN2. ((~s
T h e o r e m 4.1 is a straightforward consequence o f p r o p o s i t i o n 4.4. In order to p r o v e p r o p o s i t i o n 4.4, we need several auxiliary results. Therefore, we introduce the following functional spaces a'(ol+l (~')0, ~(~'~0))) 4/[R X 1= ( C s1+2 (~"~0, 8(~')0))) 2 X t'ol+Ect'~ "~s ~,'c'O' ~(~"~0)) X I,~'~s-1
Y
:=
e~i~ (~o, S(~o)) x (e~_2(~o, S(~o)))2 x e~_df~o, $(f~o)) x (e~+2(Fo, $(Fo))) 2 x e~+2(Fo, $(Fo)) x e~+2(T_,o,$(T-,o)) × e~+-~ (~o,
S(~o)) ×
e~+_~(~o,
S(~o)),
and for S = (e, V, p, C, w) e [R x (l?, +oo) x r°t+3(l, ,~,+1 $(I)) x • x the linear o p e r a t o r A = A(S) such that
(C,/ + 2 (Do, $(D0))) 2 we
A(S): X ~ Y (u, o, a) ~- A(S)(u, O, ~), where ~7 " U
(w" ~7)u w.~O
a(u)
Dlv
-
~0j
1
- ~ o
Pr Re
Ulro A(S)(u, 0, a) =
Olro U • n[~ o
o(u) • ~. i -
M.*
oo~
at/l%
define
1344
F. ABERGEL and C. DUPAIX
Then, we consider the following problem: Given F = (fi)/s= 1 ~ Y, find(u, 0, a) • X such that A(So)(U, 0, a) = F. This p r o b l e m can be f o r m u l a t e d as: Given F = (f/)/8 1 • Y, find (u, 0, a) • X such that - - -
Pr Re
A 0 = f3
in f~o
(12)
0 = f5
on Fo
(13)
on Zo
(14)
O0
On
f7
and V • u = fl
in Do
(15)
in D O
(16)
u = f4
o n F0
(17)
u" n = f6
o n Zo
(18)
on Eo.
(19)
- D i v a ( u ) = 2 0 j + f2
O0
a ( u ) " n - t -- Ma* ff~ + fs
Let f] be an open b o u n d e d set o f ~2. Let F be a part of 0 ~ with endpoints $(f]) = [,-)i {M(i)} • Let us suppose that there exist n e i g h b o u r h o o d s o f M (i) and functions gi such that, in the local coordinates system (M a), n(M
Stationary interfaces for Marangoni-type flow
1345
Y1 is the subspace of Y o f functions (gl, g2, g4, g6, g8) E e~+~(~o, 8(~0)) × (cls_2(~o, 8(~')0))) 2 )< CI+2tE ~ o, $(Fo)))2 × co/+2iv "+ ~-0, $(Zo)) × co/+1 "-'s-l~t E o, $(Zo)) satisfying the compatibility conditions: •
• g6(Mi) = g4" nc(Mi), • i f B ~ 7r/2,
l[
gl(Mi) - sin(B)
+)]
+
- n~ • 0~r g4 + cos(B)
l[
+
g6 - g+" ~
(
n~
+
(Mi)
0)]
+ cos(~-) t~. ~ r g+ + sin(B) Reg8 - 0T~ g6 + g4 "~t-£~n~:
(Mi),
• if B = 7r/2,
(-+t~.) ~ r
g4 (M3 =
(Reg8 - ~+g 6
+ g4" ~
+ )nz: (Mi),
where 8(f2) = U2= l [Mi} and where tv is chosen so that tv = - n ~ for fl = z#2. The p r o o f of lemma 4.6 is given in [7], and that of lemma 4.5 follows step by step that of lemma 4.6. From these two lemmas, we deduce the following corollary. COROLLARY 4.7. The assumptions on Fo and Zo are the same as in lemmas 4.5 and 4.6. (a) There exists s o with s o ~ (0, 1), such that, for all s e (0, So) and all F e Y satisfying the compatibility conditions of lemma 4.5 and 4.6, there exists a unique solution (u, 0, tr) e X of equations (12)-(19). (b) Moreover, i f f l < f l . , there exists sl > 1 such that for all s ~ (1, Sl) satisfying s _< l + 2 and all F e Y satisfying the compatibility conditions of lemmas 4.5 and 4.6, the conclusion of (a) remains true. Therefore, we deduce from the result above that A(So) is an isomorphism from X onto the subspace of Y of functions satisfying the compatibility conditions of lemmas 4.5 and 4.6. We now give another useful result. LEMMA 4.8. Let 6o > 0 be as in Section 4.4.2. For 0 < 5 < 50, S+ ~ A(S~) is a E 1 function from IR × (17, +oo) × (Bl.s(0, 1/5o) × [R × (C~+2(~ o, 8(~o))) z into L ( X , Y ) , space of linear continuous operators from X into Y.
Proof. The operator A is a linear function of (e, V, C, w). Thus, we have to prove that, for all fixed (e, V, C, w) in ~ × (I7, +oo) x R x (e~+z(f~o, 8(~o))) 2, the mapping .A(') = A(e, V , . , C, w): (Bt'~(O, 1/~o) --. L ( X , Y)
p ~ ~(p), is C ~.
1346
F. ABERGELand C. DUPAIX
This mapping can be written as p ~. z - l ( p )
~. d ( p ) .
Thus, we first consider the mapping p ~ x-l(p). Let (P.)n ~ N be a sequence of 63t's(0, 1/~o) converging in Cs+l t+3 (I, 8(1)) to p belonging to (Bt's(0, 1/~0). We have (x-l(P")
- x-I(P))(x'Y)
=
(
6~.(x,y)(p
o
- p.)(x,y)
)
'
with ~ . bounded in es÷l(I, t+3 $(I)) independently of n. Thus, there exists d > 0 such that [X-I(P.) - X-l(p)let++l(o6,s(n,)) <- d i p .
-
Plel;~(,,s(n),
and thus 63t's(0, 1/6o) --' ~-~,+l~r°l+3(f~,8(QD))2 p ~" x - ' ( p ) ,
is a Lipschitz continuous function. Moreover, and in the same way, it is easily seen that x - l ( p . ) - X - ~ ( p ) = L ( p . - p) + ]p. - ple~+](z,z(O)~(pn - p), where L(.) is a linear operator and where ~(p. - p) goes to 0 in (Cs+I t+3 (f2~, 8(f~)))2 as p tends to Pn in ¢ol+3/r 8(i)). Using for example the results of [5], we infer that x"S+1 \~) 63t,s(0, 1/~0 ) ~ (es+l t+3 (~')8 ' 8(~"~8)))2 p ~ Z-I(p) is ~1, We now consider the mapping 1+31 (~'~, ~(~'~,)))2 ..._}L ( X , (es+
Y)
x - l ( p ) ~ ~Zl(p). -~ depends on X - ~ ( p ) through its Jacobian matrix ~q-~ in a polynomial way. Then, the fact that the function g - l ( p ) ~ ~-1 is smooth from ~--~+ltr°t+3(f~6,$(f~)))2 into (Cs_ lt+2 (f~, $(~)))4 makes the proof of lemma 4.8 complete. We can now prove proposition 4.4. The mapping 5: can be written
S L
(Us, Os, as) L
where (Us, Os,as) is solution of A(S)(u, 0, a) = 0.
5:(S)(ns, Os, as),
Stationary interfaces for Marangoni-type flow
1347
We set Qs := (Us, Os, as). Thus, we first want to prove that there exists an open neighbour[ r $(I)) × ~ × (es1+2 ( f ~ , g(f~)))z such that the mapping hood Wo of SOin ~ × (V, +oo) x gol+3 ,~,+1~-, ~1 is of class C 1 on Wo. Let ~ := R × (17, +oo) × 6~1.s(0, 1/C~o) × R × ( e ,~+z( f ~ , 8(~)))2. Let Qso be the solution of A(So) • Qso = 0. Let 17 ~ R +*. For S ~ 'It such that IS - Solar < r/, let Qs be the solution of A ( S ) • Qs = 0. Then A(So) .
[QSo - Osl =
( A ( S ) - A ( S o ) ) . Os
so that, using lemma 4.8,
A(So)" [Qso - Qs]
=
DA(So)(S - So)" [Qs - Oso] + DA(So)(S +
Is
Sol~¢(s
-
-
-
So)" Qso
So),
where DA(So) is the derivative o f A at point So and ~(S - So) tends to zero in Y when S tends to So in '1/. Thus, since A(So) ~ L ( X , Y), S ,-. A ( S ) is a C 1 function, and (Qso - Qs) remains bounded in X , we deduce that, when S tends to So in '11, then A(So). [Qso - Qs] goes to zero in
L(X, Y). Thus, since A-I(So) remains bounded in L ( Y , X ) thanks to corollary 4.7, we deduce that Qs - Qso = IS - Solat~(S - So). Thus, Qs = Qs - A-I(So)(DA(So)( S - So)" Qso) + IS - S0[~t((S - So), where ((S - So) tends to zero in )~ when S tends to So in qL, so that Dffx(So) = 0 and there exists an open neighbourhood I41oo f So in ~ in which S ~ Qs is C 1. Using the expression of ~F(S) • Qs, it is now easily seen that the mapping Qs ~ ~F(S) • Qs is ~ 1 Taking into account the fact that (u o, 00, ao) = (0, 0, 0), we can see that the derivative of 5: at So evaluated at (p, C, ~¢), denoted by D(p. c,,o $(So)(P, C, #), has the following expression IT
.
1
~2
"no no + ~rr(P~f[ + gx + Dwf(Cv))
- Ma*(T(~-~
,, ,,41 + ~,~/~
-Oo(DpH(g)'p+
r" n o n o dx
I
o
~o
l
l p41 + ~ a x -1 V--~'
DwH(~¢)))-C
1348
F. ABERGEL and C. D U P A I X
where we have set v = Dpv(p) + D~v(@), T = Dp T(p) + Dw T(@), z = Dpr(p) + D.r(@), and where the derivative o f the m e a n curvature H with respect to p at ~, denoted by D p H ( g ) . p, is given by D p H ( g ) . p = _ 1 Pxx + ~2 + (1 + ~ ) 2 P x
\-O+-gx2-~
+ (1 + g ~ ) 2 ] P "
(v, T, r) are the solution o f the linearized p r o b l e m V •v = 0
in ~o
-Div(z) - 2Tj = 0
in f~o
1 - -
-
Pr Re
v=0
(Plso)
A T =
and
in ~o
0
T=0
onFo
v • no = 0
on E o
0T = 0 Üno
on E o OT
r-
no.
to -
Ma*~
Oto
= 0
on E o .
Thus, we are n o w going to prove that Do,,c,w)5(So), defined above, is an isomorphism f r o m e~++~(I, 8(1)) × [P x (e~+2(f~o, 8(~'~0))) 2 o n t o C~_+](I, 8(I)) × ~ x ~ × (C~ + 2(~ o, 8(~'~0))) 2 X [~2. Therefore, we n o w show that, under the assumption that, Ma*FrOo ~ 2 / f o r all i = 1 . . . . . K, tr 8(I)) x R × Ct+2/~ there exists a unique solution (p, C, w) e cot+3 --.s+lv, s ~ o, 8(~o)) 2 satisfying L l ( z , T ) + Dp£(p) + Lz(w) - C = hi
vx~I
"t" n o n o dx ~-. C 1 I
f
o
F,o
f -11
p~/1
v-w=h
÷
= c2 2
together with the condition
(px + P 1exexx + ~,zxJ +
( D w f ( W ) ) = a+l
at x = +1,
Stationary interfaces for Marangoni-type flow
1349
where (v, T, r) satisfies (Plso), and where we have set r)
=
.no " n 0
-
Ma
*T( gx
'x
--
,
1 -z Ma*~DpH(p), Dp£(p) = -Fr px/1 + gx + 1
L2(w) = ~rrDw f ( w ) + Ma*OoDwH(w). First,we remark that for all ( h i , C1, C2, h2, al, a_l) E C~_+~(I, 8(1)) x [R × R × (~+2(~'~0, 8(~'~0)))2 X [R2,
there exists a unique solution (v, T, r) satisfying the system (Plso) (v, T, r) = (0, 0, constant). Moreover, the condition Jg0 r" n o • n o dx = C1 yields z = (C1/Mes(Eo))Id, where Id stands for the identity matrix of [R4. Thus, v = 0 together with equation v - w = h 2 , gives w = - h 2 , SO that in order to prove proposition 4.4, we only need to study
Dp£(p) - C = h
f
l px~- + -1
Ip
x÷
vx ~ I
(20)
= C2
gxgxx \ l +gx/
(21)
,,
for/=
(22)
1,2
with x~ = - 1 Xz = +1.
However, assuming that p is known, then the equation (20) gives p equation (21) gives C as the implicit solution of
i
1 (Dp£)-l(h
C)(x)qI
+ gx2(X) d x =
=
(Dp£)-l(h
+
C), so that
C2.
-1
Thus, it only remains to prove, for all function h ~ "c°t+t/I ~ - 1 ~ , 8(I)) and all (al, a2) ~ ~2, the col+3tr 8(1)) satisfying existence of a unique function p ~ ,~s+~-,
x + P l ~ +g~" "~'x t i )"= a i
fori=
1,2
with Xl = - 1 X2 = + 1 but this is precisely the result given by the proposition 4.3. This ends the p r o o f of proposition 4.4.
1350
F. ABERGEL and C. DUPAIX REFERENCES
1. COWLEY S. J. & DAVIS S. H., Viscous thermocapillary convection at high Marangoni number, J. Fluid. Mech. 135, 175-188 (1983). 2. MANNEVILLE P., Structures Dissipatives, Chaos et Turbulence. Collection Al~a Saclay. 3. SCHWABE D., Surface-Tension Driven Flow in Crystal Growth Melts, Crystals Growth, Properties and Applications, Vol. 11. Springer, Berlin (1988). 4. ABERGEL F., A geometric approach to the study of stationary free surface flows for viscous liquids, Proc. R. Soc. Edinb. (1993). 5. ABERGEL F. & BONA J. L., A mathematical theory for viscous, free surface flows over a perturbed plane, Archs ration. Mech. Analysis 118, 71-93 (1992). 6. SOLONNIKOV V. A., Solvability of a problem on a plane motion of a heavy viscous incompressible capillary liquid partially filling a container, Math. USSR Izves. 14(1), 193-221 (1980). 7. SOLONNIKOV V. A., On the Stokes equations in domains with nonsmooth boundaries and on viscous incompressible flow with a free surface, in Nonlinear Partial Differential Equations and Applications. Coll~ge de France Seminar III, Paris (1980). 8. LAGUNOVA M.V., Solvability of the plane problem of thermocapiUary convection, J. Soy. Math. 45(1), 1130-1140 (1989). 9. BOUILHAC I., Ecoulement Marangoni-Rayleigh induit par un flux thermique sur la surface libre, Th~se de l'Universit6 Paris 11 (1991). 10. ALLAIN G., R61e de la tension superficielle dans la convection de B6nard, Math. Modelling numer. Analysis 24(2), 153-175 (1990). 11. MAZ'YA V. G. & PLAMENEVSKII B. A., Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points, Trans. Am. math. Soc. 123, 89-108 (1984). 12. LANDAU L. & LIFCHITZ E., Physique Th~orique. M~canique des Fluides. MIR, Moscow (1971). 13. FINN R., Equilibrium Capillary Surfaces, Grundlehren der Mathematischen Wissenschaften, Vol. 284. Springer, Berlin. 14. GERHARDT C., Boundary value problems for surfaces of prescribed mean curvature, J. Math. pures appL 58, 75-109 (1979). 15. CHURCHILL R. V., Operational Mathematics. McGraw-Hill, New York. 16. INCE E. L., Ordinary Differential Equations. Dover Publications, Inc., New York. 17. CODDINGTON E. A. & LEVINSON N., Theory of Ordinary Differential Equations. McGraw-Hill, New York.