On Sharp Interface Limits for Diffuse Interface Models for Two-Phase Flows

TL;DR

This paper investigates the behavior of diffuse interface models for two-phase flows as the interface thickness parameter approaches zero, establishing conditions under which solutions converge to sharp interface models or deviate from classical laws.

Contribution

It provides a rigorous analysis of the sharp interface limit for diffuse interface models, identifying how different mobility scalings affect the limiting behavior.

Findings

Weak solutions converge to varifold solutions when mobility remains positive or decreases slowly.

Fast decay of mobility leads to solutions that do not satisfy the Young-Laplace law.

Radially symmetric solutions exhibit different limiting behaviors depending on mobility scaling.

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 \epsilon>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 \epsilon 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 \epsilon^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.