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

TL;DR

This paper establishes the existence of global weak solutions for a complex Navier-Stokes-Cahn-Hilliard system modeling two-phase flows with thermo-induced Marangoni effects, including uniqueness in 2D under certain conditions.

Contribution

It proves the existence of global weak solutions for a thermo-induced two-phase flow model with variable parameters and singular potentials, advancing mathematical understanding of such systems.

Findings

Existence of global weak solutions in 2D and 3D.

Uniqueness of solutions in 2D with matched densities.

Handling of singular potentials and variable coefficients.

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.