Degenerate onset of convection in vertical porous cylinders

Abstract

The fourth-order Darcy–Benard eigenvalue problem for the onset of thermal convection in vertical porous cylinders is investigated. With cylinder walls either impermeable/adiabatic or open/conducting, the thermomechanical fourth-order problem over the cylinder’s cross-section decouples into a second-order eigenvalue problem, so that each perturbation variable is governed by a single Helmholtz equation. The eigenvalues for the decoupled Helmholtz equation give discrete sample points on the continuous dispersion relation curve, where the smallest sampled Rayleigh number represents the convection onset. Exact analytical solutions are given for a triangular cross-section and circular geometries.