On the use and misuse of the Oberbeck-Boussinesq approximation

TL;DR

This paper reviews the Oberbeck-Boussinesq approximation, highlights common misuses that lead to incorrect conclusions, and discusses its basis as an asymptotic theory, emphasizing the importance of proper application.

Contribution

It clarifies the proper use of the Oberbeck-Boussinesq approximation and illustrates potential pitfalls through simple examples, emphasizing the need for careful application.

Findings

Misuse can violate mass conservation principles.

Proper application avoids paradoxes and incorrect conclusions.

The approximation is an asymptotic limit as the product of thermal expansion and temperature difference approaches zero.

Abstract

The Oberbeck-Boussinesq approximation is the most widely employed theoretical scheme for the study of natural or mixed convection flows. However, the misunderstanding of this approximated framework is a possibility that may cause the emergence of paradoxes or, at least, incorrect conclusions. In this note, the basic features of the Oberbeck-Boussinesq approximation are briefly recalled and three simple examples where this theoretical scheme may be misused are provided. Such misuses of the approximation lead to erroneous conclusions that, in the examples presented in this note, entail violations of the principle of mass conservation. A discussion about the Oberbeck-Boussinesq approximation as an asymptotic theory obtained by letting the product of the thermal expansion coefficient and the reference temperature difference tend to zero is also presented.