The Boussinesq approximation for buoyant flows

TL;DR

This paper provides a simplified, rigorous derivation of the Boussinesq approximation equations for buoyant flows, emphasizing a local balance approach and considering viscous dissipation effects.

Contribution

It introduces a simplified asymptotic analysis method for deriving Boussinesq equations applicable to general fluids, including non-Newtonian, with detailed consideration of viscous dissipation.

Findings

Derivation of Boussinesq equations using local balance equations.

Inclusion of viscous dissipation effects in the approximation.

Applicability to general, non-Newtonian fluids.

Abstract

The aim of this communication is to present a simplified, yet rigorous, deduction of the Boussinesq approximated governing equations for buoyant flows. In order to carry out the core deduction procedure, a simplified version of the manifold asymptotic analyses available in the literature is discussed. The method adopted in this study is focussed on the local balance equations valid for a general, not necessarily Newtonian, fluid. The analysis is carried out by demonstrating the leading order terms in the governing equations for the asymptotic limit which characterises the approximation. The role played by the effect of viscous dissipation is also taken into account.