Thermal properties in surface-tension-driven convection

We discuss the approximations that may be applied to the convective problem of a horizontal layer of liquid in contact with an air layer, both enclosed between conducting walls. Assuming that heat flows across the air mostly by conduction (conducting-air hypothesis) the two-fluid problem reduces to the usual Bénard-Marangoni (BM) problem provided the spatial variations of the temperature in the thermal boundary conditions are considered. This approximation is the minimal model to compare with well-controlled BM experiments. The form of the average temperature profiles suggests the reference temperature that ought to be taken in nondimensional parameters that describe these phenomena. We also discuss how the Biot number could be estimated from the Nusselt number and the interfacial temperature field measurements even far from convective threshold. A linear stability analysis is performed with the correct thermal boundary condition. It gives thresholds that slightly differ from those obtained previously. These values are compared with recent experimental findings. All these facts will be useful in performing weakly nonlinear analyses and in planning future experiments on this instability.