Convective Instability in a Darcy Flow Heated from Below with Internal Heat Generation

Abstract

The linear stability of a Darcy flow through an infinitely wide horizontal channel is here investigated. In particular, the paper is focused on the onset of thermal convection through a convective instability. The Oberbeck–Boussinesq approximation is employed to model the buoyancy term inside Darcy’s law. The channel is impermeable and heated from below by an isoflux condition. A uniform internal heat source is imposed. A steady solution is found and used as basic state for the stability analysis. This basic state is composed of a pressure gradient term and a buoyancy force term. The rotation symmetry around the vertical axis allows an important simplification: the two-dimensional case is treated here instead of the full three-dimensional case without any loss of generality. The normal modes method is employed to perform the linear stability analysis. The resulting eigenvalue problem is solved analytically for vanishing wavenumbers and numerically otherwise. Analytical solutions are used to validate the numerical procedure. Critical Rayleigh numbers for the onset of thermal convection prove that the most unstable modes are longitudinal. Temporal growth rates for longitudinal modes are calculated under slightly supercritical conditions to identify the fundamental characteristics of the most dominant mode which will be observed in experimental or numerical simulations under the same parametric conditions.