On the Rayleigh--B\' enard convection problem for rotating fluids

TL;DR

This paper demonstrates that the Oberbeck--Boussinesq approximation cannot be derived as a singular limit of the Navier--Stokes--Fourier system in rotating fluids, challenging assumptions in thermally driven fluid models.

Contribution

It provides a rigorous analysis showing the non-existence of the Oberbeck--Boussinesq approximation as a singular limit in rotating coordinate systems.

Findings

The standard Oberbeck--Boussinesq approximation cannot be obtained from Navier--Stokes--Fourier equations in rotation.

The buoyancy force in rotating fluids involves both gravitational and centrifugal forces.

The result impacts modeling approaches for thermally driven rotating fluid systems.

Abstract

In contrast with a large variety of conventional models of thermally driven fluids, we show that the standard Oberbeck--Boussinesq approximation \emph{cannot} be obtained as a singular limit of the Navier--Stokes--Fourier system in the rotational coordinate system, with the buoyancy force proportional to the sum of the gravitational and centrifugal forces multiplied by the temperature variation.