Role of interface deformations in Benard-Marangoni instability

This work is concerned with a linear analysis of natural thermoconvection in a two-layer system formed by a liquid surmounted by an upper immiscible gas layer (Bénard-Marangoni problem). Emphasis is put on the role of the air layer on the thermoconvective instability. It is shown that the motion inside the gas phase can be disregarded provided its thickness is smaller than the thickness of the liquid layer. The gas can then be modelled as a purely conductive medium. Another important problem discussed here is the role of surface deformations on the onset of convection. In that respect, the use of Boussinesq's approximation is discussed and its limits of validity are specified. The main results are the following. First, it is concluded that overstability cannot be observed in ordinary fluids under realistic experimental conditions. Besides, it is shown that, among the relatively larger number of parameters appearing in the problem, the viscosity is by far the most important. Moreover, new dimensionless numbers are introduced to better apprehend the physical context. The experimental conditions required to observe the surface zero-wave number instability are also determined, as well as the conditions under which gravity effects may be neglected on earth. Finally it is examined under which circumstances the interface can be considered as remaining flat.