Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced Marangoni effects

TL;DR

This paper investigates the mathematical properties and long-term behavior of a diffuse-interface model describing two viscous incompressible fluids with temperature-driven Marangoni effects, including existence and stability of solutions.

Contribution

It establishes the well-posedness and long-time dynamics of a coupled Navier-Stokes, phase-field, and energy system with thermo-induced Marangoni effects.

Findings

Energy inequality demonstrating dissipative nature

Existence of weak and strong solutions

Analysis of stability and long-time behavior

Abstract

In this paper we study the well-posedness and long-time dynamics of a diffuse-interface model for the mixture of two viscous incompressible Newtonian fluids with thermo-induced Marangoni effects. The governing system consists of modified Navier--Stokes equations coupled with phase-field and energy transport equations. We first derive an energy inequality that illustrates the dissipative nature of the system under the assumption that the initial temperature variation is properly small. Then we establish the existence of weak/strong solutions via the energy method and discuss the long-time dynamics as well as stability of the system.