Rigorous derivation of the Oberbeck-Boussinesq approximation revealing unexpected term

TL;DR

This paper rigorously derives the Oberbeck-Boussinesq approximation for a compressible viscous heat-conducting fluid, revealing an unexpected non-local boundary condition for temperature in the asymptotic limit.

Contribution

It provides a rigorous mathematical derivation of the Oberbeck-Boussinesq system with novel non-local boundary conditions, differing from previous assumptions.

Findings

Asymptotic limit identified as Oberbeck-Boussinesq system

Revealed unexpected non-local boundary condition for temperature

Contrasts with Neumann boundary conditions case

Abstract

We consider a general compressible viscous and heat conducting fluid confined between two parallel plates and heated from the bottom. The time evolution of the fluid is described by the Navier--Stokes--Fourier system considered in the regime of low Mach and Froude numbers suitably interrelated. Surprisingly and differently to the case of Neumann boundary conditions for the temperature, the asymptotic limit is identified as the Oberbeck--Boussinesq system supplemented with non--local boundary conditions for the temperature deviation.