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Global Weak Solutions of a Thermodynamically Consistent Diffuse Interface Model for Nonhomogeneous Incompressible Two-phase Flows with a Soluble Surfactant

Bohan Ouyang, Maurizio Grasselli, Hao Wu

TL;DR

The study analyzes a thermodynamically consistent diffuse-interface model for two-phase incompressible flows with unmatched densities and a soluble surfactant, coupling a nonhomogeneous Navier–Stokes system to a two-field Cahn–Hilliard system with singular potentials and variable mobilities. It proves the global existence of weak solutions in two regimes: (i) nondegenerate mobilities via a semi-implicit time discretization and a Leray–Schauder fixed-point approach, and (ii) degenerate mobilities through a novel ε-regularization of mobilities and potentials followed by a careful limit passage that preserves physical bounds and energy-dissipation. The analysis leverages an energy-entropy framework and mass conservation to obtain uniform estimates and to justify the convergence to weak solutions, including a pressure reconstruction. The results extend previous nonhomogeneous AGG-type models to include a soluble surfactant, unmatched densities, and general degenerate mobilities, providing a rigorous foundation for simulations and further study of long-time behavior and regularity in these complex multiphase systems.

Abstract

We study a thermodynamically consistent diffuse interface model that describes the motion of a two-phase flow of two viscous incompressible Newtonian fluids with unmatched densities and a soluble surfactant in a bounded domain of two or three dimensions. The resulting hydrodynamic system consists of a nonhomogeneous Navier-Stokes system for the (volume averaged) velocity $\mathbf{u}$ and a coupled Cahn-Hilliard system for the phase-field variables $φ$ and $ψ$ that represent the difference in volume fractions of the binary fluids and the surfactant concentration, respectively. For the initial boundary value problem with physically relevant singular potentials subject to a no-slip boundary condition for the fluid velocity and homogeneous Neumann boundary conditions for the phase-field variables and the chemical potentials, we first establish the existence of global weak solutions in the case of non-degenerate mobilities based on a suitable semi-implicit time discretization. Next, we prove the existence of global weak solutions for a class of general degenerate mobilities, with the aid of a new type of approximations for both the mobilities and the singular parts of the potential densities.

Global Weak Solutions of a Thermodynamically Consistent Diffuse Interface Model for Nonhomogeneous Incompressible Two-phase Flows with a Soluble Surfactant

TL;DR

The study analyzes a thermodynamically consistent diffuse-interface model for two-phase incompressible flows with unmatched densities and a soluble surfactant, coupling a nonhomogeneous Navier–Stokes system to a two-field Cahn–Hilliard system with singular potentials and variable mobilities. It proves the global existence of weak solutions in two regimes: (i) nondegenerate mobilities via a semi-implicit time discretization and a Leray–Schauder fixed-point approach, and (ii) degenerate mobilities through a novel ε-regularization of mobilities and potentials followed by a careful limit passage that preserves physical bounds and energy-dissipation. The analysis leverages an energy-entropy framework and mass conservation to obtain uniform estimates and to justify the convergence to weak solutions, including a pressure reconstruction. The results extend previous nonhomogeneous AGG-type models to include a soluble surfactant, unmatched densities, and general degenerate mobilities, providing a rigorous foundation for simulations and further study of long-time behavior and regularity in these complex multiphase systems.

Abstract

We study a thermodynamically consistent diffuse interface model that describes the motion of a two-phase flow of two viscous incompressible Newtonian fluids with unmatched densities and a soluble surfactant in a bounded domain of two or three dimensions. The resulting hydrodynamic system consists of a nonhomogeneous Navier-Stokes system for the (volume averaged) velocity and a coupled Cahn-Hilliard system for the phase-field variables and that represent the difference in volume fractions of the binary fluids and the surfactant concentration, respectively. For the initial boundary value problem with physically relevant singular potentials subject to a no-slip boundary condition for the fluid velocity and homogeneous Neumann boundary conditions for the phase-field variables and the chemical potentials, we first establish the existence of global weak solutions in the case of non-degenerate mobilities based on a suitable semi-implicit time discretization. Next, we prove the existence of global weak solutions for a class of general degenerate mobilities, with the aid of a new type of approximations for both the mobilities and the singular parts of the potential densities.

Paper Structure

This paper contains 13 sections, 7 theorems, 305 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries.
  4. Basic assumptions.
  5. Statement of results.
  6. Proof of Theorem \ref{['wsnschndme']}: the Case of Non-degenerate Mobilities
  7. Implicit time discretization.
  8. Existence of a global weak solution.
  9. Proof of Theorem \ref{['wsnschdme']}: the Case of Degenerate Mobilities
  10. The approximate problem.
  11. Passage to the limit as $\epsilon\to 0$.
  12. Existence of a Pressure Field
  13. Estimate for the Approximate Potential Function

Key Result

theorem 1

Suppose that the assumptions $(\mathbf{H0})$, $(\mathbf{H1})$, $(\mathbf{H2})$ and $(\mathbf{H3})$ are satisfied. For any initial data $\mathbf{u}_0 \in \bm{L}^2_{\sigma}(\Omega)$, $\psi_0, \phi_0 \in V$ with $(\phi_0,\psi_0) \in [-1,1]\times[0,1]$ almost everywhere in $\Omega$, $\overline{\phi_0} \ then the weak solution $( \mathbf{u},\phi,\psi,\mu_{\phi},\mu_{\psi} )$ satisfies

Theorems & Definitions (29)

  • remark 1
  • remark 2
  • remark 3
  • definition 1
  • theorem 1
  • remark 4
  • definition 2
  • theorem 2
  • remark 5
  • remark 6
  • ...and 19 more