On the existence of traveling waves in the 3D Boussinesq system

TL;DR

This paper proves the existence of non-planar traveling wave solutions in the 3D Boussinesq system within certain geometrical and physical parameter regimes, extending previous work on stratified flows and flames.

Contribution

It establishes the existence of non-planar traveling waves in the 3D Boussinesq system under specific boundary conditions and geometric constraints, including explicit conditions involving physical parameters.

Findings

Non-planar traveling waves exist in 3D Boussinesq flows with certain geometries.

Existence depends on relations between channel geometry and physical parameters like Prandtl and Rayleigh numbers.

Traveling waves can also exist when advection terms are neglected.

Abstract

We extend earlier work on traveling waves in premixed flames in a gravitationally stratified medium, subject to the Boussinesq approximation. For three-dimensional channels not aligned with the gravity direction and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected.