We study Bénard-Marangoni instability in a system formed by a horizontal liquid layer and its overlying vapor. The liquid is lying on a hot rigid plate and the vapor is bounded by a cold parallel plate. A pump maintains a reduced pressure in the vapor layer and evacuates the vapor. This investigation is undertaken within the classical quasisteady approximation for both the vapor and the liquid phases. The two layers are separated by a deformable interface. Temporarily frozen temperature and velocity distributions are employed at each instant for the stability analysis, limited to infinitesimal disturbances (linear regime). We use irreversible thermodynamics to model the phase change under interfacial nonequilibrium. Within this description, the interface appears as a barrier for transport of both heat and mass. Hence, in contrast with previous studies, we consider the possibility of a temperature jump across the interface, as recently measured experimentally. The stability analysis shows that the interfacial resistances to heat and mass transfer have a destabilizing influence compared to an interface that is in thermodynamic equilibrium. The role of the fluctuations in the vapor phase on the onset of instability is discussed. The conditions to reduce the system to a one phase model are also established. Finally, the influence of the evaporation parameters and of the presence of an inert gas on the marginal stability curves is discussed.

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We propose a theoretical study of Marangoni driven convection in an evaporating liquid layer surmounted by an inert gas-vapor mixture. After reduction of the full two layer problem to a one-sided model we use a Galerkin-Eckhaus method leading to a finite set of amplitude equations for the weakly nonlinear analysis of the problem. We analyse the stability of the roll, square and hexagonal patterns emerging above the linear stability threshold for a water-air and for an ethanol-air system.

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We propose an extension of the Galerkin-Eckhaus method in order to study thermoconvective instabilities not only in the weakly nonlinear régime, but also farther from the threshold. This extension consists in an appropriate choice of the basis functions used to expand the unknowns and in an increase of the number of amplitude equations taken into account. The rigidfree Rayleigh-Bénard problem with heat-conducting boundaries and the Marangoni-Bénard problem are considered in detail before generalizing to all thermoconvective instabilities. The validity of our approach is proven by comparing its results with a multiple scale method and with a finite element method. A very good agreement between all methods is found in the weakly nonlinear regime. Farther from the threshold, the agreement between our extended Galerkin-Eckhaus method and the finite element approach is also obtained with appropriate basis functions whose choice depends crucially on the value of the Prandtl number and also on the boundary conditions imposed at the top and bottom of the system. 157