The Oberbeck--Boussinesq Approximation as a Constitutive Limit

TL;DR

This paper derives the Oberbeck--Boussinesq approximation as a limiting case of the full compressible viscous fluid equations by taking the thermal expansion and compressibility coefficients to zero.

Contribution

It provides a rigorous derivation of the Oberbeck--Boussinesq approximation as a constitutive limit from the full compressible fluid system.

Findings

The approximation is obtained as the limit when thermal expansion and compressibility tend to zero.

The derivation uses the Gibbs free energy formulation in variables v, θ, and p.

The approach clarifies the assumptions underlying the classical approximation.

Abstract

We derive the usual Oberbeck--Boussinesq approximation as a constitutive limit of the full system describing the motion of an compressible linearly viscous fluid. To this end the starting system is written, using the Gibbs free energy, in the variables $\mathbf v, \theta$ and $p$. The Oberbeck--Boussinesq system is then obtained as the thermal expansion coefficient $\alpha$ and the isothermal compressibility coefficient $\beta$ tend to zero.