Application of Wiedeburg Linearization for Solving the Stability Problem of a Two-Layer Mixture with Concentration-Dependent Diffusion

The problem of stability of an isothermal system of two miscible fluids in a gravity field is considered. The fluids are aqueous solutions of nonreacting substances with different diffusion coefficients. At the initial instant, the solutions are separated in space by an infinitely thin horizontal contact surface. This configuration can easily be implemented experimentally, although it is more difficult for theoretical analysis, because the concentration profiles evolve in time. It is assumed that the initial configuration of the system is statically stable. After the start of evolution, the solutions begin to mix, penetrating each other and providing conditions for the development of convective instability of double diffusion. An important complicating factor of the problem is the functional dependence of the diffusion coefficients of the solutions on their concentrations. In recent years, this effect has been intensely studied, because its significant influence on the convective stability has been proven experimentally. In this study, it is assumed for simplicity that the diffusion coefficients of the solutions depend linearly on concentration. A mathematical formulation of the stability problem for a mixture includes the equation of motion within the Darcy–Boussinesq approximation, the continuity equation, and two transport equations for the substance concentrations. The solution to this problem in the absence of concentration-dependent diffusion is well known from the literature. However, the consideration of this dependence yields nonlinear-diffusion equations, which can be solved in the general case only numerically. To find an approximate analytical solution, we suggest applying the method of diffusion-equation linearization, proposed by Wiedeburg in 1890. This method is well known in the theory of thermal conductivity, although it had originally been developed specifically for solutions of substances. It is shown that, in our case, the conditions for convective stability of the base state can be obtained in the analytical form. A comparative analysis of the discrepancy between the Wiedeburg solution and the numerical solution is performed. A stability map is obtained based on the analytical solution. It shows how the concentration-dependent diffusion affects the mixture stability.