The Benard-Marangoni thermocapillary instability problem

Physically, there are two main mechanism responsible for driving the instability in coupled buoyancy (Benard) and thermocapillary (Marangoni) convection problem of a weakly expansible viscous liquid layer, bounded on the bottom by a heated solid surface and on the top by a free-surface subject to a temperature-dependent surface tension. The first one is the density variation generated by the thermal expansion of the liquid, the second cause of instability 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 Benard experiments (the so-called Bi nard-Marangoni (BM) problem). Indeed, for a thin layer we show that, it is not consistent to consider simultaneously both effects, and in Section 3 we formulate an alternative concerning the role of the buoyancy. In fact, it is necessary to consider two fundamentally distinct problems: the first problem is the classical shallow convection problem for a non-deformable upper surface with a partial account of the Marangoni effect (RBM problem), the second one is the full BM problem for a deformable free-surface without the buoyancy effect. We shall be mostly concerned with thermocapillary BM instabilities problem on a free-falling vertical film, since most experiments and theories have focused on the latter (in fact, wave dynamics on an inclined plane is quite analogous). For a thin film we consider three main situations in the relation with the magnitude of the characteristic Reynolds number (Re) and we derive various model equations. These model equations are analyzed from various point 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-time. Finally, some recent numerical results are also presented.