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A unified framework for N-phase Navier-Stokes Cahn-Hilliard Allen-Cahn mixture models with non-matching densities

M. F. P. ten Eikelder

TL;DR

The paper addresses the challenge of unifying $N$-phase diffuse-interface flows with non-matching densities by deriving a single momentum equation coupled to $N$ constituent mass balances from continuum mixture theory. It presents an energy-dissipative framework that is invariant to the choice of fundamental variables and provides constitutive closures for stress, peculiar velocities, diffusion fluxes, and mass transfer via mobility matrices and a Lagrange multiplier $\lambda$, ensuring reduction-consistency. Key contributions include establishing equivalence and connections to existing binary and $N$-phase models (e.g., Dong 2018 and Class-II), proving energy-dissipation compatibility, and outlining equilibrium structure and reformulations in pressure and free-energy form. The framework offers a principled pathway to design, analyze, and numerically simulate $N$-phase diffuse-interface flows with non-matching densities, enabling systematic comparisons and robust numerical schemes across multiple phase configurations.

Abstract

Over the past few decades, numerous N-phase incompressible diffuse-interface flow models with non-matching densities have been proposed. Despite aiming to describe the same physics, these models are generally distinct, and an overarching modeling framework is absent. This paper provides a unified framework for N-phase incompressible Navier-Stokes Cahn-Hilliard Allen-Cahn mixture models with a single momentum equation. The framework naturally emerges from continuum mixture theory, exhibits an energy-dissipative structure, and is invariant to the choice of fundamental variables. This opens the door to exploring connections between existing N-phase models and facilitates the computation of N-phase flow models rooted in continuum mixture theory.

A unified framework for N-phase Navier-Stokes Cahn-Hilliard Allen-Cahn mixture models with non-matching densities

TL;DR

The paper addresses the challenge of unifying -phase diffuse-interface flows with non-matching densities by deriving a single momentum equation coupled to constituent mass balances from continuum mixture theory. It presents an energy-dissipative framework that is invariant to the choice of fundamental variables and provides constitutive closures for stress, peculiar velocities, diffusion fluxes, and mass transfer via mobility matrices and a Lagrange multiplier , ensuring reduction-consistency. Key contributions include establishing equivalence and connections to existing binary and -phase models (e.g., Dong 2018 and Class-II), proving energy-dissipation compatibility, and outlining equilibrium structure and reformulations in pressure and free-energy form. The framework offers a principled pathway to design, analyze, and numerically simulate -phase diffuse-interface flows with non-matching densities, enabling systematic comparisons and robust numerical schemes across multiple phase configurations.

Abstract

Over the past few decades, numerous N-phase incompressible diffuse-interface flow models with non-matching densities have been proposed. Despite aiming to describe the same physics, these models are generally distinct, and an overarching modeling framework is absent. This paper provides a unified framework for N-phase incompressible Navier-Stokes Cahn-Hilliard Allen-Cahn mixture models with a single momentum equation. The framework naturally emerges from continuum mixture theory, exhibits an energy-dissipative structure, and is invariant to the choice of fundamental variables. This opens the door to exploring connections between existing N-phase models and facilitates the computation of N-phase flow models rooted in continuum mixture theory.
Paper Structure (27 sections, 20 theorems, 155 equations, 2 figures)

This paper contains 27 sections, 20 theorems, 155 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Objective and main results
  4. Plan of the paper
  5. Continuum mixture theory
  6. Preliminaries
  7. Constituent balance laws
  8. Mixture balance laws
  9. Constitutive modeling
  10. Assumptions and modeling choices
  11. Modeling restriction
  12. Alternative free energy classes
  13. Selection of constitutive models
  14. Model characteristics
  15. Alternative formulations
  16. ...and 12 more sections

Key Result

Lemma 2.4

[lemma]lem: mass av vs vol av The mass-averaged and volume-averaged velocity variables are related via:

Figures (2)

  • Figure 1: Invariance of the unified framework, both at the level of balance laws (Bal. Laws) and, after closure, at the level of mixture models (Mix. Model).
  • Figure 2: Situation sketch continuum mixture theory.

Theorems & Definitions (48)

  • Remark 2.1: Incompressibility $N$-phase model
  • Remark 2.2: Alternative definitions incompressible mixtures
  • Remark 2.3: Terminology peculiar velocities
  • Lemma 2.4: Relation mass-averaged and volume-averaged velocities
  • proof
  • Lemma 2.5: Relation scaled peculiar velocities
  • proof
  • Remark 3.1: Classification
  • Remark 3.2: Energy-dissipation postulate
  • Remark 3.3: Reduced free energy class
  • ...and 38 more