Stability bounds in a problem of convection with uniform internal heat source

TL;DR

This paper provides the first rigorous linear stability analysis of convection with uniform internal heat sources, revealing how such heat sources destabilize the system, using Fourier series methods.

Contribution

It introduces a novel linear stability framework for convection problems with internal heat sources, previously only studied numerically or experimentally.

Findings

Heat sources have a destabilizing effect on convection stability.

First rigorous linear analysis of this convection problem.

Numerical results support the destabilizing influence.

Abstract

Motion in the atmosphere or mantle convection are two among phenomena of natural convection induced by internal heat sources. They bifurcate from the conduction state as a result of its loss of stability. In spite of their importance, due to the occurrence of variable coefficients in the nonlinear partial differential equations governing the evolution of the perturbations around the basic equilibrium, so far these phenomena were treated mostly numerically and experimentally. No rigorous study is known. In this paper we realize for the first time such a linear study for the eigenvalue problem associated with those equations for a convection problem with an uniform internal heat source in a horizontal fluid layer bounded by two rigid walls. Our method uses Fourier series expansions for the unknown functions. Numerical results and graphs are given showing a destabilizing effect of the presence of the heat source.