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

TL;DR

This work analyzes the onset of instability for the 2D Oberbeck-Boussinesq flow in a horizontal annulus by casting the stability threshold as a variational problem and then as a compact self-adjoint eigenproblem. A spectral framework based on Laplace, Stokes, and Bilaplacian eigenfunctions is developed, enabling both theoretical existence results (via Euler–Lagrange equations) and a stable numerical scheme. The critical Rayleigh number ${\mathrm Ra}_c$ is obtained from the largest eigenvalue of a compact operator, with finite-dimensional approximations computed through determinant-based methods, and alternative stream-function formulations provided. Numerical results illustrate convergence for selected gap-parameters $\mathcal{A}$ and demonstrate the practicality of the spectral approach in predicting the convection onset in annular geometries. The findings offer a rigorous, spectrum-based pathway for analyzing 2D and informing 3D stability considerations in narrow- and wide-gap annular flows.

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.

Figures (1)

• Figure 1: Twodimensional flow region -- temperature boundary values $T_i>T_o$.

Theorems & Definitions (19)

• Lemma 1
• Theorem 1
• Lemma 2
• Lemma 3
• Theorem 2
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Theorem 3