Diffuse Interface Models for Two-Phase Flows with Phase Transition: Modeling and Existence of Weak Solutions

TL;DR

This paper develops and analyzes diffuse interface models for two-phase flows with phase transition, proving the existence of weak solutions for both dynamic and quasi-stationary systems, advancing the mathematical understanding of such complex fluid interactions.

Contribution

It introduces two new quasi-incompressible diffuse interface models with singular free energies, extending existing models and providing rigorous proofs of weak solution existence.

Findings

Existence of weak solutions for the Navier--Stokes/Cahn--Hilliard system with source terms.

Existence of weak solutions for the quasi-stationary Stokes/Cahn--Hilliard system with phase transition.

Generalization of previous models to include phase transition and unmatched densities.

Abstract

The flow of two macroscopically immiscible, viscous, incompressible fluids with unmatched densities is studied, where a transfer of mass between the constituents by phase transition is taken into account. To this end, two quasi-incompressible diffuse interface models with singular free energies are analyzed, differing primarily in their velocity averaging. Firstly, to generalize a model by Abels, Garcke, and Gr\"un, a thermodynamically consistent system of Navier--Stokes/Cahn--Hilliard type with source terms is derived in a framework of continuum fluid dynamics, followed by a proof of existence of weak solutions to the latter. Secondly, the quasi-stationary version of a model by Aki, Dreyer, Giesselmann, and Kraus is investigated analytically, with existence of weak solutions being established for the resulting quasi-stationary Stokes system coupled to a Cahn--Hilliard equation with a source term.