CONVECTIVE FLOW OF A BINGHAM FLUID IN INTERNALLY HEATED POROUS CAVITIES

We study the onset of convection within a rectangular porous cavity which is saturated with a Bingham fluid and subjected to a uniform internal heat generation. When such a cavity is saturated by a Newtonian fluid, convection takes place at all nonzero values of the Darcy–Rayleigh number, Ra. In such cases convection takes the form of two contrarotating cells with flow down the cold sidewalls when Ra first increases from zero. However, when the enclosure is saturated by a Bingham fluid, then we find that the cavity remains stagnant until the Darcy–Rayleigh number is sufficiently large that buoyancy overcomes the yield threshold. Numerical solutions are obtained using a second-order accurate finite-difference methodology where convergence is accelerated using line relaxation. The presence of the yield surfaces, which mark the boundaries of stagnant regions, is modelled by means of a regularization of the yield threshold. It is found that the critical value of Ra above which convection arises depends roughly linearly on the value of Rb, which may be described as a convective porous Bingham number. When the cavity has a sufficiently large aspect ratio, the fluid admits more than one stable steady-state solution.