Marginal regions for the solute Bénard problem with many types of boundary conditions

A large number of variants of the Bénard problem (with a solute, rotating, subject to mag- netic field, etc.) have been extensively studied. Despite this, new interesting results can be obtained imposing very general yet physically relevant boundary conditions. In this frame- work, we develop a technique to analytically compute the marginal region in parameter space. We investigate the thermal stability of a fluid layer salted from below, subject to finite slip on velocity and Robin conditions on temperature and solute concentration. We write analytical conditions for the onset of stationary convection, obtain simplified formulas for particularly symmetric cases, and draw the associated (convective) marginal regions in some significant cases. Moreover, we describe the analytical conditions for the onset of overstability, and use such equations to numerically draw the associated (overstable) marginal region. We finally perform an asymptotic analysis for small wave numbers.