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

Helmut Abels, Harald Garcke, Julia Wittmann

TL;DR

The paper develops and analyzes two thermodynamically consistent diffuse-interface models for two-phase, incompressible flows with unmatched densities and phase transitions. Model I employs a volume-averaged velocity, coupling a Navier–Stokes-like momentum equation to a Cahn–Hilliard equation with source terms, and is proven to admit global weak solutions under singular free energies; Model II studies a quasi-stationary, mass-averaged (LT) variant, proving weak solvability for the resulting Stokes–Cahn–Hilliard system with a modified chemical potential $\omega$. The authors derive the models from continuum thermodynamics, establish energy-dissipation inequalities, and use implicit time discretization with Leray–Schauder fixed-point arguments to obtain convergence to time-continuous weak solutions. A reformulation to the mass-averaged velocity framework clarifies the coupling and dissipation structure, with separate weak-solution results for the quasi-stationary regime. Overall, the work provides rigorous existence theory for phase-transition-enhanced diffuse-interface models with density contrasts, laying a solid analytic foundation for further study and numerical approximation.

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 Grün, 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.

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

TL;DR

The paper develops and analyzes two thermodynamically consistent diffuse-interface models for two-phase, incompressible flows with unmatched densities and phase transitions. Model I employs a volume-averaged velocity, coupling a Navier–Stokes-like momentum equation to a Cahn–Hilliard equation with source terms, and is proven to admit global weak solutions under singular free energies; Model II studies a quasi-stationary, mass-averaged (LT) variant, proving weak solvability for the resulting Stokes–Cahn–Hilliard system with a modified chemical potential . The authors derive the models from continuum thermodynamics, establish energy-dissipation inequalities, and use implicit time discretization with Leray–Schauder fixed-point arguments to obtain convergence to time-continuous weak solutions. A reformulation to the mass-averaged velocity framework clarifies the coupling and dissipation structure, with separate weak-solution results for the quasi-stationary regime. Overall, the work provides rigorous existence theory for phase-transition-enhanced diffuse-interface models with density contrasts, laying a solid analytic foundation for further study and numerical approximation.

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 Grün, 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.
Paper Structure (16 sections, 11 theorems, 227 equations)

This paper contains 16 sections, 11 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. General Assumptions.
  5. Preliminary Results
  6. Derivation of a Model with Volume Averaged Velocity
  7. Existence of Weak Solutions to the Model with Volume Averaged Velocity
  8. Definition of Weak Solutions
  9. Implicit Time Discretization
  10. Proof of Theorem \ref{['thm:AGG']}
  11. Energy Inequality
  12. Reformulation of the Model with Mass Averaged Velocity
  13. Existence of Weak Solutions to the Quasi-Stationary Model with Mass Averaged Velocity
  14. Definition of Weak Solutions
  15. Implicit Time Discretization
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\mathbf{v}_0\in \mathrm L^2(\Omega)^d$ and $\varphi_0\in \mathrm H^1(\Omega)$ with $| \varphi_0 |\leq1$ a.e. in $\Omega$ and $\langle \varphi_0 \rangle \in (-1,1)$. Under the assumptions ass:domain--ass:coeffs below, there exists a weak solution $(\mathbf{v},\lambda,\mu,\varphi)$ of eq:AGG in t

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Definition 4.1
  • Remark 4.2
  • Definition 4.3
  • ...and 20 more