Sharp Interface Limits for Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids

Abstract

We consider the mathematical relation between diffuse interface and sharp interface models for the flow of two viscous, incompressible Newtonian fluids like oil and water. In diffuse interface models a partial mixing of the macroscopically immiscible fluids on a small length scale ɛ > 0 and diffusion of the mass particles are taken into account. These models are capable to describe such two-phase flows beyond the occurrence of topological singularities of the interface due to collision or droplet formation. Both for theoretical and numerical purposes a deeper understanding of the limit ɛ → 0 in dependence of the scaling of the mobility coefficient m ɛ is of interest. Here the mobility is the inverse of the Peclet number and controls the strength of the diffusion. We discuss several rigorous mathematical results on convergence and non-convergence of solutions of diffuse interface to sharp interface models in dependence of the scaling of the mobility.