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Global Weak Solutions of a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase flows with Thermo-induced Marangoni Effects

Lingxi Chen, Hao Wu

Abstract

We study a diffuse-interface model that describes the dynamics of two-phase incompressible flows driven by the thermo-induced Marangoni effect. The hydrodynamic system consists of the Navier-Stokes equations for the fluid velocity, the convective Cahn-Hilliard equation for the phase-field variable, and a convective heat equation for the (relative) temperature. For the initial-boundary value problem with variable viscosity, mobility, thermal diffusivity, and a physically relevant singular potential, we establish the existence of global weak solutions in two and three dimensions. When the spatial dimension is two, we also prove the uniqueness of weak solutions for the case with matched densities under suitable assumptions on the initial temperature, mobility, and thermal diffusivity.

Global Weak Solutions of a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase flows with Thermo-induced Marangoni Effects

Abstract

We study a diffuse-interface model that describes the dynamics of two-phase incompressible flows driven by the thermo-induced Marangoni effect. The hydrodynamic system consists of the Navier-Stokes equations for the fluid velocity, the convective Cahn-Hilliard equation for the phase-field variable, and a convective heat equation for the (relative) temperature. For the initial-boundary value problem with variable viscosity, mobility, thermal diffusivity, and a physically relevant singular potential, we establish the existence of global weak solutions in two and three dimensions. When the spatial dimension is two, we also prove the uniqueness of weak solutions for the case with matched densities under suitable assumptions on the initial temperature, mobility, and thermal diffusivity.
Paper Structure (11 sections, 8 theorems, 233 equations)

This paper contains 11 sections, 8 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notation
  4. Statement of results
  5. Existence of Global Weak Solutions
  6. Preliminaries
  7. The partially implicit time-discretization scheme
  8. Existence of weak solutions to the continuous problem
  9. Uniqueness in Two Dimensions
  10. An auxiliary problem
  11. Proof of Theorem \ref{['weaksolution-2d']}

Key Result

Theorem 2.1

Let $d=2,3$. Suppose that the assumptions $\mathbf{(A0)}$--$\mathbf{(A5)}$ are satisfied. For any initial data $\boldsymbol{u}_0 \in \bm{H}_{\sigma}$, $\phi_0 \in H^{1}(\Omega)$, $\|\phi_0\|_{L^{\infty}(\Omega)} \leqslant 1$, $|\overline{\phi_0}|<1$, $\theta_0 \in L^{\infty}(\Omega)$ and the boundar where $p \in [2,\infty)$ if $d=2$, and $p=6$ if $d=3$. The solution fulfills the following weak for

Theorems & Definitions (26)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.2
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.1
  • ...and 16 more