Determination of the stability boundary of a two-layer system of miscible liquids with linear diffusion laws

Abstract

The authors consider the problem of determining the stability boundary of a two-layer system of miscible liquids placed in a gravity field. Liquids are aqueous solutions of non-reacting substances with different diffusion coefficients, which are linear functions of concentrations. At the very beginning of the evolution, the solutions 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 since the base state of the system is non-stationary. Once brought into contact, the solutions begin to mix penetrating each other and creating conditions for the development of the double-diffusive instability since the initial configuration of the system is assumed to be statically stable. The problem of the convective instability of a mixture includes the equation of motion written in the Darcy and Boussinesq approximations, the continuity equation, and two transport equations for the concentrations. We apply the linearization method suggested by Wiedeburg (1890) to find a closed-form solution to the non-stationary base state problem including concentration-dependent diffusion laws for species. We derive analytical expressions for neutral stability curves and study corrections introduced by nonlinear diffusion to the stability analysis.