Homologous onset of double layer convection

The onset of convection in two superimposed fluid layers of the same height is considered. It is found that the neutral curve for R(a) for the onset Rayleigh number R in dependence on the wave number a is an invariant of a multidimensional parameter space of property ratios of the system even though the corresponding convection solutions may vary strongly with these property ratios. For each neutral curve R(a) two manifolds of solutions are found one of which can be understood on the basis of symmetry properties of the system, while the other does not exhibit simple symmetry features. In particular the neutral curves R(a) for various single Rayleigh-Bénard convection layers are shown to correspond to two two-dimensional manifolds of solutions. Analytical expressions for the latter are derived in the case of outer stress-free boundary conditions.