On existence and uniqueness of solutions to a Boussinesq system with nonlinear and mixed boundary conditions

TL;DR

This paper investigates the existence and uniqueness of solutions to a Boussinesq system with complex boundary conditions, introducing a new artificial heat transfer condition and demonstrating its advantages through numerical tests.

Contribution

It introduces a novel artificial boundary condition for heat transfer in Boussinesq systems and proves solution existence and uniqueness under this framework.

Findings

Proved existence and uniqueness of weak solutions.

Demonstrated improved numerical performance with the new boundary condition.

Validated the approach through numerical experiments.

Abstract

We study a Boussinesq system in a bounded domain with an outlet boundary portion where fluid can leave or re-enter. On this boundary part, we consider a do-nothing condition for the fluid flow, and a new artificial condition for the heat transfer that couples nonlinearly the fluid velocity and temperature. The latter can be further adjusted if convective or conductive phenomena are dominant. We prove existence and, in some cases, uniqueness of weak solutions to stationary and evolutionary problems by a fixed point strategy under suitable assumptions on the data. A variety of numerical tests shows the improved performance of the new artificial condition with respect to other standard choices in the literature.