Bénard-Marangoni Convection in Two Layered Liquids

Wayne A. Tokaruk(*), T. C. A. Molteno(†), and Stephen W. Morris(‡) Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 1A7
Abstract

We describe experiments on Bénard-Marangoni convection in horizontal layers of two immiscible liquids. Unlike previous experiments, which used gases as the upper fluid, we find a square planform close to onset which undergoes a secondary bifurcation to rolls at higher temperature differences. The scale of the convection pattern is that of the thinner lower fluid layer for which buoyancy and surface tension forces are comparable. The wavenumber of the pattern near onset agrees with the prediction of the linear stability analysis for the full two-layer problem. The square planform is in qualitative agreement with recent one- and two-layer nonlinear theories, which fail however to predict the transition to rolls.

Submitted to Physical Review Letters , June 29, 1998. See also http://mobydick.physics.utoronto.ca.

pacs:
47.54.+r,47.20.Dr

Convection in fluids has been a fruitful system for the study of nonlinear, nonequilibrium patterns for almost 100 years. (4); (5); (6)The original experiments of Bénard (4)used shallow layers of whale oil, heated from below and open to the air above. Many years passed before it was conclusively shown that surface tension gradients, or “Marangoni”, forces were crucial. (7); (8)In general, both Marangoni and buoyancy forces are present. (9); (10)Beginning with Pearson (8), theories of this instability have traditionally neglected the dynamics of the upper fluid (air in Bénard’s experiments), replacing it by a constant heat flux boundary condition on the lower fluid. This condition is experimentally unrealizable, however. In well-controlled experiments (11); (12); (13); (14); (15); (16), the upper fluid is bounded above by a plate, on which a constant temperature boundary condition is maintained. The heat flux across the interface is then the result of conduction and convection in both fluids, and is not constant above the onset of convection. The two-fluid interface is deformable, and its distance from the upper plate also changes the local heat flux. (16)In addition, buoyancy forces in the upper fluid may actively assist or impede convection in the lower fluid via surface stresses. (10); (17); (18)Recent experiments have found it necessary to treat the dynamics of both fluids in order the quantitatively explain the data. (16)

In order to gain insight into the more complex, but experimentally better posed, two-layer problem, we have undertaken an experimental study of convection in a system of two immiscible fluids sharing a deformable interface. To our knowledge, only one previous experiment has been done on this system, and it was restricted to locating the onset of convection, without visualization (17). On the theoretical side, the two-layer problem has been the subject of a complete linear stability analysis (10); (17); (18)and some weakly nonlinear analyses (10); (27)in certain limiting cases. The general problem, including surface deformations, should however be well within the range of modern weakly nonlinear theory and of numerical simulations.

In our experiment, both fluids were liquids, and the conditions were such that the Marangoni forces were comparable to the buoyancy forces at onset. The dimensionless heat flux across the interface is larger in our experiment than in previous studies. This is expected to have important effects in the nonlinear regime. (25); (27)We used shadowgraph imaging to visualize the pattern and studied its secondary instabilities. We measured the mean wavenumber of the patterns, and compared them to the linear stability result, calculated for the full two-layer problem (18). We could not resolve any hysteresis at the onset of convection. The first clear pattern we found above onset had a square planform. The mean wavenumber of this pattern is in good agreement with the critical value predicted from linear theory (18). The square planform persisted up to ϵ0.7similar-toitalic-ϵ0.7\epsilon\sim 0.7 , where it underwent a slightly hysteretic transition to a roll pattern. Here, ϵ=(ΔT/ΔTc)-1italic-ϵΔ𝑇Δsubscript𝑇𝑐1\epsilon=(\Delta T/\Delta T_{c})-1 , where ΔTΔ𝑇\Delta T is the temperature difference across both layers, and ΔTcΔsubscript𝑇𝑐\Delta T_{c} is its value at the onset of convection. Up to the maximum value of ϵ1.4similar-toitalic-ϵ1.4\epsilon\sim 1.4 we reached, the scale of the convection cells was determined by the depth of the lower liquid. We compare our results with two recent nonlinear theories (25); (27)and find qualitative agreement.

The lower fluid was FC-75, a low-viscosity, perfluorinated hydrocarbon (19), while the upper was water. An important parameter is the depth fraction, L=d+/d𝐿superscript𝑑𝑑L=d^{+}/d , where d+superscript𝑑d^{+} ( d𝑑d ) is the depth of the upper (lower) fluid. In this paper, we report results for L=2.18±0.04𝐿plus-or-minus2.180.04L=2.18\pm 0.04 . The dimensionless heat flux across the interface, which is uniform below the onset of convection, is given by the Biot number =Λ+/ΛLsuperscriptΛΛ𝐿{\cal B}=\Lambda^{+}/\Lambda L , where Λ+superscriptΛ\Lambda^{+} ( ΛΛ\Lambda ) is the thermal conductivity of the upper (lower) fluid (20). In contrast to previous liquid/gas experiments (11); (12); (13); (14); (15); (16), for which 0.1similar-to0.1{\cal B}\sim 0.1 , in our experiment Λ+>ΛsuperscriptΛΛ\Lambda^{+}>\Lambda so that most of the total temperature drop ΔTΔ𝑇\Delta T falls across the thinner lower layer and =4.31±0.08plus-or-minus4.310.08{\cal B}=4.31\pm 0.08 at onset.

Water, at a constant temperature T0subscript𝑇0T_{0} , bathed the top surface of a sapphire window which formed the top boundary of the cell. The base plate of the cell was

Figure 1: Shadowgraph images of the patterns observed. (a) A disordered square planform at ϵ=0.48italic-ϵ0.48\epsilon=0.48 . (b) Near the transition from squares to rolls at ϵ=0.78italic-ϵ0.78\epsilon=0.78 , coexistence is observed. (c) Roll planform at ϵ=1.38italic-ϵ1.38\epsilon=1.38 . (d) A more ordered square pattern observed upon slowly decreasing ϵitalic-ϵ\epsilon to 0.50, starting from roll pattern (c).

maintained at a temperature T1>T0subscript𝑇1subscript𝑇0T_{1}>T_{0} using a thin-film electric heater attached to its bottom surface. Both temperatures were feedback controlled to ±1plus-or-minus1\pm 1 mK. ΔT=T1-T0Δ𝑇subscript𝑇1subscript𝑇0\Delta T=T_{1}-T_{0} is the experimental control parameter. We estimated the heat flux through the cell by monitoring the heater power. The sidewall of the cell was made of plastic and included a wedge-shaped radial fin to stabilize the two-fluid contact line. The round cell had thickness D=d++d=4.06±0.01𝐷superscript𝑑𝑑plus-or-minus4.060.01D=d^{+}+d=4.06\pm 0.01 mm, and radius r=41.15±0.05𝑟plus-or-minus41.150.05r=41.15\pm 0.05 mm. The relevant aspect ratio is that of the lower fluid, Γ=r/d=32.3Γ𝑟𝑑32.3\Gamma=r/d=32.3 . The uniformity of D𝐷D was ±8μplus-or-minus8𝜇\pm 8\mu m, checked interferometrically. The cell was leveled to within ±100μplus-or-minus100𝜇\pm 100\mu rad, using an electronic bubble level.

Both fluids wet all of the surfaces, so bridged configurations of the interface were difficult to avoid. To prepare the two-layer configuration, the horizontal cell was completely filled with FC-75 which was then partially displaced by water slowly introduced above the fin. The interface was made level with the fin by visually eliminating the meniscus. The resulting metastable layering was somewhat imperfect due to the pinning of small deformations of the contact line, and because the interface was difficult to see. The filling fraction L𝐿L was determined destructively at the end of the experiment by tipping the cell on edge and measuring the interface position.

The onset of convection and the nonlinear patterns were visualized by the shadowgraph method (24), using a beam reflected off the mirrored bottom plate of the cell. This beam passed twice through each fluid and twice through the interface. The shadowgraph signal therefore contained contrast due to temperature gradients in both fluids, plus a component due to deflections at the deformed interface. The latter effect is likely to be small, since the two fluids are nearly index matched. We observed that the scale of the pattern in the upper layer was slaved to that of the lower. Thus, it was not necessary to disentangle the various contributions to the shadowgraph images in order to understand the basic planforms.

A typical run consisted of slowly increasing and decreasing ΔTΔ𝑇\Delta T , while recording images and measuring the heat flux. The onset of convection occurred at ΔTc=0.999±0.025CΔsubscript𝑇𝑐plus-or-minus0.999superscript0.025𝐶\Delta T_{c}=0.999\pm 0.025^{\circ}C . Convection entered from one side of cell as a highly disordered pattern, and eventually filled the cell. We did not observe hysteresis in the onset, presumably because of the rounding effect due to nonuniformities in the cell. The first clear patterns that emerged had a square planform. Square patterns have been observed previously in Marangoni convection (14), but only at much higher ϵitalic-ϵ\epsilon as a secondary bifurcation from a hexagonal pattern. Here, our observations are consistent with a square planform emerging at onset, but, due to rounding effects, we cannot exclude the possibility that a hexagonal pattern exists in a narrow range close to onset.

Typical patterns are shown in Figure 1. For increasing ϵitalic-ϵ\epsilon , a patchy, time-dependent square planform, as shown in Figure 1(a) was found between onset and ϵsr=0.70±0.01subscriptitalic-ϵ𝑠𝑟plus-or-minus0.700.01\epsilon_{sr}=0.70\pm 0.01 where a slightly hysteretic transition to rolls was observed. This transition could easily be located by a significant increase in the heat flux through the cell. Near the transition, we find a dynamical coexistence of squares and rolls, as shown in Figure 1(b). The rolls appeared where squares merged along shared edges, and visa versa . At values of ϵ>ϵsritalic-ϵsubscriptitalic-ϵ𝑠𝑟\epsilon>\epsilon_{sr} , the pattern was dominated by rolls, shown in Figure 1(c), with squares appearing only at grain boundaries. The more ordered roll patterns were very slowly time dependent. The finned sidewalls had a rather weak orienting effect on the rolls. As ϵitalic-ϵ\epsilon was decreased, an ordered square pattern re-emerged at ϵrs=0.67±0.02subscriptitalic-ϵ𝑟𝑠plus-or-minus0.670.02\epsilon_{rs}=0.67\pm 0.02 from the pre-existing rolls. The hysteresis ϵsr-ϵrs=0.03subscriptitalic-ϵ𝑠𝑟subscriptitalic-ϵ𝑟𝑠0.03\epsilon_{sr}-\epsilon_{rs}=0.03 was just barely resolvable from the heat flux data. Figure 1(d) shows such a square planform. In all cases, the patterns had a scale dsimilar-toabsent𝑑\sim d , the dimension of the thinner lower layer.

The characteristic timescale for convection in the lower fluid is the vertical thermal diffusion time τv=d2/κ=47subscript𝜏𝑣superscript𝑑2𝜅47\tau_{v}=d^{2}/\kappa=47 s, where κ𝜅\kappa is the thermal diffusivity. We ramped the temperature slowly compared to τvsubscript𝜏𝑣\tau_{v} , but it was impractical to increase it more slowly than the much-longer horizontal diffusion time τh=Γ2τv=14subscript𝜏superscriptΓ2subscript𝜏𝑣14\tau_{h}=\Gamma^{2}\tau_{v}=14 hrs. Thus, some of the disorder we observe on increasing ϵitalic-ϵ\epsilon may be due to the finite ramp rate. However, we also performed experiments in which ϵitalic-ϵ\epsilon was ramped very quickly to a value <ϵsrabsentsubscriptitalic-ϵ𝑠𝑟<\epsilon_{sr} , and then held constant for 6τhsimilar-toabsent6subscript𝜏\sim 6\tau_{h} . The resulting

Figure 2: The mean wavenumber kdelimited-⟨⟩𝑘\langle k\rangle , showing the linear stability boundary from Ref. (18)(solid line). Bars indicate the spread in kdelimited-⟨⟩𝑘\langle k\rangle due to disorder in the pattern. We find a square planform (square symbols) for ϵ<0.7italic-ϵ0.7\epsilon<0.7 , and rolls (circular symbols) otherwise. Open (solid) symbols indicate increasing (decreasing) ϵitalic-ϵ\epsilon .

square pattern did not anneal significantly and remained time-dependent. Thus, the disorder may be partially dynamical in origin.

We determined the wavenumber of the patterns from Fourier analysis. We found the azimuthally averaged structure function of a 2562superscript2562256^{2} pixel region of the centre of the cell. The peak in the structure function gave the mean wavenumber kdelimited-⟨⟩𝑘\langle k\rangle . Fig. 2compares kdelimited-⟨⟩𝑘\langle k\rangle with the results of linear stability theory (18). The dimensionless wavenumber k𝑘k is scaled by the depth of the bottom fluid d𝑑d . The value of kdelimited-⟨⟩𝑘\langle k\rangle at small ϵitalic-ϵ\epsilon agrees with the critical value kc=2.38subscript𝑘𝑐2.38k_{c}=2.38 from the linear theory. The more ordered square patterns observed for decreasing ϵitalic-ϵ\epsilon have slightly larger kdelimited-⟨⟩𝑘\langle k\rangle than those of increasing ϵitalic-ϵ\epsilon .

To evaluate the linear stability boundary shown in Fig. 2, all of the material parameters are needed. These are, for the lower fluid, the kinematic viscosity ν𝜈\nu , the density ρ𝜌\rho , and κ=Λ/ρCP𝜅Λ𝜌subscript𝐶𝑃\kappa=\Lambda/\rho C_{P} , where CPsubscript𝐶𝑃C_{P} is the heat capacity, as well as the corresponding quantities for the upper fluid and the interfacial surface tension σ𝜎\sigma . All of these are assumed to be independent of T𝑇T , according to the usual Boussinesq approximation, except σ𝜎\sigma and the two densities, which are linearly temperature dependent. Thus, we also require the thermal expansion coefficient β=(1/ρ)ρ/T𝛽1𝜌𝜌𝑇\beta=(1/\rho)\partial\rho/\partial T for each fluid and the temperature derivative of the interfacial surface tension γ=-σ/T𝛾𝜎𝑇\gamma=-\partial\sigma/\partial T . The negative sign is included so that γ>0𝛾0\gamma>0 . All of the parameters are known, except σ𝜎\sigma and γ𝛾\gamma . Using Antonow’s rule (21), we estimated σ𝜎\sigma to be equal to the difference of the surface tensions of the two fluids, measured against air. This approximation suffices, because the neutral stability boundary is very insensitive to σ𝜎\sigma , which only enters into the small surface deformations (22). On the other hand, the remaining parameter γ𝛾\gamma is crucial to the Marangoni mechanism. We fixed γ𝛾\gamma by requiring that the linear theory reproduce the correct measured ΔTcΔsubscript𝑇𝑐\Delta T_{c} . The result is γ=0.047±0.003𝛾plus-or-minus0.0470.003\gamma=0.047\pm 0.003 dynes/cm K. This is consistent to within a factor of two with the difference of the known values of γ𝛾\gamma for each fluid, measured against air (i.e. with Antonow’s rule (21)applied to σ/T𝜎𝑇\partial\sigma/\partial T ).

One- and two-layer theories of Bénard-Marangoni convection traditionally scale the problem using the temperature difference ΔTLΔsuperscript𝑇𝐿\Delta T^{L} across the lower fluid. Below onset,

ΔTL=(1+ΛL/Λ+)-1ΔT=(1+1/)-1ΔT.Δsuperscript𝑇𝐿superscript1Λ𝐿superscriptΛ1Δ𝑇superscript111Δ𝑇\Delta T^{L}=(1+\Lambda L/\Lambda^{+})^{-1}\Delta T=(1+1/{\cal B})^{-1}\Delta T. (1)

This scaling is only approximate above onset, where convection effectively increases the average thermal conductivity of each layer, and also introduces temperature variations on the interface (10); (23). The often-used assumption of a constant {\cal B} is problematical above onset for the same reasons. (20); (23)However, in order to make contact with theories using this scaling, it is customary to adopt Eqn. 1even above onset. (11)The seriousness of this approximation can be appreciated from heat flux measurements. (23)The total effective thermal conductivity of the cell at the highest ϵitalic-ϵ\epsilon was only about 8% larger than that below onset. This sets an upper bound on the validity of Eqn. 1.

Assuming Eqn. 1, one can define the Rayleigh and Marangoni numbers {\cal R} and {\cal M} of the lower layer as follows;

=[gβd3κν]ΔTL   =[dγρκν]ΔTL,formulae-sequencedelimited-[]𝑔𝛽superscript𝑑3𝜅𝜈Δsuperscript𝑇𝐿delimited-[]𝑑𝛾𝜌𝜅𝜈Δsuperscript𝑇𝐿{\cal R}=\bigg{[}\frac{g\beta d^{3}}{\kappa\nu}\bigg{]}\Delta T^{L}{~{}~{}~{}~% {}~{}~{}~{}}{\cal M}=\bigg{[}\frac{d\gamma}{\rho\kappa\nu}\bigg{]}\Delta T^{L}\,, (2)

where g𝑔g is the acceleration due to gravity. {\cal R} and {\cal M} are not independent, so it is convenient to define two related parameters (26),

α=[1+MM0R0R]-1   λ=RR0+MM0.formulae-sequence𝛼superscriptdelimited-[]1𝑀subscript𝑀0subscript𝑅0𝑅1𝜆𝑅subscript𝑅0𝑀subscript𝑀0\alpha=\bigg{[}1+\frac{M}{M_{0}}~{}\frac{R_{0}}{R}\bigg{]}^{-1}{~{}~{}~{}~{}~{% }~{}}\lambda=\frac{R}{R_{0}}+\frac{M}{M_{0}}\,. (3)

Here, 0subscript0{\cal R}_{0} ( 0subscript0{\cal M}_{0} ) is the critical value of {\cal R} ( {\cal M} ) in the absence of Marangoni (buoyancy) forces. In general, in the two-layer problem 0subscript0{\cal R}_{0} and 0subscript0{\cal M}_{0} will differ from the usual critical values for any one-layer model of the lower fluid. They can be determined from the full linear theory of the coupled, two-layer system, as described below. α𝛼\alpha is fixed by the choice of fluids and the depth d𝑑d , and can be considered as the fraction of the total forcing due to buoyancy, while λ𝜆\lambda is proportional to the total temperature difference ΔTΔ𝑇\Delta T .

Figure 3: The Marangoni number {\cal M} vs. the Rayleigh number {\cal R} for the lower fluid, showing the neutral stability boundary (thick solid line) from the complete two-layer analysis of Ref. (18). Open (solid) symbols show data for increasing (decreasing) ϵitalic-ϵ\epsilon . Square (circular) symbols represent points where the square (roll) planform was observed.

Fig. 3shows a plot of {\cal M} vs. {\cal R} . The linear stability boundary for the coupled, two-layer system at constant L𝐿L separates the conduction regime near the origin from the convecting regime. Its intercepts are the critical values 0subscript0{\cal R}_{0} and 0subscript0{\cal M}_{0} of each number in the absence of the complementary effect. The boundary is calculated by holding L𝐿L and the material parameters constant and varying d𝑑d . One point on this line corresponds to the experimental value of d𝑑d . α𝛼\alpha varies from 0 on the {\cal M} axis to 1 on the {\cal R} axis. Above the line α=1/2𝛼12\alpha=1/2 , Marangoni forces dominate; below, buoyancy. For our experimental conditions, α=0.58𝛼0.58\alpha=0.58 , so that the two effects are nearly equal. Assuming Eqn. 1holds, increasing λϵproportional-to𝜆italic-ϵ\lambda\propto\epsilon corresponds to moving outward along the radial line α=0.58𝛼0.58\alpha=0.58 . The symbols show the values of {\cal M} and {\cal R} that we reached, and the point where the planform changed from squares to rolls. It would be interesting to map the stability boundaries of various patterns in the (,)({\cal R},{\cal M}) plane by experimentally varying α𝛼\alpha and λ𝜆\lambda independently. This can be done in principle by varying ϵitalic-ϵ\epsilon and d𝑑d , keeping L𝐿L fixed.

Two recent nonlinear theories are relevant to our observations (25); (27), although neither treats the full two-layer problem. Regnier et. al. (25)included both surface tension and buoyancy in a one-layer model with no surface deformations and a constant {\cal B} boundary condition on the upper surface. They derived a set of amplitude equations which show various regimes of hexagonal, square and roll planforms above onset. Their main result is that there exists a critical value αcsubscript𝛼𝑐\alpha_{c} above which a hexagonal pattern at onset gives way to squares for α<αc𝛼subscript𝛼𝑐\alpha<\alpha_{c} or to rolls for α>αc𝛼subscript𝛼𝑐\alpha>\alpha_{c} . Unfortunately, they only present results for <22{\cal B}<2 , where our 4similar-to4{\cal B}\sim 4 . The trends however suggest that as {\cal B} is increased in their model, αc1subscript𝛼𝑐1\alpha_{c}\rightarrow 1 , favouring squares. In the same limit, the range above onset over which the hexagonal pattern exists becomes very narrow. These two predictions are in qualitative agreement with our observations at low ϵitalic-ϵ\epsilon .

Golovin et. al. (27)treated the full two-layer problem, but included Marangoni forces only. They derived amplitude equations, including the effect of surface deformations, but neglecting stresses on the interface on the assumption that the upper fluid was a gas. Interestingly, when the heat transfer through the upper fluid (i.e. {\cal B} ) is increased, a square pattern is predicted at onset. This is also consistent with our observations. Neither of these nonlinear theories predicts the square to roll transition we observed, however. Nor do the amplitude equations contain the gradient terms that would be necessary to describe disordered patterns of the kind we observe.

In summary, we have performed experiments on thermal convection in a system of two layered liquids under conditions where both buoyancy and Marangoni forces are comparable. We found a square planform of convection just above onset, which underwent a transition to a roll pattern at higher control parameter. We compared the wavenumber of the patterns to the linear stability boundary for the full two-layer coupled problem. We found quantitative agreement between the wavenumber of the pattern just above onset and the critical wavenumber kcsubscript𝑘𝑐k_{c} from linear theory. Two nonlinear theories, originally proposed to explain gas over liquid experiments, can be applied to our case. They qualitatively predict the square patterns found near onset, but not the observed transition to rolls.

We thank Stephen VanHook, Harry Swinney and Zahir Daya for useful discussions. This research was supported by the Natural Sciences and Engineering Research Council of Canada and the Walter C. Sumner Memorial Fellowship.

References

  • (*) wayne@mobydick.physics.utoronto.ca
  • (†) tim@videoscript.com
  • (‡) smorris@physics.utoronto.ca
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