A Particular Solutions for a Two-Phase Model with a Sharp Interface

Abstract

Two-phase models can be used to describe the dynamics of mixed materials and can be applied to many physical and biological phenomena. For example, these types of models have been used to describe the dynamics of cancer, biofilms, cytoplasm, and hydrogels. Frequently the physical domain separates into a region of mixed material immersed in a region of pure fluid solvent. Previous works have found a perturbation solution to capture the front velocity at the initial time of contact between the polymer network and pure solvent, then approximated the solution to the sharp-interface at other points in time. The primary purpose of this work is to use a symmetry transformation to capture an exact solution to this two-phase problem with asharp-interface. This solution is useful for a variety of reasons. First, the exact solution replicates the numeric results, but it also captures the dynamics of the volume profile at the boundary between phases for arbitrary time scales. Also, the solution accounts for dispersion of the network further away from the boundary. Further, our findings suggest that an infinite number of exact solutions of various classes exist for the two-phase system, which may give further insights into the behaviors of the general two-phase model.