Exact and approximate solutions to the double-diffusive Marangoni–Bénard problem with cross-diffusive terms

Abstract

We discuss the linear stability of a cross-doubly-diffusive fluid layer with surface tension variation along the free surface. Two limiting cases of the mass flux basic state are considered in the presence of non-zero Soret and Dufour diffusivities. The first case, which has remained largely unexplored, is one where a temperature difference, ΔT¯, and a concentration difference, ΔC¯, are both imposed across the layer. The second case, which has greater significance to thermosolutal systems, is that where the imposed ΔT¯ across the layer induces a ΔC¯. We rescale the problem in the absence of buoyancy, which leads to a more concise representation of neutral stability results in and near the limit of zero gravity. We obtain exact solutions for stationary stability in both cases. One important consequence of the exact solutions is the validation of recently published numerical results in the limit of zero gravity. Moreover, the precise location of asymptotes in relevant parameter (Smc, Mac) space are computed from exact solutions. Both numerical and exact solutions are used to further examine stability behaviour. We also derive algebraic expressions for stationary stability, oscillatory stability, frequency, and codimension two point from a one-term Galerkin approximation. The one-term solutions qualitatively reflect the stability behaviour of the system over the parameter ranges in our investigation. A practical consequence is that the nature of the stability (oscillatory or stationary) for a given set of parameter values can be determined approximately, without solving the numerical eigenvalue problem.