Traveling wave solutions to a general incompressible Navier-Stokes-Fourier system with free boundary

TL;DR

This paper investigates traveling wave solutions in a generalized, heat-conducting, incompressible Navier-Stokes system with a free boundary, incorporating temperature-dependent properties and external forces, establishing well-posedness under small data conditions.

Contribution

It introduces a novel analysis of traveling wave solutions for a generalized Navier-Stokes-Fourier system with free boundary and temperature-dependent coefficients, including Marangoni effects.

Findings

Existence of unique solutions under small data assumptions

Development of a Sobolev space well-posedness theory

Inclusion of temperature-dependent viscosity and capillarity effects

Abstract

We study traveling wave solutions to the free boundary problem associated to a generalized Navier-Stokes Fourier system, which models a viscous, incompressible, heat-conducting fluid. The fluid is assumed to occupy a horizontally infinite strip-like domain with flat rigid bottom and moving upper surface. The fluid is acted upon by gravity as well as external sources of bulk force and boundary stress and an external heat source. Additionally, we allow for temperature-dependent viscosity and capillary coefficients, the latter of which gives rise to Marangoni stresses on the free surface. We develop a small data well-posedness theory in Sobolev spaces that shows that if the sources of force, stress, and heat are small, then there exists a unique solution depending continuously on these data.