A unified framework for Navier-Stokes Cahn-Hilliard models with non-matching densities

TL;DR

This paper unifies various diffuse-interface Navier-Stokes Cahn-Hilliard models with non-matching densities into a single framework based on continuum mixture theory, energy laws, and constitutive choices, clarifying their relationships.

Contribution

It introduces a comprehensive unified framework for non-matching density Navier-Stokes Cahn-Hilliard models, based on fundamental principles, consolidating existing models and guiding future developments.

Findings

Single system of balance laws for all models

Unified energy-dissipation law derived

Clarifies differences through constitutive choices

Abstract

Over the last decades, many diffuse-interface Navier-Stokes Cahn-Hilliard models with non-matching densities have appeared in the literature. These models claim to describe the same physical phenomena, yet they are distinct from one another. The overarching objective of this work is to bring all of these models together by laying down a unified framework of Navier-Stokes Cahn-Hilliard models with non-zero mass fluxes. Our development is based on three unifying principles: (1) there is only one system of balance laws based on continuum mixture theory that describes the physical model, (2) there is only one natural energy-dissipation law that leads to quasi-incompressible Navier-Stokes Cahn-Hilliard models, (3) variations between the models only appear in the constitutive choices. The framework presented in this work now completes the fundamental exploration of alternate non-matching density Navier-Stokes Cahn-Hilliard models that utilize a single momentum equation for the mixture velocity, but leaves open room for further sophistication in the energy functional and constitutive dependence.