Deformational instability of a plane interface with transfer of matter. Part 2.—Non-oscillatory and oscillatory modes with linear and exponential concentration profiles

Abstract

Dispersion relations between the real and imaginary components of the complex time constant and the wavevector are evaluated by computer on a previously proposed model for interfacial turbulence. The model describes deformational instability of a plane liquid–liquid interface with a perpendicular linear concentration profile of a third diffusing component. The dispersion relations found are compared with the relations calculated by an earlier model of Sternling and Scriven in which the interface is assumed flat and in which the gravity, the surface diffusion, the mass of the adsorbed mass of surfactant and the dependence on the absolute value of the unperturbed interfacial tension were neglected. In many cases of interfacial turbulence no differences between the two models are found for realistic values of the parameters, and both models are shown to be in qualitative agreement with reported data for the system ethylene glycol + ethyl acetate with acetic acid as diffusing surfactant. Dispersion curves depend strongly on surface viscosity. In cases of diffusion from the phase with the higher diffusion coefficient to the phase with the lower, both models exhibit under suitable circumstances oscillatory instabilities with overstability critical boundaries. The dispersion curves for the two models differ for low values of the surface tension. Preliminary computer results are obtained for exponential concentration profiles—being approximations to error function profiles—and compared with data for ethylene glycol + ethyl acetate + acetic acid. The main difference between a linear and an exponential profile is that two critical boundaries, one critical boundary or even no critical boundary of exchange of stabilities are found in the latter case with decreasing thickness of the diffusion zone. Finally, it is found that the Rayleigh–Taylor instability—contained in our model as a special case—is in practice independent of the slope of the concentration profile and the value of the surface viscosity, with regard not only to the critical wavevector, but also the whole course of the dispersion curve.