The effect of rotation on the Rayleigh-Bénard stability threshold

Abstract

The standard method used to solve the Rayleigh-Bénard linear stability problem for a rotating fluid leads to a complex expression which can only be evaluated numerically. Here the problem is solved by a different method similar to that used in a recent paper on the non-rotating case [A. Prosperetti, “A simple analytic approximation to the Rayleigh-Bénard stability threshold,” Phys. Fluids 23, 124101 (2011)10.1063/1.3662466]. In principle the method leads to an exact result which is not simpler than the standard one. Its value lies in the fact that it is possible to obtain from it an approximate explicit analytic expression for the dependence of the Rayleigh number on the wave number of the perturbation and the rate of rotation at marginal stability conditions. Where the error can be compared with exact results in the literature, it is found not to exceed a few percent over a very broad Taylor number range. The relative simplicity of the approach permits us, among others, to account for the effects of a finite thermal conductivity of the plates, which have not been studied before.