Convection in a Thin Liquid Layer with Mixed Thermal Boundary Conditions: Benard-Marangoni Convection

We investigate by means of linear stability analysis, the onset of convection in a horizontal liquid layer heated from below, confined below by a rigid plane and above with a free surface whose surface tension linearly depends on temperature. Consideration is given to a variety of thermal boundary conditions at the bounding surfaces. We use a combination of analytical and numerical techniques to obtain a detailed description of marginal stability curves. It is established numerically that the principle of exchange of stabilities is valid. The numerical results are presented for a wide range of the values of the parameters characterizing the nature of thermal boundary conditions. Attention is focused on a situation which is of possible practical interest, where value of the parameter characterizing the thermal condition at the upper boundary varies inversely to the parameter characterizing the thermal condition at the lower boundary, and find that increasing values of the parameter of the lower boundary has destabilizing effect. In this case, we also distinguish ranges of the parameter in which formation of convection cells of increasing or decreasing sizes occur at the onset of convection.