Non-linear stability analysis of a Darcy flow with viscous dissipation

The nonlinear convective instability of a flow in a fluid saturated impermeable and rectangular porous channel of arbitrary aspect ratio is here investigated by taking into account the effect of viscous dis- sipation. Darcy’s law and Oberbeck-Boussinesq approximation are assumed. The vertical boundaries are assumed to be adiabatic and the horizontal boundaries are taken to be isothermal with the cold face placed on top. The system is characterised by two sources of thermal instability: the buoyancy activated by the non trivial temperature distribution due to the internal heat generation by the viscous dissipation and the buoyancy triggered by the non linear temperature distribution due to the temper- ature gap between the horizontal boundaries. The novel feature introduced in the present paper is the fully nonlinear approach to the stability analysis. The results obtained by the linear stability analysis are here used as a reference. The purpose of this paper is to analyse the system with the aim of finding possible subcritical instabilities. The technique employed in order to investigate the nonlinear prob- lem is the generalized integral transform technique. The computational task relative to the integral transformation procedure and the solution of the ordinary differential equations obtained are carried out by Mathematica 9