Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

TL;DR

This paper introduces a thermodynamically consistent diffuse interface model for incompressible two-phase flows with different densities, deriving sharp interface limits and ensuring energy dissipation and frame indifference.

Contribution

It presents a novel diffuse interface model based on continuum mechanics that accounts for different densities and soluble species, with rigorous derivation of sharp interface limits.

Findings

Model satisfies local and global dissipation inequalities

Derivation of classical and new sharp interface models

All models fulfill natural energy inequalities

Abstract

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.