Marginal stability and traveling fronts in two-phase mixtures.

Mixtures of materials that move relative to each other arise in a variety of applications, especially in biophysical problems where the mixture consists of materials with different material properties. The variety of applications leads to a bewildering array of multiphase models, each with slightly different behaviors and interpretations, depending on the application. Some of the behaviors include phase separation, traveling waves, and linear instabilities. Because of the variability of the predicted behaviors, there has been considerable attention paid to minimal models to determine the fundamental solutions, bifurcations, and instabilities. In this paper we describe a new solution for the simplest two-phase system where both phases are dominated by viscous forces, one-phase response to osmotic forces, and the phases interact through a drag term. The system develops a traveling front separating an unstable, uniform solution from a patterned, phase separated solution. We seek the velocity of the traveling front and show that, for large diffusion, marginal stability gives a simple and accurate prediction for the velocity. For smaller diffusion constants, the front is "pushed," and the linear prediction fails.