On the hydrodynamic and magnetohydrodynamic stability of an inclined layer heated from below

In this paper we investigate the stability of either the rest state (Bénard problem) or parallel laminar flows in hydrodynamics and magnetohydrodynamics for inclined layers heated from below. In particular, we numerically investigate the linear instability under general three-dimensional perturbations of the basic state, and we also give conditions for the nonlinear stability. For particular basic states, a linear instability analysis shows that the critical Rayleigh numbers can be obtained for longitudinal perturbations (which can be studied analytically both, in the linear and nonlinear case) or for transversal perturbations depending on the inclination of the layer (and of course also on the velocity of the boundaries and the applied magnetic field). Remarkable is the presence of points of codimension-two: Particular values of the critical Rayleigh numbers obtained for specific angles of inclination for which there exist two equally destabilising perturbations, one longitudinal and the other transversal.