Interfacial stability in a two-layer Bénard problem

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Abstract
A linear stability analysis of the Bénard problem for two layers of different fluids lying on top of each other and bounded by free surfaces is considered. The fluids are assumed to be similar and perturbation methods are used to calculate the interfacial eigenvalue in closed form. The case of the Rayleigh number and wavenumber of the disturbance being close to the first criticality of the one‐fluid Bénard problem has been investigated in a previous paper [Phys. Fluids 2 8, 2699 (1985)], and was found to exhibit both overstability and convective instability. In this paper, the Rayleigh number is assumed to be less than that of the first criticality of the one‐fluid problem, and in this situation, overstability does not occur. An unexpected result is that by an appropriate choice of parameters, it is possible to find linearly stable arrangements with the more dense fluid on top.