On sharp interface limits for diffuse interface models for two-phase flows

Abstract

We discuss the sharp interface limit of a diffuse interface model for a two-phase flow of two partly miscible viscous Newtonian fluids of different densities, when a certain parameter $ϵ>0$ related to the interface thickness tends to zero. In the case that the mobility stays positive or tends to zero slower than linearly in $ϵ$ we will prove that weak solutions tend to varifold solutions of a corresponding sharp interface model. But, if the mobility tends to zero faster than ${ϵ}^3$ we will show that certain radially symmetric solutions tend to functions, which will not satisfy the Young-Laplace law at the interface in the limit.