Communications in Mathematical Sciences

Volume 11 (2013)

Number 2

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

Pages: 603 – 633

DOI: https://dx.doi.org/10.4310/CMS.2013.v11.n2.a15

Authors

Hao Wu (School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, China)

Xiang Xu (Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

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 the 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 and discuss the long-time behavior as well as the stability of the system.

Keywords

phase-field model, Navier–Stokes equations, Marangoni effects, well-posedness, long-time dynamics

2010 Mathematics Subject Classification

35K55, 35Q35, 76D05

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Published 7 December 2012