Linear stability of buoyant convection in horizontally heated porous layers

This paper investigates the onset and nature of natural convection instabilities within elongated cavities filled with a Newtonian, incompressible fluid saturating a homogeneous and isotropic porous medium. The study begins by deriving an exact analytical solution for the base flow, which takes the form of a one-dimensional motion between two horizontal, thermally conducting parallel plates. To capture inertial effects beyond the Darcy regime, the base flow is subsequently extended through numerical simulations incorporating the nonlinear Forchheimer correction. All analyses are carried out under the assumption of an effective Prandtl number equal to unity (Pr=1) and perfectly conducting thermal boundary conditions. A comprehensive linear stability analysis is then performed to assess how the porous medium characteristics, namely, the Darcy number Da, porosity ε, and the Forchheimer drag coefficient CF, influence the emergence and nature of flow instabilities. The results reveal a rich spectrum of convective modes, comprising both stationary (ST-2D and SL-3D) and oscillatory (OT-2D and OL-3D) behaviors. The existence and structure of these disturbances are shown to depend sensitively on the interplay between diffusive, inertial, and buoyant effects modulated by the governing parameters Da, ε, and CF. To elucidate the physical mechanisms underlying the bifurcations, an energy budget analysis is conducted at the critical thresholds. This reveals that the instabilities may be driven by shear, buoyancy, or a synergistic combination of both, depending on the mode and parameter regime. Altogether, the findings provide new insight into the intricate dynamics of convective flows in porous media, with implications for both fundamental science and a variety of practical applications.