The onset of steady Bénard-Marangoni convection in a two-layer system of conducting fluid in the presence of a uniform magnetic field

Abstract

The onset of steady Benard-Marangoni convection in two horizontal liquid layers of electrically conducting immiscible fluids subjected to a uniform vertical magnetic field and temperature gradient is analysed by means of a combination of analytical and numerical techniques. The free surface can be either deformable or nondeformable and the interface between the fluids is always assumed to be flat. The effect of the lower layer on the critical values of Rayleigh, Marangoni and wave numbers for the onset of steady convection is investigated. When the free surface is nondeformable, the critical parameters for the onset of pure Marangoni convection are increased, whereas for the onset of pure Benard convection they are decreased compared to the single-layer model. The results for a single-layer and for two-layers are qualitatively similar for Benard-Marangoni convection when the free surface is deformable. All disturbances can be stabilized with sufficiently strong magnetic field when the free surface is nondeformable. If the free surface is allowed to deform and gravity waves are excluded, then the layers are always unstable to disturbances with sufficiently small wave number with magnetic field. Inclusion of gravity waves has a stabilizing effect on certain disturbances of small wave number in the presence of weak or moderate magnetic field.