Influence of thermal boundary conditions on the stability of thermocapillary-driven convection at low Prandtl numbers

We analyze the effect of various thermal boundary conditions on the linear stability of surface-tension-driven flow in an unbounded liquid layer subject to a longitudinal temperature gradient. An original approach is devised to estimate the critical instability parameters. The order of magnitude estimates are used to solve the problem asymptotically for small Prandtl numbers. The instability is shown to be essentially determined by the thermal boundary conditions. For insulating boundaries the critical wavenumber scales as k_c ∼ Pr^1/2 meaning that the most unstable wave is considerably longer than the depth of the layer. When the bottom is well conducting, the critical wavelength is comparable to the depth of the layer. For the case of insulating bottom and non-adiabatic free surface the critical wavenumber depends on the Biot number as k_c ∼ Bi^1/2. Even a weak thermal coupling between the free surface and the ambient medium such that Bi ∼ Pr can significantly influence the instability threshold.