Deformational instability of a plane interface with transfer of matter. Part 1.—Non-oscillatory critical states with a linear concentration profile

Abstract

General laws of conservation of mass and momentum are formulated for a moving and arbitrarily deformed interface in local equilibrium with the adjacent, immiscible bulk liquids in which a third component is distributed. The equations are linearised for the case of small deformations from a plane interface.

The stability of a plane interface with a perpendicular, linear concentration gradient with respect to deformation is investigated by means of linear, hydrodynamic stability theory. A general dispersion relation between the complex time constant of perturbation ω and the wavenumber k is obtained. An explicit solution for the exchange of stability boundary is found. For spontaneous, interfacial deformation to occur, the diffusion has to be directed from the liquid with the smallest value for the diffusion coefficient of the third component to the liquid with the greatest. The direction of the gravitational field has no importance in the linear theory. All wavenumbers below a certain critical number k_cr will be unstable for a fixed difference of slopes of the concentration profiles. On the other hand, when k is fixed, the difference of slopes has to exceed an instability threshold which increases with the bulk and surface viscosities and the diffusion coefficients, and decreases with increasing value of the coupling coefficient between the surface pressure and the surface concentration of the third component.