Linear Stability Analysis of Mixed Convection Problem In A Porous Rectangular Duct of Finite Extent

This paper deals with mixed convection (MC) in a fluid saturating a porous rectangular duct heated from below. In particular, we study the transition towards transverse 2D rolls appearing at low Reynolds and Rayleigh numbers. We assume that the horizontal surfaces of the cavity are kept at constant and different temperatures, while the other lateral walls are supposed either adiabatic or maintained at a given constant temperature. The purpose of this paper is to study the effect of longitudinal confinement on the MC flow stability problem and to numerically highlight the effect of the existing of thermal input area before the settling of the transverse 2D rolls. We predict numerically the supercritical Rayleigh for the onset of convection, based on the linear stability analysis, using both Galerkin and finite element methods. Since the flow is confined, the aspect ratio effect was taken into account and analyzed. It is demonstrated that, the critical Rayleigh number depends on the Peclet number and the length of the domain and tends toward the classical natural convection critical Rayleigh value ad infinitum. The two cases rigid rigid (R-R) and free-free (F-F) dynamic boundary condition were compared.

The particular case of high Darcy number corresponds to the clear fluid were the problem under consideration became the classical Poiseuille Rayleigh-Benard problem (PRB). A quantitative and qualitative comparison with between of clear fluid duct and a foam of Da=10⁻⁵ is done and the entrance domain effect is also analysed.