Adv. SpaceRes.Vol. 8, No. 12,pp. (12)293—(12)304,1988 0273—1177/88$0.00+ .50Pnntedin GreatBritain. All rights reserved. Copyright© 1989 COSPAR

SURFACE TENSION-DRIVEN FLOWS INHORIZONTAL LIQUID METAL LAYERS

H. Ben Hadid,* B. Roux,* P. Laure,** P. Tison,***D. Camel*** and J. J. Favier****Institut de MécaniquedesFluides, UM34 du CNRS, 1 rue Honnorat,

F—13003Marseille, France* *Laboratoire de Mathématiques,Universitéde Nice, France* ** Laboratoire d’Etude de la Solidification, Centre d’EtudesNucléaires,

85X, F-38041 Grenoble, France

ABSTRACF

The problem of surface tension-drivenflows in horizontal liquid layers has been studiedexperimentally, and theoretically by direct numerical simulation and small perturbationanalysis. We focus our attention on situations in which the depth of the fluid (liquid tin;small Prandtl number, Pr=0.015) is small enough to ensure the predominanceof thesurface tension forces over those due to the buoyancy. The surface velocity has beenexperimentally obtained for liquid tin layer with various aspectratio (length to height) inthe range 5<A<83. The thermal gradients are ranged from 5 to 40°K/cm.In the numericalstudy, the Navier-Stokesand energy equationsare solved by an efficient finite differencetechnique. The parametersgoverning the flow behaviour in the liquid are varied todetermine their effects on thermocapillary convection: the Reynolds number 10<Re<2104and the aspectratio 2<A<25; with Pr kept constant at Pr=0.015. The linear eigenequationresulting from small spatial disturbancesof the Couette flow solution is solved using anTau-Chebyshevapproximation. A notable feature of the theoretical study is the totallydifferent end circulations. In the region near the cold wall a multicell structure is evident.This agreeswith the eigensolutionwhich is of complex type, indicating spatial periodicity.In the hot wall region the flow is acceleratedto reach the velocity value for the fully-developed Couette flow which is reached under conditions such as Re/A<20. The transitionfrom viscous to boundary layer regime occurs for a critical value (Re/A)c of nearly about200, as deduced from the numerical and experimental results.

INTRODUCFION

Liquid metal flow due to surface tension forces is relevant to many industrial processessuch as welding, casting, or crystal growth. This type of convective flow is known to occurin liquid layer systems with free surfaces (Levich /1/) and becomes a dominantmechanism under certain conditions, such as in low-gravity environments and in thinlayers. In fact, under these conditions the buoyancy forces are reduced but the surfacetension forces will still exist and play a dominant role in the fluid flow (see Villers andPlatten /2/).

The primary interest of the present work is to investigate the flows generatedby surface-tension gradients in liquid layers. The available works relevant to this subject have beendiscussedby Schwabe/3/ and Ostrach /4/. A review of thermocapillary instabilities hasbeen given by Davis /5/.

Experimentsperformed in drop towers by Ostrach and Pradhan/6/ are focused on theeffects of surface tension gradient forces which becomesdominant in reduced gravity.The velocity close to the surface is as large as the velocity generatedby buoyancy atnormal gravity. It should be noted that, in general, buoyancy and thermocapillary driven

(12)293

(12)294 H. Ben Hadid etal.

convection will occur simultaneouslyin systems with free surface, when experimentsareperformed in l-g environment. Schwabe and Scharmann/7/ carried out experimentsunder 1-g in a rectangular open quartz boat filled with NaNO3-melt. The authors found

that even in moderate temperature gradients of about 10°Kcm1, thermocapillaryconvection dominates over buoyancy convection near the free surface. The flow in thefree surface is from hot to cold wall. The highest velocity is at the free surface anddecreasesquickly towards the interior of the melt. By lowering the height, H, of arectangular cavity filled with water-n-heptanol,from H=l4mm to 3.3 mm, Villers andPlatten /2/ showed the dominanceof surface tension effects on the buoyancy effects inthin layers. They also showed that when buoyancy and surface tension forces balanceeach other (intermediate values of H) the two kinds of convection exist creating twocounter-rotating cells with an upper cell driven by surfacetension forces and a lower oneby buoyancy forces.

At present, there is a lack of experimental data such as velocity and temperaturefield inliquid metals with low Prandtl number. This is probably due to the problems encounteredin this kind of fluid such as opacity which prevent direct observationsand wettability ofthe solid substract. Numerical simulation is of the one way to obtain quantitativeinformations.

Becauseof the importance of these types of flows in material processingthere have beenrecently a large number of numerical studies. Simulations of convective flows in squarecavity driven by simultaneousbuoyancy and thermocapillary effects have been done byBergmanand Ramadhyani/8/; noticeable influence of surface tension in liquid with Pr=5have been shown. Bergman and Keller /9/ gave computational results for liquid metalalloys (Al-Sn) in rectangularcavities with aspectratios of 0.5, 1. and 2. The results showthat the liquid velocities are enhancedwhen surface tension forces act in the same senseas buoyancy forces but the flow patterns and isotherms remain qualitatively unchanged.While in the counteractingcase, there is a change in the flow structure and subsequentlyin local and overall heat transfer rates. The free surface deformations accompanyingfluidflows have been studied numerically by Strani and Piva /10/ and Cuvelier /11/ for flowsin a square cavity. An asymptotic theory has been developed by Sen and Davis /12/ andStrani and Piva /13/ for thermocapillary-drivenflow and values of the aspect ratio A—*oo.

The characteristic features of the solutions given in /13/, for a small crispation number,are that the parallel flow in the central region can be establishedif A�4. Moreover, thefollowing conditions must be fulfiled by the governing parameters:Reynolds-Marangoni

number Re�100 and Marangoni number Ma=Re Pr�”I 1 0~,so that the maximum variationbetween the endwalls is less than 10%. Ben Hadid and Roux /14/ gave the flow structureof strong thermocapillary convection in shallow cavities with flat free surface (lowcrispation number). The computationsconcern three aspect ratios, A=4, 12.5 and 25, andvarious Re number, l<Re<500. In particular the results of surface velocity yield universalvelocity distribution and a condition under which the fully-developed Couette flow wasobtained. High Marangoni number convection in a square cavity have been studied byZebib et al/iS! where they compute the flow motion for l02*z(Re)Zebib=(2 A Re)<5l04.The features of their results are that a change of the nature of the flow regime toboundary layer occurs for (Re)Zebib greater than about l0~. They also develop an

asymptotic theory for Re —~ oo which is found consistentwith their numerical results. Thelocation of the maximum of the streamfunction Wmax , of the single cell was founddependenton the value of Pr number. It is close to the cold corner for Pr<1, while for Pr>lit is near the hot wall. This is mainly due to the temperaturegradients along the uppersurfacewhich is induced by the flow. This coupling effect is higher for higher Pr.

The present study deals with thermocapillary flow in shallow cavities filled with lowPrandtl number liquid (metal). The upper boundary is a free surface. Each of the twovertical walls is maintained at a constant temperature, Tl and T2, so that a constanttemperaturedifference is imposed on the liquid zone. The horizontal walls are consideredas perfectly conducting with a linear temperatureprofile.

SurfaceTension-DrivenFlows (12)295

As soon as Tl�T2, a thermocapillary-driven flow is generateddue to the surface tensionforces resulting from the change in the surface tension, a, along the upper horizontalsurface. The flow strength is characterized by the Marangoni number defined asMa=-aa/aT Gh H2/(pvic), where v and ic are the kinematic viscosity and the thermaldiffusivity, respectively;H is the height and L the length of the cavity ; and Gh= AT/L is

the horizontal temperaturegradient, where EsT= Tl-T2

HYPOTHESISAND MATHEMATICAL MODEL

We consider a low-Prandt-numberfluid (liquid metal) in a rectangular open cavity. Theboundary conditions at the free surfaces are complex when they involve the unknowngaz-liquid shape interface (see Strani et al /10/, Cuvelier /il/ and Sen and Davis /12/).This complication is strongly reduced when the free surface is assumedflat and subjectedto a surfacetension, a, related to the temperatureby a linear law : a=a~[1-’y(T-T0)]. Thedensity p is directly related to the temperaturethrough the Boussinesqapproximation by

= P0 [1-fl (Ti-To)] where the subscript 0 refers to a referencecondition T0=(T1+T2)/2,

and fI is the thermal expansioncoefficient.

The governing system is given by the Navier-Stokes and energy equations. When thevorticity and the streamfunction are taken as dependent variables, the resultinggoverning system can be written in the general form as follows:

a2c a2c ae

+ ~~]-ZBT~ (1)

a~ a2~ a2~,= ~—~- +~+~ (2)

ae ae ae a2e a2e~+TI[u ~+ v ~]= TD [~~+ ~] (3)

v=-~ and u=~ (4)

The constantsin the equations (1)-(3) depend on the choice of the scaling factors. WhenH2 V

we use H, and Re as scaling factors for length, time and velocity, respectively,

while the temperaturescale is taken as 0ref = where ~T= Ti-T2 , the above

coefficients are the following

ZI=Re ZD=1; ZBT= 0 ; TI=Re; TD=l/Pr

The problem involves numerous parameters;the effects of only some of them have beenstudied. The numerical study deals with stationary flows in shallow cavities with A=i2.5and 25 and various Re. We considerin all cases Pr=0.015 which correspondsto liquid tinused in the experimental Bridgman system of Camel and Tison /23/ For length L=2.5 cm

and height 0.2< H<4 mm the range of Re in these experimentsis from 10 to iø~.Theparameterrange studied herein covers these experimental conditions.

ASYMPTOTIC SOLUTIONAT FINITE H AND L -~

This limiting behaviour has been studied by Yih /16/ and Levich and Krylov /1/. Anexact solution for the equations of the fluid motion induced by surface tension gradientshave been given in the general case (deforming upper surface). Following the arguments

(12)296 H. Ben Hadidetal.

of /16/, in low flow regime (e.g small Re) and in the case of finite H and infinite L (A—~oo)

h(y)and taking into account the total mass flux, j v(x) dx, as well as the equilibrium condition

at the free surface which is ~ =-~ , the expressionfor the velocity and the depth

variation along the layer are:

iaa l~Pv(x,y)= (~)x - ~ s— [2 h(y) - x] x (5)

3 ~ah(y)= H pgH2 (~)y + 1] 1/2 (6)

where H is the height of the layer at x=0. As an exampleof surfaceelevation variation, ifwe consider liquid tin with the following parameters:cavity length L=2.S cm and heightH=0.4 cm, liquid density p=7 g cm3 and coefficient of the surface tension aa/aT =-0.07

erg cm-3 K1, at y=L the departurefrom the flat surface is, from (6), of about 0.06%,

which is negligeable. In the case of flat surface (h(y)=H and ~ constant) we can simply

take the velocity in the central part solely depending on the vertical coordinate x. Theexact expressionfor v(x) is then given by v(x)= (3x-2) x/4 which is the Coutte flowsolution (see Birikh /17/). For a case of combined buoyancy and thermocapillary-drivenflow, Kirdyashkin /18/ made comparison between the analytical velocity andtemperatureprofiles with those measured experimentally in cavities with 7�A�90 andfilled with water, ethyl alcohol 4�Pr�5.5.The experimental investigationshave shown thatthe flow has a largely plane-parallel pattern and the author does not point out anyvariations of the height of the layer.

BOUNDARY CONDITIONS

- On the rigid vertical and horizontal walls, x=0, y=O and y=A ; we have u=v=0- On the upper free surface(x=1) which is assumedto be flat: u=0 and the contribution of

thermocapillary forces in the convective flow enters into the governing equation via thetangential stress boundary condition as

av ~O~ =-~ equilibrium condition (Levich /19/) (7)

- On the isothermalvertical walls : O(x,0) = A and 9(x,A) = 0-The horizontal walls are consideredperfectly conducting: O(O,y)=O(l,y)A-y

PERTURBATIONANALYSIS

The problem under considerationis studied by means of perturbationanalysis used earlyby Bey /20/. The liquid is consideredisothermal and subjected to a constant surfacestress ‘r~acting along the free surface. We search for the stationary solution of the abovegoverning equations (1) and (2) which can be presentedin term of streamfunction as

Re[NJy(V2W)~ WxCV2v)yI~V4v (8)

The boundary conditions become:- On the rigid wall (x=O, y=0 and y=A): ~,= ~i~= iji~=O- On the free surface(x=1) : ~= qc),=O and ~c,~=l

and the Couette solution becomes:‘wc =-x2(x-i)/4

Let us define 4, a small perturbationof the Couette flow ‘Vc; it is governedby the following

SurfaceTension-DrivenFlows (12)297

linearized equation

Re [4,y(Wc)xxx - (Wc)x (V24,)~]= (9)

with 4,(O,y)=O; 4’x(O,y)=O; 4,(l,y)=O; 4’xx(l’Y)°

We search for a solution of the linear equation (9) in the form 4,=f(x) e ~y with ~=~t+iowhere I.L and 0 are unknown functions of Re; ~i representingthe exponential decay of the

disturbance, and co its wavenumber in the y-direction. The following perturbationequation is then obtained

f(4)÷2?~2f(2)+?~4f=-Re~ ~2f(2) - ~~3)f] (10)

with four boundaryconditions on f(x) ; f(O)=0; ~~0) 0; f(l)=0; f(2)(l)0

Then we have an eigenvalueproblem in ?~which is solved by a Tau method (a version ofGalerkin method in which the boundary conditions are to be explicitly imposed).We usedChebyshevpolynomials as basic functions. More detailed informations about this methodand the resolution of this eigenvalue equation are available in papers of Orszag /21/ andBrenier et al /22/.

RESULTSAND DISCUSSION

Results of the eigenequation

Before commenting the feature of equation (9), we give some precisions about thedisturbances.They are consideredgrowing in the y-direction if the real part, js, is positive;negative value is associated to a disturbance growing in opposite direction. In thefollowing, only the smallest absolute values of ~ have been considered,as they give themost effective exponential decay of the disturbances.The different solution branches(with real or complex ?~)are plotted against Re in Fig.2. From this figure we note that forlow Re there are two symmetric complex solutions A.. The solution with positif ~ becomesreal at Re= 15.45. Thus, for Re�15.45 the structure of the two end regions becomesdifferent giving a difference between the two end flows. For large Reynolds number wefound that as Il.~Istrongly decreasesfor increasing Re, it is better to represent the product~t Re. For downwind disturbances(in the cold region) the eigensolutionplotted in Fig.3 iscomplex indicating that the solution is oscillatory in space. As we will show later thesetwo results are well verified by direct numerical calculations (see Fig.4 and Fig.5).

Numerical results

We investigate the case of a constantPrandtl number, Pr=0.OlS and two large aspectratios, 12.5 and 25, for various values of Re. The flow structureis given by the plot of thestreamlines.We will first consider the results obtained for aspect ratio 12.5. In Fig.4 wecan see the streamline pattern for Re ranging from 66 to 2.6 l0~ the flow is notsymmetric with respect to the vertical midplane even for low values of the Reynoldsnumber. When the value of Re is under650 one cell is formed at the right cold wall of the

aacavity. In the presentcase ( negative ~ and positive Re) the fluid motion is from the hot

wall to cold wall as the fluid at the surface moves in the direction of increasing surfacetension. As Re increasesa second cell develops and a small counter-rotatingcell rises atthe bottom horizontal wall. When Rereaches3333 a third co-rotating cell is formed in thecold region and there are two small counter-rotatingcells at the bottom. As Re is furtherincreasedthe flow structure in the right part of the cavity remains with three cells. Thereis only an increaseof the strength of the convective flow. The resulting effect on thecounter-rotatingcells is that the former increasesin size. The counter-rotatingcell (the

JASR O:12—T

(12)298 H. BenHadidetal.

outer with respect to the cold wall) continues to expandas Re increasesand reachesthehot left wall at Re about 1.3 i04 (Ma=Re Pr=200). In the hot wall a substantialchangeinthe thickness of the surface layer can be noticed.A shallow cavity, with a large aspect ratio A=25 has also been consideredin order toanswer the question of whether the general characteristic features of the flow structureobserved at A=12.5 are still maintained under these conditions. The streamline patternspresentedin Fig.S for Re number ranging from 66 to 2 1O~are consistentwith the one forA=12.5. They show that the flow is not symmetric with respect to the vertical midplaneeven for Re as low as 300. We find that most of the flow variation occurs at the colderwall where successivecells appearas Re is increased.The cells induced by the strong flowmotion at the interface for Re=2104 are five: four cells with nearly the same size and thefifth much smaller. Counter rotating-cellsare also observedat the bottom in the most leftpart of the cavity. Moreover, for increasing Re a strong motion remains confined in layerswith small depth under the free surface and one can recognize the tendency for thehorizontal extent of this layer to shorten near the hot wall (left side).

The behaviour of the surface velocity as a function of the horizontal coordinate ispresentedin Fig.6.a and Fig.6.b for A=12.5 and 25, respectively, for various Re. Fromthese curves the role of Re is more clearly displayed and three behaviours can bedistinguished.In the first part of the curves the surface velocity is acceleratedto reachthe asymptotic value of 0.25 correspondingto a fully-developed Couette flow while it isdeceleratedin the last part of the curves. An intermediateregion, where the velocity hasa constantvalue 0.25 can be observeddepending on the value of Re. This limiting value ofRe is found to strongly dependon the aspectratio A (Re<300 for A=12.5 and Re<600 forA=25). We note also that the length of the region where the velocity is acceleratedaugmentsas Re is increased.More details about the conditions of the observability of theintermediateregion can be found in a previous paper of Ben Hadid and Roux /14/.

Experimental results

Experimentshave been done by Camel and Tison /23/ with a liquid tin layer (Pr=0.015).Various aspect ratios were investigated. The dimensions of the rectangular boat are:length, L=2.5cm, width, W=lcm and layer thickness ranged from 0.3mm to 4mm. Thelateral aspectsratios (Az=width/height) are 33 and 25 for low Re (Re<lOO) and 2.5 forhigh Re (Re>7103). From the experimental data plotted in Fig.8 we can clearly observethat at low Re/A<10 the surface velocity remainsnearly constant about 0.25 while at highRe/A>102 the surface velocity varies as Re/A to the power of one-thirds. Thus, we canconclude that there is a transition from viscous flow to boundary layer flow, as Reynoldsnumber exceededa critical value. It is of interest to note that in these experimentsnooscillations have been evident which is not contradictory to the predicted conditions forthe appearanceof the instabilities calculated by Smith and Davis /24,25/. For example, atPr=O.015 and from the stability analysis calculated by these authors,the critical Reynoldsnumber, Rec at which the instability sets up is Rec=600, the perturbations then propagate

with an angle ± 770 with respect to (oy) and the critical wave length A.~ vane as

2.6 Pr1/2 and is 21 times the height H. Thus, these types of instabilities arethreedimensionaland can be observed only in very wide cavities w>>H. These conditionsare not verified in these experiments. For a review of thermocapillary instability thereader is referred to papers /5,24,25/.

CORRELATIONOFDATAS

As it was indicated in the discussionabove the value of the surface velocity dependsonthe physical and geometrical dimensionlessparametersof the fluid flows. An order ofmagnitude analysis is used for estimating the values of the surface velocity and obtainingqualitative features of the flows. The procedure follows that, according to the aboveresults, for high Re a boundary flow will occur. Thus, considering the force balance(viscous-inertia) and the equilibrium condition at the free surface one can estimate thethickness ~ of the fluid flow and the velocity at the surface

SurfaceTension-DrivenFlows (12)299

v*= yl/3 Re1/3, (ha, b)

In particular, for a cavity with aspect ratio A, the maximum surface velocity is reachednear the end wall (cold wall in this case), thus the maximun surface velocity can bewritten as follow

vmax (_~-~)-1/3 (12)

In Fig.7 the numerical surface velocity at the middle of the cavity is displayed as afunction of Re/A and for various A. The primary observation is that the surface velocityfor A=2 is below the asymptotic Couette-velocity value, 0.25, even for Re as low as 10.This result is consistent with that of Strani et al /13/ giving the condition of theobservability of the parallel flow which is A�4. Secondly the surface velocity curves aresimilar and quantitatively, they present the same behaviour independent of A. Thecurves indicate the existence of two distinct behaviour for high and low Re/A rangesrespectively.This urged us to correlateall the surfacevelocity results at the middle of thecavity. Such correlationsgiven in terms of Re/A are

for Re/A < 20 v(x=1,A/2) 0,25 (13a)

and for Re/A > 200 v(x=1,A/2) 0,95 (~~y1/3 (13b)

Thus, the above relationships for the surface velocity relate two dimensionlessparameters:physical Re, and geometrical A. Relation (13a) defines the condition of theobservability of the Couette flow and relation (l3b) gives the condition for the transitionto the boundary flow regime. The part of the curves corresponding to range 20’zRe/A<200is the transition regime which is due to the effect of the confinement. From Fig.7, we canextrapolate the critical value, (Re/A)c, which divides the two ranges,it is about 50. Whenwe use the order of magnitude analysis and by equating the relationships giving thesurface velocity in low Re value(viscousflow regime v=0.25) and high Re value (boundaryflow regime), one can easily find that (Re/A)c” 43~This result is in good agreementwiththat deduced from the numerical results.

DISCUSSIONOFTHE PREVIOUSRESULTS

The computationalresults of Strani et al /13/ gave insight on the behaviour of the surfacevelocity for insulated upper free surface and higher value of Pr number, Pr=1. In theirfigures 10, 16 and 18, the horizontal surfacevelocity distribution is displayed for variousRe, one can see that the surface velocity decreasesin the central region and increasesnear the cold wall. At high values of Re (Re5trani=A Re=3000) a secondmaximum appears

in the vicinity of the hot wall. The former maximum is not found in our previouscomputations with upper conducting free surface and Pr=O.015 (Ben Hadid and Roux/14/). This trend of the surface velocity is owing to the local decreasesof the surfacetemperaturegradients (insulated upper surface) enhencedby the effect of the relativelyhigh value of Pr. The two maximum appearing in the horizontal surface velocitydistribution are consistent with surface temperatureprofiles displayed in figures 24.a,cand d where the high thermal gradients are located in the corner with the highest at thecold one. In this region the larger temperaturegradient generatesa local increasein thedriving force leading to local increase in the surface velocity. Note that, the samequalitative behaviour can be observedin the paper of Zebib et al /15/. In fact, one cansee that in their Figure 7, a second maximum arises for relatively low value of Re=20 andbecamemore important as Re increases.The computation was made for a value of Prandtlnumber equal to 50. These results give insights on the significant interaction of fluidflows and thermal fields in such configuration.

(12)300 H. Ben Hadid et a!.

CONcLUSIONThe numerical results presentedherein have shown the effect of the Re number on theflow structure in large cavity. Some insights have bee given by perturbationmethod andconfirmed by direct numerical simulation. The main features of these studies are thatfrom direct simulation the flow structure exibits first, three distinct regions two endwallregions; and an intermidiate one (where the velocity has Couette velocity profile). Thislater disappears for increasing Re leading to the coalescenceof the two end-wallstructure. A two end flow structure is evident in a cellular flow in the cold wall regionand a thin layer with large velocity close to the free surface. In the remaining region acounter-rotating flow prevails for large Re values. Some of the described trend of thesurface velocity are in good agreementwith the experimental results achieved withprevailing surface driven flow. The qualitative agreementmay be observed but the slightdiscrepancycan be explained by considering the relevant effect of the strong confinementas the higher values of Re are obtained for the smallestaspect ratio A=5. In the range ofinvestigatedRe, three flow regimes were evident by the two numerical and experimentalmethods. These regimesare divided by the magnitude of Re/A. A critical (Re/A)c for thetransition to boundary layer regime is found to be of about nearly 200. While in theparallel flow, viscous regime prevails up to the limiting value of Re/A<20. The range ofRe/A between the two values is identified as the transition regime. This study gives someinsight on thermocapillary-driven flows and more accurate investigations are neededwith quantitative and qualitative experimental data for further theoretical andexperimental comparisons. Free surface deformation as well as the combined buoyancyand thermocapillary-driven flow needs more detailed investigations as it is shown, fromthe literature, to have a large importance in determining the nature of the convectiveflow.

Acknowledgments

The computationswere carried out on the cray2 of the C.C.V.R using IBM facilities ofC.N.U.S.C and P.A.Saint-Charles.Researchwas supported by C.N.E.S (Division MicrogravitdFondamentaleet Appliqude).REFREN~ES

1. V. G. Levich and V.S. Krylov, Surface-Tension-DrivenPhenomena,Ann. Rev. Fluid Mech 1, 293-316(1969)

2. D. Villers and J. K. Platten, Separationof Marangoni convection from gravitationalconvection in earth experiments.,PCH Physico Chemical Hydrodynamics,8,173-l83(1987)

3. D. Schwabe,Marangonieffects in crystal growth melts., PCH Physico ChemicalHydrodynamics, 4, 263-280 (1981)

4. S. Ostrach, Low gravity fluid flows, Ann Rev. Fluid Mech. 14, 313 (1982)

5. S.H. Davis, Thermocapillary instabilities, Ann. Rev. Fluid Mech 19, 403-435 (1987)

6. S. Ostrach and A. Pradhan,Surface tension inducedconvection at reducedgravity.,AAIAJ 16, 419-424 (1978)

7. D. Schwabe and A. Scharmann,The magnitudeof thermocapillary convection in largermelt volumes, Adv. SnaceRes, 1, 13-16 (1981)

8. T.L. Bergman and S. Ramadhyani, combined buoyancy-andthermocapillary-drivenconvection in open squre cavities, Numerical. Heat Transfer.,9, 441-451 (1986)

9. T.L. Bergmanand J.R. Keller, combined buoyancy-surfacetension flow in liquidmetals, Num. Heat Trans. 13, 49-63 (1988)

SurfaceTension-DrivenFlows (12)301

10. M. Strani and R. Piva, Surface tension driven flows in micro-gravity conditions,Int. J. for numercal Methods in Fluids, 2, 367-386 (1982)

11. C. Cuvelier and J.M. Driessen,Thermocapillaryfree boundariesin crystal growth,J. Fluid Mech. 169, 1-26 (1986)

12. A.K. Sen and S.H. Davis, Steadythermocapillaryflows in two-dimensionalslots.J. Fluid Mech. 121, 163 (1982)

13. M. Strani, R. Piva and Graziani. G, termocapillary convection in a rectangularcavity:asymptotic theory and numerical simulation, J. Fluid Mech. 130, 347-376 (1983)

14. H. Ben Hadid and B. Roux, Thermocapillaryconvection in long horizontal layersoflow-Prandtl number melts ubject to an horizontal temperaturegradient,J. Fluid Mech.(to be published).

15. A. Zebib, G.M. Homsy andE. Meiburg, Phyics of Fluids, High Marangoninumberconvection in square cavity28, 3467-3476 (1985)

16. C.S. Yih, Fluid motion induced by surface-tensionvariation, The Physics of Fluids,3, 477-480 (1968)

17. R.V. Birikh, Thermocapillary convection in horizontal layer of liquid, J. Appl. Mech

.

Tech. Phys,7, 43 (1966)

18. A.G. Kirdyashkin, Thermogravitationaland thermocapillary flows in a horizontalliquid layer under the conductions of a horizontal temperaturegradient,Int. J. Heat Mass Transfer, 27, 1205-1218 (1984)

19. V.0. Levich, Motion inducedby capillarity. In PhysicochemicalHydrodynamics, p372. PrenticeHall, EnglewoodCliffs, NJ (1962)

20. J.A.T. Bey, Numerical solutions of the steady-statevorticity equation in rectangularbasins, J. fluid Mech, 26, part. 3, 577-598 (1966)

21. S.A. Orszag ,Accurate solution of Orr-Sommerfeld stability equation,J. Fluid Mech. 50, 166-208 (1971)

22. Brenier, B. Roux and P. Bontoux., Comparaisondes méthodesTau-Galerkindansl’étude de stabilitd des mouvementsde convection naturelle, Probldme des valeurspropre parasites,J. Méc. Thdo. Appli., 5, 95-119 (1986)

23. D. Camel, P. Tison and J.J. Favier, Marangoniflow regimesin liquid metals,~ta Astronautica,13, 723-726 (1986)

24. M.K. Smith and S.H. Davis, Instabilities of dynamic thermocapillaryliquid layers.Parti. Convective instabilities J. Fluid Mech~132, 119 (1983)

25. M.K. Smith and S.H. Davis, Instabilities of dynamic thermocapillaryliquid layers.Part2. Surface wave instabilities J. Fluid Mech. 132, 145 (1983)

(12)302 H. BenHadideta!.

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H ___________________________________

T=Ti T=T2Ic

00 L

Fiq.1 , Two-dimensional model30

0uS0

_l 0< 20

> 0zuJ0______ _______ no _______ _______ _______w is

W DODI

a a a on o nanmnIL &a Ao 0 ~

•••~ •.. .• UU. U U U UU •UU•— • U

- af~’~ a A £ } Complex elgenvaluesaw a~a £ } Real eigenvalues

aa.20

.30

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Fig.2200

________________ aaa a A A AAAA AAL&A a a a a a a a aa) A

£

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A eigensolutiofla

z aw

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0iol io2 io~ Re IO~

Fig. 3

SurfaceTension-DrivenFlows (12)303

(a) ~‘ :: ____________________________

~

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Fig.5

(12)304 H. Ben Hadid etal.

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