Natural convection in the horizontal annulus: critical Rayleigh number for the steady problem

TL;DR

This paper determines the critical Rayleigh number at which steady natural convection becomes unstable in a 2D annular domain, using a combination of analytical and numerical methods.

Contribution

It introduces a precise functional analytical framework and a numerical scheme to compute the critical Rayleigh number for the stability analysis of natural convection in an annulus.

Findings

Calculated the critical Rayleigh number for the annular domain.

Developed a new analytical framework for stability analysis.

Provided numerical results for the stability threshold.

Abstract

For the 2D Oberbeck-Boussinesq system in an annulus we are looking for the critical Rayleigh number for which the (nonzero) basic flow loses stability. For this we consider the corresponding Euler-Lagrange equations and construct a precise functional analytical frame for the Laplace- and the Stokes problem as well as the Bilaplacian operator in this domain. With this frame and the right set of basis functions it is then possible to construct and apply a numerical scheme providing the critical Rayleigh number.