Well-posedness of a diffuse-interface model for two-phase incompressible flows with thermo-induced Marangoni effect

TL;DR

This paper establishes the well-posedness of a complex diffuse-interface model for two-phase incompressible flows with temperature-dependent surface tension, viscosity, and thermal diffusivity, including local and global solutions.

Contribution

It proves existence and uniqueness of solutions for the model in both two and three dimensions, extending understanding of thermo-induced Marangoni effects in fluid dynamics.

Findings

Existence and uniqueness of local strong solutions in 2D and 3D.

Global weak solutions in 2D under temperature bounds.

Existence and uniqueness of global strong solutions in 2D.

Abstract

We investigate a non-isothermal diffuse-interface model that describes the dynamics of two-phase incompressible flows with thermo-induced Marangoni effect. The governing PDE system consists of the Navier--Stokes equations coupled with convective phase-field and energy transport equations, in which the surface tension, fluid viscosity and thermal diffusivity are temperature dependent functions. First, we establish the existence and uniqueness of local strong solutions when the spatial dimension is two and three. Then in the two dimensional case, assuming that the $L^\infty$-norm of the initial temperature is suitably bounded with respect to the coefficients of the system, we prove the existence of global weak solutions as well as the existence and uniqueness of global strong solutions.