The Benard–Marangoni thermocapillary-instability problem

Physically, there are two main mechanisms responsible for driving the instability in the coupled buoyancy (Benard) and thermocapillary (Marangoni) convection problem for a weakly expansible viscous liquid layer bounded from below by a heated solid surface and on the top by a free surface subject to a temperature-dependent surface tension. The first mechanism is density variation generated by the thermal expansion of the liquid; the second results from the surface-tension gradients due to temperature fluctuations along the upper free-surface. In the present paper we consider only the second effect as in the Benard experiments [the so-called Benard–Marangoni (BM) problem]. Indeed, for a thin layer we show that it is not consistent to consider both effects simultaneously, and we formulate an alternative concerning the role of buoyancy. In fact, it is necessary to consider two fundamentally distinct problems: the classical shallow-convection problem for a non-deformable upper surface with partial account of the Marangoni effect (the RBM problem), and the full BM problem for a deformable free surface without the buoyancy effect. We shall be mostly concerned with the thermocapillary BM instabilities problem on a free-falling vertical film, since most experiments and theories have focused on this (in fact, wave dynamics on an inclined plane is quite analogous). For a thin film we consider three main situations in relation to the magnitude of the characteristic Reynolds number (Re) and we derive various model equations. These model equations are analyzed from various points of view but the central intent of this paper is to elucidate the role of the Marangoni number on the evolution of the free surface in space and time. Finally, some recent numerical results are presented.