Stability in the rotating Bénard problem with Newton–Robin and fixed heat flux boundary conditions

We study the classical Bénard system, with and without rotation, in a variety of boundary conditions both for the velocity field and the temperature field. For the temperature field, we consider Newton–Robin boundary conditions, and fixed heat fluxes (also known as “insulating” boundary conditions). A range of stability estimates, with linear instability methods, are presented, in rotating and non-rotating systems. In the limit case of assigned heat fluxes, we find that the critical wave number is asymptotically equal to zero only up to a threshold of rotation speed, dependent on the boundary conditions on the velocity field. This appears, as far as we know, to be a new result, that perhaps could be tested experimentally. Sufficient conditions for the validity of the principle of exchange of stabilities (PES) under various boundary conditions are given. For stress-free b.c., and fixed heat fluxes, the condition obtained is also optimal. Some overstability results, for fixed heat fluxes, are presented.