Stability analysis of the perturbed rest state and of the finite amplitude steady double-diffusive convection in a shallow porous enclosure Mahmoud Mamou * Institute for Aerospace Research, National Research Council Canada, Ottawa, Ontario, Canada K1A 0R6 Received 4 March 2002; received in revised form 17 September 2002 Abstract Transition from rest and steady convective states to oscillatory flows is investigated in a shallow porous enclosure subject to vertical thermal and solutal gradients. Various combinations of Dirichlet and Neumann thermal and solutal boundary conditions is considered. The unsteady form of Hazen–Darcy law with the Boussinesq approximation is used to model the convective flow through the porous medium. The governing and perturbation equations are solved nu- merically using finite element method. The threshold of transition, which characterizes the transition from steady to oscillatory finite amplitude flows, and the threshold of overstability (Hopf bifurcation), which characterizes the tran- sition from the rest to oscillatory state, are obtained for a wide range of the governing parameters. The porosity and the acceleration parameter of the porous medium have a strong effect on the thresholds of transition and overstability. An increase in the acceleration parameter and the normalized porosity delays the onset of overstability and the transition to oscillatory finite amplitude flows. For Neumann boundary conditions type, the wavenumber is zero at the onset of overstabilities and finite at the transition threshold. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Porous layer; Overstabilities; Transition; Mixed boundary conditions; Stability analysis; Finite element method 1. Introduction Combined heat and mass transfer by natural convection in porous materials is a field of great interest for many researchers and engineers. Its applications include many diversified fields such as transport of moisture in fibrous in- sulation and contaminants in saturated soil. Underground disposal of nuclear or non-nuclear wastes, food processing, metallurgy and chemistry are also some disciplines where combined heat and solute transfer in multi-component fluids are involved. In double-diffusive convection, when the thermal and solutal buoyancy forces are comparable and opposing each other, a rich variety of phenomena can occur (here, the word solutal refers to solute or mass transfer effects). These phenomena include the possible existence of multiple steady and unsteady state solutions for the same set of governing parameters, subcritical and oscillatory flows, traveling waves and asymmetric flow patterns (see, for example, Nield [1], Taunton et al. [2], Mamou et al. [3] and Mamou and Vasseur [4]). These convective phenomena are caused by the fact that the thermal and solutal diffusivities are different from each other, which leads to different time scales for the heat and solute transfer. Therefore, the heat and solute transfers can get out of phase and cause the different convective phenomena mentioned above. The difference in time scales can also occur when the solid material of the porous medium * Tel.: +1-613-998-9953; fax: +1-613-998-1281. E-mail address: Mahmoud.Mamou@nrc-cnrc.gc.ca (M. Mamou). 0017-9310/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0017-9310(02)00523-9 International Journal of Heat and Mass Transfer 46 (2003) 2263–2277 www.elsevier.com/locate/ijhmt