INTERFACIAL TURBULENCE IN EVAPORATING LIQUIDS : THEORY AND PRELIMINARY RESULTS OF THE

ITEL-MASER 9 SOUNDING ROCKET EXPERIMENT

P. Colinet1, L. Joannes1, C.S. Iorio1, B. Haut2, M. Bestehorn3, G. Lebon4 and J.C. Legros1

1 Université Libre de Bruxelles, Service de Chimie Physique E.P., CP 165/62, 50 av. F.D. Roosevelt, 1050 Bruxelles, Belgium.

2 Université Libre de Bruxelles, Service de Génie Chimique, CP 165/67, 50 av. F.D. Roosevelt, 1050 Bruxelles, Belgium.

3 Brandenburgische Technische Universität Cottbus, Lehrstuhl für Theoretische Physik, Erich-Weinert-Straβe 1, 03046 Cottbus, Germany.

4 Université de Liège, Institut d’astrophysique et de géophysique, Building B5, Sart-Tilman, B-4000 Liège, Belgium.

ABSTRACT Evaporation of a pure liquid into a inert gas is studied theoretically and experimentally. In contrast with the case where the gas phase is made of pure vapor, the thermocapillary (Marangoni) effect strongly destabilizes the system, and results in intensive and often chaotic forms of interfacial convection. Theoretically, a generalized one-sided model is proposed, which allows to solve thermo-hydrodynamic equations in the liquid phase only, still taking into account relevant effects in the gas phase. The equivalent heat transfer coefficient (Biot number) to be incorporated in this one-sided model appears to be high, which results in an acceleration of transitions to polygonal chaotic patterns. Chaotic interfacial patterns driven by the Marangoni effect have indeed been observed during the ITEL-Maser 9 sounding rocket experiment flown in March 2002, in preparation of the CIMEX (Convection and Interfacial Mass Exchange) experiment foreseen for the International Space Station. INTRODUCTION

Evaporative convection is important in a number of applications (for reviews, see Berg et al., 1966; Molenkamp, 1998), such as drying of paint films, thin-film evaporators, spray drying, … Most often, the liquid evaporates into an inert gas (such as air), and there is some limitation of the evaporation rate arising from diffusion of the vapour through the latter. However, as explained in this paper, the presence of an inert gas also strongly favours surface-tension-driven instabilities in the liquid, which can enhance the heat transfer through the liquid phase, and hence the evaporation rate. The question therefore is to determine what will be the net effect of the inert gas, and as a preliminary step, we will estimate hereafter its effect on the surface-tension-driven instability thresholds.

Most of the classical papers on evaporative convection driven by the Marangoni effect assume that the gas phase is pure (see e.g. Palmer, 1976; Prosperetti and Plesset, 1984; Burelbach et al., 1988), and focus on other destabilising effects such as departure of the interface from chemical potential equilibrium (i.e. from

the Clausius-Clapeyron coexistence boundary), surface deformability and vapour recoil (important at very low pressure only). While these two last effects are neglected a priori in our analysis (not too thin layers and moderate evaporation rates), the first one is incorporated, but appears to remain negligible in the case where the gas phase contains a second component. Note that a recent paper (Ha and Lai, 1998) also considers a second component in the gas, but the liquid and gas phases are assumed semi-infinite, which is justified at the beginning of the evaporation process only (penetration theory).

Although when a liquid evaporates into an inert gas, motions of the latter at some distance of the interface can be important (e.g. in spray drying and thin-film evaporators, there is a relative velocity between the evaporating liquid and the ambient gas, and in the ITEL experiment described hereafter a flow of inert gas is imposed along the surface), the effect of this relative flow will be neglected as a first approximation, in order to address the importance of other effects on the development of surface-tension-driven instabilities. Although a relative velocity of the gas with respect to the liquid induces boundary layers (viscous, thermal and concentrational), these are roughly modelled here, similarly to the classical stagnant film theory (Bird et al., 1960), by assuming that the gas phase is perfectly mixed at some distance from the interface. THEORETICAL MODELING

The system considered is sketched in Figure 1. It consists in a layer of pure volatile liquid, in contact

with a stagnant gas layer. Note that because of lack of space, we will only outline here the main steps and hypotheses of the mathematical description, and present some of the essential results. The reader is referred to Colinet et al. (2002), and Haut and Colinet (2003) for details of the theoretical analysis.

Fig. 1. Geometry of the problem and basic assumptions

In our theoretical description (Colinet et al., 2002), we assumed that the Boussinesq approximation is

valid and that the gas may be taken as perfect and incompressible. Buoyancy effects have been neglected (not too large depths or micro-gravity conditions). Note that a priori, as the gas phase is a non-isothermal mixture, the Soret effect could be important, especially in the mass-diffusion-limited regime (see below). This effect has therefore also been included, but shown to remain small for most practical purposes.

Depending on the operating conditions (initial temperatures, pressures, concentration of vapour in gas phase), the system may either be at equilibrium, or undergo evaporation or condensation, starting in the immediate neighbourhood of the interface. For short times, it can therefore be expected that particular boundary conditions at the bottom rigid plate and at the top boundary are unimportant. A self-similar solution (penetration theory) then applies, as shown by Ha and Lai (1998). Then, for long enough times, the temperature and concentration profiles in both phases reach the top and bottom boundaries, and accordingly their influence cannot be neglected anymore. The regime reached in this limit is assumed to be quasi-stationary, as discussed below.

Our theory (Colinet et al., 2002; Haut and Colinet, 2003) is based on mass conservation for incompressible fluids (solenoidal velocity fields), Navier-Stokes equations, and energy conservation

(neglecting viscous heating effects) in both liquid and gas phases. In addition, the conservation of the mass of volatile component must also be expressed in the gas. These seven balance equations must be complemented by boundary conditions at the presumably flat interface )(thz = , expressing : i) the temperature continuity, ii) the energy conservation including latent heat consumption, iii) the no-slip condition, iv) the tangential stress balance including the Marangoni effect, v) the total mass conservation, vi) the species conservation, vii) a kinetic condition (Hertz-Knudsen non-equilibrium law) expressing that the evaporative mass flux is proportional to the difference between the saturation pressure (given by the Clausius-Clapeyron curve) and the actual partial pressure of vapor in the gas, assumed to be a perfect mixture. Finally, at the bottom plate z=0, the no-slip condition applies and the temperature is fixed at 0ll TT = , while at the upper boundary

)()( tthz δ+= , the horizontal velocity is assumed to vanish, the total pressure is fixed at pg0, as well as the temperature (fixed at 0gg TT = ) and the vapor mass fraction ( 0gg NN = ).

Note that our definition of the Marangoni number is

plll c

hLTMa

κµσ 0)/( ∂∂−= (1)

where )/( T∂∂σ is the variation of surface tension with temperature (usually negative), L is the latent heat,

0h is the initial depth of liquid, lµ is the liquid dynamic viscosity, lκ is the liquid thermal diffusivity, and

plc is the specific heat of the liquid. Accordingly, Ma has a fixed value for a given fluid, provided the initial height 0h is given. Quasi-Steady Reference Solution

A quasi-steady state can be determined when there is no horizontal flow, and when thermal and concentration fields only depend on the vertical coordinate z. In this quasi-steady regime, we can set to zero all the time-derivatives in transport equations, which can then be solved as a function of the slowly varying liquid thickness h(t). Still, it is necessary to determine the evaporation flux )(hJJ = ( dtdh /−= in our choice of units) numerically, because transport effects in the gas phase generate some exponential terms in the temperature and concentration profiles. However, except for very high evaporation rates, the latter transport effects are small, and the profiles are actually quasi-linear (assumption of small Peclet numbers in the gas). In this case, a satisfactory approximation of the dimensionless evaporation rate (reduced by

0/ hllκρ , where lρ is the liquid volumic mass) in the reference state is given by

)('

)(

0

0

TphQrLe

TpJ

s

sref

++=

ρκδ (2)

which is restricted, for clarity, to the case 000 TTT gl == , 00 =gN , and for negligible thermal conduction and Soret effect in the gas. The latter assumption is valid for the conditions of interest here, but might have to be relaxed, e.g. when the thermal gradient is large in the gas. In the expression (2), )( 0Tps and )(' 0Tp s are the saturation pressure and its temperature-derivative (at temperature T0), respectively reduced by the total gas pressure pg0 and by pg0 cpl/L, Le is the gas Lewis number (ratio of mass and heat diffusivities), iv MMr /= is the ratio of molecular masses of vapor and inert gas, and κρ, are the gas-to-liquid ratios of densities and thermal diffusivities, respectively. Finally, the dimensionless number Q quantifies the constitutive resistance to evaporation : both limiting cases of local equilibrium ( 0→Q ) and non-evaporative interface ( ∞→Q ) are covered by this expression. Actually, Q is inversely proportional to the accommodation coefficient (of the Hertz-Knudsen law), and turns out to be small in practice.

The expression (2) allows a clear distinction between the possible evaporation regimes. The three terms at the denominator represent respectively the three possible limitating mechanisms : mass diffusion in the gas (remember that δ is the gas-to-liquid ratio of thicknesses, and the Lewis number Le is proportional to the isothermal diffusion coefficient in the gas), interfacial resistance to evaporation quantified by Q, and

thermal diffusion in the liquid (needed to compensate for the latent heat consumption at the interface). Note that in general, the interfacial resistance Q is negligible compared to the effects of heat/mass bulk diffusion, apart when the accommodation coefficient is very small (e.g. impurities on the interface).

Fig. 2. Effects of a variation of temperature (left) and depth of the gas phase (right) on the evaporation rate for ethanol in contact with nitrogen at 0,5,1 00 === gg Nmmhatmp . The liquid depth h is scaled by h0. In each case, the exact solution (full curves) is compared to the approximate analytical solution (dashed curves) for small evaporation rates (small thermal and concentrational Peclet numbers in the gas). Linear Stability Analysis

For conciseness, we do not either reproduce here the details of the linear stability analysis of the quasi-steady reference state (Colinet et al., 2002). After superposing perturbations of velocity, temperature, pressure and mass fraction to the reference state, the equations and boundary conditions are linearized with respect to the perturbations, i.e. assuming the latter to be infinitesimal. Then, each perturbation is Fourier-decomposed, i.e. written in terms of normal modes proportional to [ ]tikx σ+exp , where k is the wavenumber

and σ the growth rate. This leads to an eigenvalue problem for the growth rate σ of the perturbations, which can in principle be determined as a function of the Marangoni number Ma (and of k). In order to determine the threshold of stability of the quasi-steady reference state (assumed frozen in time), we rather directly set σ=0, and solve the complete eigenvalue problem for the critical value of Ma above which instability occurs. Remember that the actual value of the Marangoni number is fixed for a given liquid (and depth), so that we can directly decide whether the reference state is stable or unstable by comparing the value (1) to the critical value determined from the eigenvalue problem.

The stability of the full two-phase problem has also been compared to the stability determined in the frame of the assumption of small Peclet number (high diffusivities) in the gas. In this case, provided the depth of the gas is not too large, one may consider a simplified version of the full problem, which consists in slaving adiabatically the perturbations in the gas phase to the dynamics of perturbations in the liquid. Actually, the velocity field in the gas becomes unimportant (assuming that the dynamic viscosity of the gas is negligible compared to that of the liquid) and doesn’t need to be solved for, while heat and species balance equations reduce to Laplace equations 0=∆=∆ gg NT , whose solution can in principle be found as a function of the liquid quantities, using boundary conditions at the interface and on the top boundary. This leads to a generalized « one-sided » problem (Burelbach et al., 1988; Colinet et al., 2001a; Margerit et al., 2001; Colinet et al., 2002; Haut and Colinet, 2003), allowing to account for the effect of evaporation via a generalized heat transfer boundary condition (effective interfacial heat transfer coefficient). It is worth emphasizing already that both the exact two-layer approach and the one-sided model agree very well for small enough J, i.e. when Peclet numbers are small enough.

Denoting by λ the gas-to-liquid ratio of thermal conductivities, we find that the dimensionless heat transfer coefficient at the interface, i.e. the Biot number, is given by

)()coth()( kBikkkBi Ev+= δλ (3) which is found to be dependent on the wavenumber k of the perturbation. Our expression (3) therefore represents a non-local generalisation of other expressions found in the literature (Colinet et al. 2001a; Margerit et al., 2001), and incorporates the effect of evaporation through the second term )(kBiEv , given by

)coth(

1

))1((

)(')(

,

2,

0

δκρ kkLeJ

N

Nrr

rQ

TpkBi

ref

refg

refg

sEv

+−

×−+

+= (4)

where Ng,ref and Jref are respectively the mass fraction near the interface and the evaporation flux in the reference state, and other symbols have already been defined. Note that the contribution (4) is found to be much larger, in general, than the first term in Eq. (3), which quantifies the effect of heat conduction in the gas (Colinet et al., 2001a). The effective Biot number )(kBiEv is represented in Figure 3 for the ethanol-

nitrogen system, together with the corresponding neutral stability curves Ma(k) found by solving the one-sided eigenvalue problem. Note that the latter has the structure of the Pearson’s problem (Pearson, 1958), whose solution may then straightforwardly be used, leading to

( )( ))3sinh()sinh(3)cosh(4

)2sinh(2)sinh()()cosh(16)(

3

1

kkkk

kkkkBikkkkMa

−+−+= −β (5)

where β is the temperature gradient in the liquid [in units of L/(cpl h0)], generated by evaporation. In Figure 3, we also compare the critical values of the Marangoni number Ma to those obtained from

the exact full two-layer problem. This allows to conclude that for the conditions considered here, the one-sided model turns out to be a very satisfactory approximation of the full stability problem. Note that it has also been checked that surface deformation can indeed be neglected in the conditions considered here.

Fig. 3. Comparison between exact stability results (full curves) and results of the generalized one-sided model (dashed curves), in the case of ethanol evaporating in nitrogen (initial depth mmh 50 = ), for various experimental conditions (see legend). In the right plot, the supercriticality cMaMa /=ε (where cMa is the critical value of the Marangoni number, i.e. the minimal value of Ma in the left plot) is plotted against the depth ratio δ of gas and liquid, and shows that the system is strongly unstable on a wide range of δ.

It may thus be expected from the above analysis that for a 5mm-deep ethanol layer, whatever the effective value of the stagnant gas thickness (except if it is much larger or much smaller than the liquid

depth) the supercriticality cMaMa /=ε will generally be much higher than unity, even in the absence of buoyancy.

In fact, while the Biot number is generally found to be of order unity (in the neighbourhood of the critical wavenumber minimizing the curve of Ma) in the presence of an inert gas, it is much larger when the gas phase is pure (Colinet et al., 2001a), as also seen for 1, →refgN in Eq. (4). As the critical Marangoni

number increases proportionally to Bi when the latter is large, this shows that the case of a pure vapor phase is much more stable than the case considered here. Actually, this is clearly due to the fact that in a one-component system, the interface temperature is bound to remain close to the boiling temperature at the given pressure of the gas (apart for some small pressure fluctuations and non-equilibrium effects quantified by Q). A second component in the gas allows for partial pressure fluctuations, hence for larger deviations with respect to the saturation temperature and corresponding Marangoni stresses along the interface. Nonlinear Regimes

Direct numerical simulations of the 3D nonlinear balance equations and boundary conditions has been performed for the one-sided model, at varying Biot number (Merkt and Bestehorn, 2003). It has been found that for increasing Biot numbers, the transition between hexagonal structures and squares occurs at lower supercriticality ε, and that a transition to chaotic polygonal patterns takes place when further increasing ε. A snapshot of a time-dependent regime is presented in Figure 4 (left), together with a pattern obtained (Colinet et al., 2001a) from a model of thermocapillary convection at infinite Prandtl number Pr, and very high ε (right).

Fig. 4. Snapshots of free surface temperature fields. Left: Direct 3D numerics (Bi=5, Pr=6, ε=3), Right: 2D model (Bi=0, Pr>>ε>>1). Both polygonal patterns are chaotic, and convection cells are limited by sharp and straight ripples, colder than their center.

It is not our purpose here to discuss these models in details, but merely to illustrate that in both cases,

the polygonal cells are delimitated by sharp (colder) regions named thermal ripples in the following, and that the latter evolve chaotically, by a continuous process of creation of new cells (cell splitting) and destruction of smaller cells (cell collapse). Although a qualitative agreement exists with some of the experimental results we now present, further modeling will be needed to fully explain the latter. EXPERIMENTAL RESULTS

A preliminary step in the preparation of the CIMEX-1 (Convection and Interfacial Mass Exchange) experiment foreseen onboard the International Space Station, has been the ITEL (Interfacial Turbulence in Evaporating Liquids) experiment, flown onboard the MASER-9 sounding rocket in March 2002. During the 6 min micro-gravity phase, ethyl alcohol (5 mm deep) has been evaporated in a closed cell, using both

reflection-Schlieren and 3D temperature field reconstruction via optical tomography. The latter diagnostic uses interferometry in six directions parallel to the liquid/gas interface, to measure integrated optical paths along different directions. The six projections are then grabbed and treated numerically via some reconstruction algorithm, to determine the 3D temperature field within the liquid layer. More details about the set-up implemented in the ITEL flight module, in collaboration with SSC (Swedish Space Corporation), can be found in Joannes et al. (2000) and Colinet et al. (2001b). The experiment principle and diagnostics are sketched in Figure 5. The liquid filling procedure, successfully performed in micro-gravity, is described in Figure 6.

Schlieren

Interferometry (tomography)

Fig. 5. Main functionalities of the ITEL experimental cell (left) and picture of the complete ITEL module (right) flown onboard the MASER-9 sounding rocket. The ITEL flight module was built by SSC (Swedish Space Corporation), while the optical diagnostics (Schlieren using reflection on bottom mirror, and 3D re-construction of the liquid temperature field by optical tomography) were designed by Lambda-X.

0 s

3.2 s

4.8 s

6.4 s Fig. 6. Filling of the cell in micro-gravity. Left : sketch of donut-shape liquid zone during filling. Right: 3 of the 6 tomographic views at 4 different moments, and relative time from first image. Ethyl alcohol first fills the groove between aluminum and glass, then wets the aluminum plate, the circular glass walls, and the stainless steel foil, up to an anti-wetting barrier painted on its top part. As liquid is further injected, the central part fills, and the free surface finally reaches a quasi-flat position. At this moment, liquid is injected more slowly, and the injection motor is controlled by an automatic algorithm based on Schlieren images.

Note that due to the short micro-gravity time (6 min) of MASER, only transient effects can be investigated (ripple formation). It indeed appeared that the development of thermal ripples occurs on a sufficiently fast time scale (see Figure 7), while the overall organization of the pattern typically takes a much longer time (the vertical diffusion time is of the order of min4.4/2

0 =lh κ ), and will be studied in the future CIMEX-1 experiment (which will also investigate other liquids, and allow to obtain maps of regimes as a function of pressure, temperature, gas flow rate, and liquid depth).

Unfortunately, even though the ITEL module behaved perfectly during all mission simulation tests, and even during countdown, a software parameter used to control the pressure within the experimental cell changed just before launch, for a currently unidentified reason. This off-nominal pressure control was responsible for an overpressure in the cell during most of the flight, which greatly reduced the evaporation rate, and hence the interfacial turbulence. Despite of this failure, the post-flight analysis of the results shows that all other functionalities of the module worked quite satisfactorily (such as optical tomography, Schlieren, automatic surface flatness control, thermal regulation, …), and allowed to clearly demonstrate the occurrence of interfacial turbulence in micro-gravity.

Indeed, after several attempts to control the experiment run by manual tele-commands, it has been possible, just before the end of the micro-gravity phase, to maintain the aperture of the outlet valve, allowing the nitrogen to flow over the liquid surface at about 600 ml/min, and the pressure to decrease down to about 620 mbar within the cell (note that the temperature of the cell walls was about 25°C during all the flight). In these conditions, nice surface-tension-driven ripples were observed for about 30 s (see Figure 7), even though the pressure and flow rates could not be maintained completely steady during that period.

1

2

3

4

5

6 7 8 9 10

Fig. 7. Ten Schlieren snapshots of the 15 mm diameter evaporation zone (limited by a thin stainless steel foil covering the free surface of the liquid). Thermal ripples are continuously created, destroyed, and advected by a large-scale flow spanning the whole cell. The time step between two images is 0.2 s.

Note that as the bottom aluminium plate acts as a mirror, both surface deformations and refractive index gradients within the liquid contribute to the Schlieren image. The latter contribution is dominant in general, except for larger surface deformations due to under-filling or over-filling of the cell. In fact, the Schlieren view was also grabbed on-board, analyzed in real-time via a dedicated algorithm, and used to control the liquid injection system such as to correct for the evaporation rate, which allowed to maintain the free surface flat (up to about 50 µm) during the whole flight, from the moment of liquid injection in the cell.

As also predicted by our models of thermocapillary convection at high Marangoni number described in the previous section, the thermal ripples observed in the experiment are indeed quite localized and appear to sharpen as the evaporation rate (i.e. the Marangoni number) increases. Previous ground experiments (Colinet et al., 2001b) had also shown that the ripples indeed appear at the zones of convergence of the free surface flow, i.e. at the cold zones of the free surface. There is also a nice agreement as far as some dynamical processes are concerned, namely polygonal convection cells limited by cold ripples occasionally split into smaller cells, while other smaller cells disappear, being compressed by larger ones. However, the agreement remains only qualitative for several reasons (bad pressure control within the experimental cell during flight, micro-gravity time not sufficiently long to reach permanent regimes).

Moreover, an important effect not yet included in the theoretical model consists in the gas flow imposed along the free surface, and resulting viscous, thermal and concentrational boundary layers developing from the attack border (rather, an averaged boundary layer thickness has been used in the previous sections). Our preliminary theoretical and numerical analysis shows that due to these effects, and mostly to the concentrational boundary layer, the evaporation rate is larger on the zone of the free surface were the gas flow arrives first, and the temperature is consequently smaller there. This temperature inhomogeneity is indeed observed during the ITEL experiment, as demonstrated by tomographic reconstructions of the three-dimensional temperature field, a typical example being presented in Figure 8.

Fig. 8. Three slices parallel to the free surface, at different depths (deeper from left to right, 1.5 mm between each slice), of the 3D temperature field reconstructed from the interferometric projections. Colder regions are lighter, hotter regions darker. The maximal temperature difference (left plot) is about 1.1°C.

A consequence of this temperature inhomogeneity is a strong thermocapillary flow, opposed to the direction of the gas flow, and advecting the thermal ripples towards the attack border. This effect is visible in Figure 7, although not as strikingly as on the recorded flight movies. Finally, another effect still under study is the tangential stress induced by the gas on the liquid, which competes with the thermally induced thermocapillary flow, and could explained some of the patterns observed during ground experiments. This will be reported elsewhere. CONCLUSIONS

In this paper, we have first summarized recent theoretical results describing evaporation of a pure liquid layer into an inert gas, and focused on the role of several effects on the development of surface-tension-driven instabilities in the liquid. Linear stability analysis of a quasi-steady reference state has shown that the system is strongly destabilized by the presence of an inert gas. It has also been emphasized that at moderate evaporation rates, the real two-layer system may be studied using a one-layer model, with a generalized heat transfer coefficient taking into account thermal conduction in the gas and evaporation. The dimensionless value of this coefficient (i.e. the Biot number) turns out to be much higher than for non-evaporating liquids, which results in an acceleration of the nonlinear transitions to polygonal chaotic patterns.

Experimentally, the ITEL (Interfacial Turbulence in Evaporating Liquids) micro-gravity experiment has allowed to visualize chaotic patterns in a layer of ethyl alcohol under a flux of nitrogen. Although there is some qualitative agreement between the observed polygonal convection cells and theory (sharpness of cold ripples, cell splitting and collapse), further analysis is needed to fully explain the phenomenon of interfacial turbulence, and to validate the generalized one-sided model in more extreme conditions (including mechanically forced flows and boundary layers in the gas).

Moreover, due to a technical problem during the ITEL experiment, the cell pressure and gas flow rates fluctuated too much for reliable quantitative results to be obtained. Further experiments will thus be needed to obtain a sufficient number of steady experimental conditions, and prepare the future CIMEX

(Convection and Interfacial Mass Exchange) experiment onbard ISS. While important (and quantitative) results will certainly be obtained if ITEL is allowed to re-fly, we believe that the feasibility of the CIMEX experiment and its visualization has been clearly demonstrated during the MASER-9 flight. In particular, optical tomography has allowed to measure 3D temperature fields within the liquid, and to understand the origin of large-scale thermocapillary flows advecting the chaotic polygonal ripples. ACKNOWLEDGEMENTS

The authors are extremely grateful to the SSC team (K. Löth, H. Schneider, B. Larsson, O. Janson) for their invaluable help, and to W. Herfs (ESTEC), G. Frohberg (TU Berlin), O. Dupont (Lambda-X) and P. Queeckers (ULB). Part of this work has been supported by the PRODEX Programme of the Belgian Office of Scientific, Technical and Cultural Affairs, by the ICOPAC (Interfacial Convection and Phase Change) Network funded by the European Union, and by the CIMEX-MAP research programme of the European Space Agency. P.C. and B.H. acknowledge financial support from the Fonds National de la Recherche Scientifique. REFERENCES Berg, J.C., A.A. Acrivos, and M. Boudart, Evaporative convection, Adv. Chem. Eng., 6, 61-123, 1966. Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Wiley, New York, USA, 1960. Burelbach, J.P., S.G. Bankoff, and S.H. Davis, Nonlinear stability of evaporating/condensing liquid films,

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