Application of Wiedeburg linearization for solving the stability problem of a two-layer mixture with concentration-dependent diffusion

Abstract

In this paper, we consider the problem of the stability of an isothermal system of two miscible fluids in a gravity field. Fluids are aqueous solutions of non-reacting substances with different diffusion coefficients. At the very beginning, the solutions uniformly fill half-spaces, which are separated from each other by an infinitely thin horizontal contact surface. Such a configuration can be easily realized experimentally, although it is more difficult for theoretical analysis. We assume that the initial configuration of the system is statically stable. After the start of evolution, the solutions begin to mix, penetrating each other, and creating conditions for the development of the convective instability of double diffusion. An important complicating factor of the problem is the functional dependence of the diffusion coefficients of solutions on their concentration. In recent years, this effect has been actively studied, since its significant influence on convective stability has been proven experimentally. For simplicity, we assume that the diffusion coefficients of solutions depend linearly on concentration. The problem of the stability of a mixture includes the equation of motion in the Darcy and Boussinesq approximation, the continuity equation, and two transport equations for the concentrations. The solution to this problem in the absence of the effect of concentration-dependent diffusion is well known from the literature. If we take into accountsuch a dependence, then we have to deal with the problem of nonlinear diffusion, which can only be solved numerically. To find an analytical solution to the problem, we propose to apply the method of preliminary linearization of Wiedeburg (1890). The method is well known in the theory of thermal conductivity, although it was originally developed specifically for solutions of substances. In our case, we demonstrate that the conditions for convective stability of the base state can be obtained analytically. The comparative analysis of the discrepancy between the Wiedeburg solution and the numerical solution is given. Based on the closed-form analytical solution, we obtain a stability map for the problem under the consideration. We show how the effect of concentration-dependent diffusion affects stability.