On the Principle of Exchange of Stabilities in Rayleigh-Bénard Convection

Abstract. The problem of Rayleigh–Bénard convection with internal heat sources and a variable gravity field is treated. For the case of stress-free boundary conditions, it is proved that the principle of exchange of stabilities holds as long as the product of gravity field and the integral of the heat sources is nonnegative throughout the layer. The proof is based on the idea of a positive operator, and uses the positivity properties of Green’s function.