Benard-Marangoni convection in two thin immiscible liquid layers with a uniform magnetic field

The effect of a uniform magnetic field on B~nard-Marangoni convection in a shallow cavity, with differentially heated side walls, filled with two viscous, immiscible, incompressible an electrically conducting fluids is studied in the presence of a buoyancy force. The fluid-fluid interface and the free surface are assumed to be flat, and the driving forces for the flow are the thermocapillary and the buoyancy forces. Closed-form solutions, under thin layer approximation neglecting the side wall effects, are obtained for the stream function and the temperature. The solutions are obtained in various limiting cases, namely: absence of buoyancy force, absence of thermocapillary force and absence of magnetic field, and they coincide with the results existing in the literature. The velocity is calculated, and the resulting cell patterns are discussed for different values of A, the ratio of the temperature gradients of surface tension at the interface and the free surface and the Hartmann number. There exist four different flow regimes depending on the values of A but with reduced convection compared to the non-magnetic case for Marangoni convection. It is observed that it is possible to control the convection in the lower layer by a suitable choice of the magnetic field.