Asymptotic study and weakly nonlinear analysis at the onset of Rayleigh-Bénard convection in Hele-Shaw cell

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Asymptotic study and weakly nonlinear analysis at the onset of Rayleigh–Bénard convection in Hele–Shaw cell

Saïd Aniss, Mohamed Souhar, and Jean Pierre Brancher

Citation: Physics of Fluids (1994-present) 7, 926 (1995); doi: 10.1063/1.868568 View online: http://dx.doi.org/10.1063/1.868568

View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/7/5?ver=pdfcov Published by the AIP Publishing

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Asymptotic study and weakly nonlinear zkdysis at the onset of Rayleigh-Bbnard convection in Hek-Shaw cell

Sa’id Aniss, Mohamed Souhar, and Jean Pierre Brancher

LEiWD CNRS URA 875.2, Avenue de la For& de Haye, BP 160, 54504 Vandoeuvre-les-Nancy C&den, France

The aim of this paper is the derivation of the Ginzburg-Landau equation [as introduced by A. C.

Newell and J. A. Whitehead, J. Fluid Mech. 3$, 279 (1969)] f rom the hydrodynamic equations for an inflnite Hele-Shaw cell. The dimensional analysis and the asymptotic study allow one to distinguish two nonlinear formulations, each one depends on the order of magnitude of the Prandtl number. The first formulation corresponds to the case Pr=O(l) or Pr@l, whereas the second corresponds to the case Pr=O( c*‘), where c*<l denotes the aspect ratio of the cell. Here a weakly nonlinear analysis is performed for the two formulations. 0 1995 American Institute of Physics.

I. INTRODUCTION

Weakly nonlinear analysis has been carried out in the thermal convection problem in a fluid layer heated from below and cooled from above in order to establish the amplitude equation by means of the multiscale technique. For the free-free case; this equation has been established by Newell and Whitehead.’ The resultant envelop equation has allowed to treat the stability of the rolls system and to obtain on the one hand, the sideband instability of Eckhaus which is related to the invariance by translation in the wave-number direction; on the other hand, the zigzag instability’ which is related to the invariance by rotation. Thg amplitude equation for the rigid.-rigid case has been established by Wesfreid et aZ.;3 and recently, Cross4 compiled the coefficients -of this type of equation. A detailed study and recent references con- cerning the weakly nonlinear analysis are given by Manne- ville.5

In this ,paper, interest is attached to a Hele-Shaw con- figuration where a viscous fluid in a narrow slot between vertical parallel plates is heated from below and cooled from above. The analogy between motion in a Hele-Shaw cell and motion in a porous medium has frequently, been used to simulate porous convection. See, for example, Elder,6 Horne and O’Sullivan,7 and Hartline and Lister.s All these authors claim that their experimental results (i.e., onset of convection) are in reasonable agreement with the theoretical predic- tions of Lapwood.’ According to his theory for a horizontal, infinitely wide, saturated porous layer between two horizontal impervious walls, convection starts at a critical Rayleigh number Ra=4$.

However, the analogy between motion in a porous medium and motion in a Hele-Shaw cell has obvious limita- tions: on the one hand, in a porous medium, disturbances of three-dimensional character are the most critical ones, while for a Hele-Shaw cell only two-dimensional disturbances may exist; the discrepancies associated with these effects have been examined by Kvernvold,” Kvernvold and Tyvan,” and Frick and Clever.” On the other hand, when. the nonlinear contributions are taken into account, the advection term v-Vv does not intervene in the Darcy-Boussinesq equations of motion in a porous medium, the only term which plays a role is v-VT; while in a Hele-Shaw cell, it is shown in this paper that the advection terms can intervene in

the motion equations when Pr=O( l *‘) -8 denotes the aspect ratio of the Hele-Shaw cell.

From a procedural point of view, one is going to refor- mulate rigorously the theoretical problem of convection in a Hele-Shaw cell. In order to do this, dimensional analysis is performed by means of a judicious choice of scales so .that a perturbation parameter involving only the aspect ratio, of the cell appears. That is; searching for a first approximation corresponding to e *2=0, from the nondimensional problem by means of the asymptotic methods. With these assumptions, two nonlinear formulations can be distinguished, each one depending on the order of magnitude of Prandtl number.

The purpose of this study is to determine the influence of Prandtl number on the amplitude equation: establishing the Ginzburg-Landau equation near the critical point corr’e- sponding to the onset of convection for the case Pr=O(l) or Prsl and for the case Pr=O(e*2). .

II. PROBiEkl kORMULA~ION

Given, a Newtonian fluid contained in a, Hele-Shaw cell: 0 <y--( E, 0 ~24 1 (Fig. 1). The directions of the coor- dinate system are shown in Fig. 1. The cavity has an infinite extent in the x direction, 1 is the height and E is the distance between the two vertical planes, the values z= 9, 1 and y = 0, 1 correspond to the positions of boundaries. It is. assumed that the fluid confined in the cell is bounded vertically by two thermally insulated planes and horizontally by two perfect heat conducting planes, having constant temperatures T1 and T2, respectively, where the lower plane is the warmer.

Tl 4 x

fiG. 1. Scheme arrangement of Hele-Shaw cell.

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If T, , pe , and pe represent the temperature, pressure and density, respectively, of the fluid at the rest state, the equilibrium equations are written as

v= 0, - VP,+ peg= 0, T,=T,-(T,-TZ) ;. (1) The study of stability of this basic state consists of superim- posing perturbations on the equilibrium variables by setting:

T=T,+T’, p=pe+pr, V=O+V. Where V is the velocity field. The components of the velocity field in the X, y, and t directions are denoted by u, u, w. Before writing the motion equations describing the evolution of the velocity field, and the derivation of the temperature from the linear conducting profile, the nondimensional variables are introduced. The ge- ometry of the cavity leads to the choice of the following change of variables:

x z U V

-p=- 1’

z*= -;

1

u*= - *

Vo’

v =I) VO

w*= w. ^T’ t ⁽²⁾

Vo’ ^T*=E, ^t”=G.

In a discussion on the choice of the characteristic reference magnitudes,r3 it is shown that

vo=;, v;=; v,, to=-, l2 tva

a PO=77

where g denotes the gravitational acceleration, v the kine- matic viscosity, a the thermal diffusivity, and p the coeffi- cient of thermal expansion.

Under these assumptions and by introducing the reduced variables into the governing equations derived from the Navier-Stokes energy equations and the Boussinesq approximation, the following equilibrium equations are obtained:

dU* &L* au* Al*

-++*

at*

-fv” âx* ----SW” âY* - âz*

aP* ^d2U”

=-- ax* + EDNA+* + -yy, aY

6% dV” &J*

----+v* --+w* ___

ax* JY* az”

-+E*~A~v*+E*’ dP” a2v *

=- ay* p’

Ebb Pr-’ aw* aw* aw* aw

-+u*

at* -+v* -

ax* a* +w* -

az*

*

= - $ + E*~A~w* + a2W*

(?y,_z+T*Y

.e2 f-g-+, *

63) where A,= d2/dx”2+ d”ldz”2 designates the Laplace operator, Ra” =PgATl e’/va the Rayleigh number of the cell, Pr

= v/a the Prandtl number and E* = ~11 the aspect ratio of the cell, Ra*=O( 1) is the only case which leads to significant solutions.r3 Applying the following boundary conditions corresponding to the “free-free” case to the system of equations (4)-(8):

w*=O, T”=O at z=O,l, (9) u*=v*=w*=() , aT* g=O at y=O,l.

Subsequently, for convenience reasons the asterisks in the reduced variables of the system (4)-(8) are omitted. It is obvious that the order of magnitude of the Prandtl number must be estimated in order to be able to drop terms which are of the order CZAR. Therefore, from the equations system (4)- (8), we can distinguish two different formulations, each one depending on the order of magnitude of the Prandtl number:

(i) Pr=O(l) or Prsl, in this case the advection terms d/dt+v.Vv can be neglected from motion equations (S)-(7) in the Hele-Shaw approximation which consists in setting E *z=O. This situation is similar to the problem of thermal convection in porous medium where V-VT is the only nonlinear contribution which exists in the Darcy-Boussinesq equations.

(ii) Pr=O(c*z), in contrast with the above case and in addition to v. V T of the energy equation, the advection terms persist in the motion equations (5) and (7). In this situation, care must be taken when using similarity between motion in a Hele-Shaw cell and motion in a porous medium.

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III. PART I: Pr=O(l) OR Prsl A. Hele-Shaw approximation

A first approximation is obtained from the system of equations (4)-(8) by setting E *2=0. The solution of this first approximation is denoted by uo, vo, wo, To, and p. . From Eq. (6), it is shown that the pressure PO is independent of y.

Also from Eq. (8) and using the adiabatic condition onto the temperature, it is shown that To is independent of y . Thus the following is obtained:

PO(W), To~wj.

After neglecting terms of the order c?*’ from (5) and (7) and integrating the derived equations, one finds

Phys. Fluids, Vol. 7, No. 5, May 1995 Aniss, Souhar, and Brancher 927

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By inserting (11) into the continuity equation (4) and integrating, v. is obtained. The solution which satisfies the boundary condition vo=O at y=O, 1 is vo=O. This leads to the differential partial equation:

(12)

C. Weakly nonlinear analysis 7. Introduction

The nonlinear system of differential partial equations (12), (15) which is associated to the boundary conditions (16) may be rewritten as

It is clear that at order E*~=O, the Eq. (12) which is derived from the motion equations and from the Boussinesq approximation cannot be coupled with the energy equation. The situation is that of a pseudosingular perturbation. Therefore, the energy equation is exploited at the order rr*’ by using the accurate development:13-15

T=T,+ l ^{*“T1 .} (13)

Inserting (13) into the energy equation (8), and keeping only the terms of the order Ebb allow to obtain

a2T1 ay2

~$!+;~(~-l)

(14) After integrating (14) with respect to y, the boundary conditions (10) are applied on T, to obtain the partial differential equation:

(15) Finally, the Hele-Shaw approximation that has just been performed leads to the system of partial differential equations (12) and (15) which is associated to the boundary conditions: w. = To = 0 at z = 0, 1 expressed as

$-J!--To=To=O at z=O,l.

B. Linear analysis

The right-hand side of the energy equation (15) is neglected and corrected with (12) to obtain a system of linear differential partial equations:

A2To-g g-TO =O.

i 1 W-3)

The critical Rayleigh and wave numbers corresponding to the onset of convection are Ra,* = 48~~, qc= r. A solution of the system (17)-(18) which satisfies the boundary conditions (16) at the critical point may be written as

pO=cos(rz)eirx, (19)

To= - 27r sin( rz)eiwx. (20)

=--- dT0

at

The weakly nonlinear analysis to be carried out is based on the multiscale technique. It is convenient to retake the same heuristical considerations on the spatial and temporal scales which intervene near the onset of convection as those used by Segelr6 and by Newell and Whitehead:’ x=SX and t = #z where S is a small parameter. The derivatives with respect to the new variables are

a a d a a a

g-t~+S-> 2x z- & +a22

a.P aZ’Z’

the margiml mode is stationary =o

(23) The solution of the nonlinear problem in the neighborhood of the critical point, corresponding to the onset of convection, is developed with respect to the parameter S by

uo(x,z,a)=Sup+a2U~2)+S3U~3)+O($)

where UO=(PO,TOA

Ra*=Ra(“)+SRao)+S2Ra(2)+O(S3)

where Ra(‘) = Ra,* = 4 8 rr’ ; W>

Up, ub2)... define the structure of the nonlinear solution by its development. More precisely, ~8’ is the solution of the linear problem at the critical point (qC= r, Ra,* = 48~~).

The solution of the linear problem (17)-(18) is particularly written in the following form:

pf’= [A(X , f)e’” +A(X,F)e-‘?rx]cos( 7rz), Tb’)= -2r[A(X 7 F)eim+A(X 3 Y)e-‘7;x]sin(?rz)(26) , A(X,Y) denotes the amplitude of the fluctuation and A(X,Y) is its conjugate complex. In this analysis, it will be necessary to eliminate the secular terms, in other words, the solvability condition must be applied (Fredholm alternative).

Then, it is necessary to determine the linear adjoint problem of the original linear problem at the critical point. If the linear operator of the original problem, expressed at the critical point (qC= ~~ Ra,* = 48rr2), is denoted by so, the adjoint operator Z. is obtained by means of the transforma- tion:

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(BkJdd.e;tes the scalar product in (L’[O, l])‘, and it is de-

Then, the orthogonality of the couple (F,G) with respect to the kernel of the linear adjoint operator will be evaluated as follows:

I 1

(Ff+Gg)dz=O;

0 (27)

so is not a self-adjoint operator. Hence, the adjoint problem is written as

b,fi,+$f.$,

A2Fo+ Jg Fo+ g==o; ⁽²⁸⁾

F. and Co obey the same boundary conditions as those of the original problem:

ri'o= dz *-Fo=O at z=O,l.

The solution of the adjoint problem (31) is po=j(z)ef*x, ( f(z)=cos(7rz), Fo=g(z)efTx, where

i(z)= & sin(%-z). (29) 2. Asymptotic derivation of the Ginzburg-Landau

equation

Taking (16) into account, the boundary conditions in the terms of the expansions (24) are written as

Tl:=-- app

dZ Tg)=O at z=O,l, i=1,2,3.

Taking (23) into account, the expansions (24) and (25) are substituted formally into the nonlinear system of equations (21), (22). After ordering according to the power of S, we obtain a sequence of systems of equations:

Order & One obtains

(30) System (30) corresponds to the linear problem discussed in Sec. III B, here it is expressed at the critical point (qc,Raz) = (rr,48v2), its solution is given by (26).

Order S2: One obtains

(31) where

a2pp F2=-2-

ax ax'

The results of the order Sallow the following:

G2=4i&

X(Ae f~n+Ae-‘WX)sin(7rz)+ $ti sin(25-z).

The solution (pb2), TL2)) of the system (31) exists if the solvability condition (27) is satisfied:

s

~[j(~)F~+~(z)G,ldz=O. (32)

It could be recalled that [f(z),g(z)] is the solution of the adjoint linear problem given by (29). Equation (32) is satisfied if Ra(‘)=O. In order to obtain Tb2) andpb2), system (31) of two equations with two variables is reduced to only one equation for Ti2):

Ja,( Tg’) = - $r’A/i sin( 27rz) 9 where -A0 is defined by

a2

(33)

c&=A2(A2)+4s-” dx2. (34)

Finally, the solution of system (31) corresponding to the order S2 is

Tf)= - SAA sin(2rz),

,f)=I 24 eim- s ^e-im

v ( ax ax 1 cos( 92-z) +$A cos(2rz).

where

a2pi2) a2phl) F3= -2 --

axax ax23

(35)

(36)

G3=--2 aTp

$Ti2) ,f,2)'$1) ---,+~(!!$-T;li) ax ax

(6)

x(f$+$!)+$!+.,~)) f$! proximation (e*‘=O). Hence, setting Pr=P” Pr*, where Pr*=0(1), the equation system (4)-(8) becomes

+( !?&-T~2~) 23

au av a+4 -+-+-g=o, ax ay The results of the order 6 and 8 allow to explicit F3 and Gs .

Keeping only the secular terms, one obtains a2A .

F3=z erwx cos( TZ)

+c.c. (c.c. is the conjugate complex),

ap ^a2U

=Pr* --++*‘A~u+T

ay

l5*2 ^dv av au

d”+u -+v -+w z

at ax ay

RaC2) a2A aA 7r3

12 rA+21r z--2= -r -A2A

aY 144 = Pr* ap

- - + E*~A~v + CZAR ay

X sin( rrz) + c.c. +no secular terms. aw aw aw aw

The solution @b’), Tb3)) of (36) exists if the solvability condition is satisfied:

x+u x+v -$w --g

ap a2W

=Pr* --++*2A2w+y

I ‘(fF3+iG3)dz=0 aY

0

from which is found

aA 2

I_- 2 f-!&it

a37 ax 24 (37)

Equation (37) is the amplitude equation of Newell and Whitehead’ derived here for the problem of Rayleigh- Binard convection in a Hele-Shaw cell and for the case Pr=O(l) or P&+1. Often, this equation is called the Ginzburg-Landau equation. The nontrivial steady solution of the amplitude equation (37) is written in the form

As=SefmX,

S=q!g-2m2)1’2. (

e*2 (

dT dT aT

z+u -+v -+w x

at ax ay 1

= SZ*~ Ra* w+ E”~A~T+ 2. d2T ay

(39)

(40)

(41)

(42)

(43) Applying the following boundary conditions which correspond to the free horizontal walls and adiabatic vertical walls to this system:

w=O, T=O when z=O,l, (44)

u=v=w=o, $=O when y=O,l. (45)

In order to search a first approximation to the equation system (39)-(43), the same procedure as in Sec. III A is used, +I, vo, wo, To, and p. denote the solution of the first approximation which consists in setting E*“=O. From Eqs.

(39), (40), and (42), the following is obtained:

The study of stability of this steady state leads to the domain of Eckhaus instability:

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IV. PART II: Pr=O(e*‘) A. Hele-Shaw approximation

au0 au0 au0 au0

(38)

--g-+uo x+vo -+wo x ay

(

eo d2Uo

=pr* --+

ax 9’ i

awe ho

%+u, ?+v, dyfwo -g-

(47)

In this situation, the velocity field takes more impor- tance, since the advection term (a/at fv-Vv) does not disap- pear from Eqs. (5) and (7) after making the Hele-Shaw ap-

aPo a2wo

=Pr* -x-l--p-+T, (4%

(7)

Prom (41), it could be shown that the pressure p. depends only on x and z. Also from (43) and satisfying the adiabatic condition on the vertical walls, it could be shown that To depends only on x and z. Thus one obtains

po(x,z) and To(x,z). (49)

The energy equation at the order .s*’ is obtained by setting

T=To+e*‘T _1. (50)

Hence, introducing (50) into (43) and keeping only the terms of the order Ebb one obtains

JTo ²

dt+UO z+wo $-Ra* wo+A2To+ G.

aY (51) The Hele-Shaw approximation that just carried out leads to the system of partial differential equations (46)-(48) and (51) which is associated to the boundary conditions,

wo=To=O at z=O,l, (52)

uo=uo=wo=O at y=O,l. (53)

The term T1 of the asymptotic expansion (50) must satisfy the adiabatic condition on the vertical walls:

%=O at y=O,l. (54)

B. Weakly ndnlinear analysis

The linear dynamic of the system of equations (46)-(48) and (51) has been discussed in Sec. III B. Here, the tieakly nonlinear analysis near the critical point (qc= rr, Raz

= 48 rra) of the marginal curve corresponding to the onset of convection will be treated directly. The solution of the nonlinear system (46)-(48) and (51) is developed as for the case of “Pr=O(l), Prsl” in Sec. III C 1:

_d

d2@ dpf’

-Tg-= ax ’

$9 AaTbr)+Ra(‘) wi’)+ G=O

aY .

Iio(x,z, 8) = sip + s”@ + s3up + 0 ( $)

where uo=(po,To), (55)

Ra*=Ra(‘)+S2Ra(‘)+O(S3)

where Ra(“)=Ra*=48m’. c (56)

According to (52) and (53), the boundary conditions con- cerning the terms of each expansion are written as

*$‘)=T$)=O at z=O,l, (57)

u~i)=u~)=*~)=O at ^y=O,l, (58)

(i=1,2,3).

The integration of (63), (64), and (62) with respect to y allows to obtain the velocity field:

apb”

us”=: (y-l) F, 2Y

UO wq

(!!$Tr’).

The system (62)-(65) corresponds. to the linear study that has been already reduced to the following system:

(1) 0

230 ;{I, = o .

( Hi

Also, the second term T, of the asymptotic expansion (50) is

developed as The solution of this system is written as

Phys. Fluids, Vol. 7, No. 5, May 1995 Aniss, Souhar, and Brancher

T1(x,z,S)= STY)+ L?~T~)+ c~~T;~)+O(@). (59) Since T, satisfies the adiabatic condition on the vertical walls, then

dTI’)

-=0 at y=O,l i=(1,2,3).

aY (60)

The same slow variables in space and time are introduced as those used in the case “Pr=O(l) or P&-l:” x= 6X and t= S2E Using the new variables, the derivatives can be written as

a a d a

-+z+s -)

ax ax sr----+

the marginal mode is skionary =o

03)

From (55), (56j, (59j, and (61), the equation system (46j, (47), (48), and (51) is developed as a sequence of systems with respect to the power S. Order S: One obtains

aup aup awp

-+ -+ -=o,

ax ay az (62)

(63)

(64)

(65)

(66)

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pr)=[A(X t Y)e’q*+A(X f ~)e-iq~]cos(rz) 7

T$f)= -2r[A(X (67)

7 s) &s+A(X,Y je-@]sin( rrz).

Order S2: One obtains dufj2)

-+ au fj2’

---I- dwi2) aup

ax ay -+ az ax=O’

c3242) Pr* 2

JY

633)

=pr*( ‘$ 1 $) +@ !$!+wp) f!$, ⁽⁶⁹⁾

A2Ti2) + Ra(‘)wh2) + d2T(,2) Jr’ 1

#Thl)

= -2 -

ax ax (71)

The integration of (69) and (70) with respect to y and the results of the order S allow to obtain

Y6 Y5 Y4 Y

G-5+6-3o app d2pF)

-

ax -72- +~(~_Tb”)]+$yCv-l)(~+~), (72)

(73) Equations (72) and (73) are introduced into the continuity equation (68) then integrated in order to obtain ub2):

dTfj2) f?2p;1)

A&,2’-,-+2 B ,

ZI$? must satisfy the adherence condition at y = 0, 1, from which a first differential partial equation is obtained:

(74)

(75) If the energy equation (71) is integrated and the

obtained:

adiabatic condition on Ti2) applied, a second differential partial equation is

The system consists of the two coupled equations (75) and (76) derived from the continuity and energy equations respectively, and is written as

The expressions of (67) allow to explicit F, and G,:

i

dA . aA

F2= -2irr z er”x- z emim cos( rrz) 1

(77) ^{_ z} ^[(Azeirx

280 Pr* +A2emim)+2AA cos(2rfz)],

%($J=( a;).

(9)

The presence of secular terms is observed. The solution (T&“,P~~)) of (77) exists if the solvability condition is satisfied. In this case, this condition is satisfied and one only has to verify that

I I

(jF2+gG2)dz=0.

0

It is recalled that [f(z),g(z)] is the solution of the linear adjoint problem which is given by (29) in Sec. III C 1. In order to obtain T&‘) and pb2), the system of two equations (77) with two variables is reduced to only one equation for Tb’). Equation (77) can be formally expressed in the form

I

<&,(Tg)) = - $r5ti sin(2vz),

A0 is defined by (34) in Sec. III C 2. Finally, one obtains Tg)= - “Alli sin(2mz).

12 (78)

However, pb”) is obtained from the second equation of the system (77) :

AA cos(2rz)

3,ir2 cos(7rz)+ ~

560 Pr*

(79)

(81)

Pr*

(82)

1\2T~3)+Ra(0) wf)+ -Ra(2)wb1)-2

$77) J2$)

--

ax ax -iiF

(83) Asimilar proceeding as that of the order 6’ allows to explicit uh3’ and wb3’ with respect to the variables Tf,l), T&‘), Ts), p$l), PC?, and pb3) after integration of (90) and (91). The expressions of ~6”) and wh3) are introduced into the continuity equation

(3)

(89) that is integrated in order to obtain, afterwards, u. . The adherence condition on u a) at y = 0, 1 allows to obtain a f&t differential partial equation where pb3) is the unknown variable.

The energy equation (92) is also integrated with respect to y. The adiabatic condition on Ti3) at y=O, 1 allows to obtain a second differential partial equation where’Tb3) is the unknown variable. The development of expressions is very long but does not pose great difficulties.r8 Finally, the following system:

230($:)=( ;;).

The results of order S and L? allow to explicit F3 and G3 : a2A

F3=z e ilrX cos( VZ) + C.C. +no secular terms,

G,=[-,,(I+$) -$+~A+2~-$$(485~~7pr~~+&)jA~2A]ei~xsin(?ri) + c.c. +no secular terms.

(84)

(10)

The solvability condition associated to the system (93) is satisfied if

I l[&+gG,]=O.

0

From which the Ginzburg-Landau equation with respect to the Prandtl number’Pr=e*’ Pr* is obtained:

i 1

^{IAl2 4}

which can be written as

(85)

@6) The steady solutions are written as follows:

A,=SeimX, where

26 --

S= q-jy&R~: 2m2r2*

The linear study of stability of these stationary solutions leads to the same Eckhaus domain, as that of the case Pr

=0(l) qr PrPl. Hence, in the neighborhood of the critical point corresponding to’ the onset of convection, the Eckhaus curve is deduce! generally from the marginal curve by knowing that the &Nature’ of the first is triple that of the

second. I.

V. SUMMARY AND DjSCUSSIBN ~

The dimensional analysis and the Hele-Shaw approximation, which have been carried dut by classical asymptotic methods, have led to distinguishing two nonlinear formulations of the original problem, and eaqh one depends on the order of magnitude, of the I%andtl number. In other words, it has been seen that the Prandtl number controls the weakly nonlinear aspect thrqugh the thermavhydrodynamical character of modes. Whereas t&e linear analysis of stability giyes the same critical parameters as those’ investigated by Frick and ‘Cl&er.‘7

If the fluid possesses a Prandtl number Pr=O(l) or P&-l, and the aspect ratio of the cell is e*=O.l; in experi- ments, this corresponds @ the case of gases whose Prandtl number is qf the order of 1 and to the case of liquids whose Prandtl number ranges from 10 to lQO0. We have seen that the advection terms v.Vv can be eliminated from motion equatio,ns ‘when we set E*‘= 0, and the only nonlinear contribution comes from the term V-VT of the energy equation.

This situation is similar to that of porous mediums in which the Darcy-Boussinesq equations of motion do not contain the advection terms v.Vv. Moreover, if the perturbations in a porous me$um are t&o dimensional, the analogy between flow in it and flow in a Hele-Shaw cell is rigorous.18

In this case, the Ginzburg-Landau equation which char- acterizes the stationary rolls is given by (37). The coefficients of such equation are independent of the Prandtl number. ‘The stability of the rolls system which corresponds to the stationary solution of such equation, have led evidently only to the instability domain of Eckhaus.

In the special case when the fluid possesses a small Prandtl number Pr=b(E*2), in addition to the nonlinear contributions V-VT of the energy equatidn, the advection terms v.Vv pkrsist in equations of motion for the Hele- Shaw approximation. In this singular case, it is illusory to talk about analogy between flow in porous medium and flow in Hele-Shaw cell. .?“herefore, the Prandtl number plays a role in the determination of the amplitude equation and in- tervenes in its coefficients. In this case, the Ginzburg- Landau equation is given by (86). When e*=O.l, this case corresptinds, in practice, to liquid metals, mercury for example whose erandtl number is Pr=0.02. The stability of the rolls system leads to the same instability domain of Eckhaus as that of the first nonlinear formulation.

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