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Natural convection in a vertical slot filled with a fluid-saturated porous medium modeled by the Brinkman equation is considered. A continuum of solutions are found from capped to open ends in fully-developed two-dimensional flow. This brings to flow in a porous media some aspects of the Newtonian conduction regime results of Bühler (Heat Mass Transf 39: 631, 2003) and the convection regime results of Weidman (Heat Mass Transf 43: 103, 2006). The Brinkman model, valid for sufficiently large porosity and governed by parameter α, allows one to connect Bühler’s conduction regime flow at low α to a near-Darcy convection regime boundary-layer flow at high α. [ABSTRACT FROM AUTHOR]

Stability boundaries separating different states of bimodal convection in a box of saturated porous material with impermeable bounding faces are determined. Two opposing vertical end walls are always insulated, while the thermal conditions on the other set of opposing vertical side walls range from insulated to perfectly conducting, as measured by the Biot number based on box height. The temperature difference between the hot lower plate and the cooler upper plate provides the mechanism for instability. The eigenvalue problem for the critical Rayleigh number is solved numerically over a range of box sizes and side-wall heat transfer conditions. For small values of the distance H2 between conducting side walls, agreement with previous asymptotic analysis is obtained. New results for boxes with planform dimensions both comparable to the box height show the existence of isolated regions rich in bimodal structure. These islands are separated by broad regions having modal properties identical to those found for conducting side walls in the limit H2→0. The numerical results for O(1) box dimensions continue to exhibit the stabilizing effect of side-wall heat transfer as previously observed for thin slabs. [ABSTRACT FROM AUTHOR]

The effect of side wall heat transfer on the stability of natural convection in a vertically oriented finite slab of saturated porous material is considered. All six bounding faces are impermeable. A temperature contrast between the top and bottom horizontal surfaces provides the mechanism for destabilization. The narrow vertical end walls are perfectly insulated. The thermal conditions on the broad vertical side walls range from perfectly insulating to fully conducting as determined by the value of B, the Biot number based on slab height. An asymptotic analysis of the general solution is made in the narrow gap limit ε<1, where ε is the cross-slab width-to-height ratio. In this case the relevant heat transfer variable is B=εB, the Biot number based on the narrow slab width. In the limit ε → 0 when B=O(1), including the case B→∞ that corresponds to a linear side wall temperature profile, tall, vertical, three-dimensional, finger-like cells are found at the critical Rayleigh number Rc=π2/ε2. In the limit B → 0 corresponding to perfect insulation, one obtains two-dimensional, O(1) aspect ratio rolls with axes normal to the side walls at the critical value Rc=4π2. These two-dimensional rolls predominate only for B=O(ε2), and transition to tall narrow three-dimensional cells occurs when O(ε2)< B < O(1). [ABSTRACT FROM AUTHOR]