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CONVECTIVE FLOW OF A BINGHAM FLUID IN AN INTERNALLY-HEATED POROUS ENCLOSURE

DOI: 10.1615/ICHMT.2017.CHT-7.1590
pages 1517-1524

D. Andrew S. Rees
Department of Mechanical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

Résumé

A rectangular porous cavity is saturated with a Bingham fluid and is subjected to a uniform internal heat generation while all the bounding surfaces are held at the same fixed temperature. When the porous medium is saturated by a Newtonian fluid then convection takes place at all nonzero values of the Darcy-Rayleigh number, Ra, and at moderate values of Ra, convection takes the form of two contra-rotating cells with flow down the cold sidewalls. However, when the enclosure is saturated by a Bingham fluid, then we find that no flow takes place when the Darcy-Rayleigh number is below a critical value because buoyancy forces are too weak to overcome the yield threshold.
Numerical solutions are obtained using a second order accurate finite difference methodology where convergence is accelerated using line-relaxation and the Full Approximation Scheme multigrid method. The presence of the yield surfaces, which mark the boundaries of stagnant regions, is modelled by means of a regularisation of the yield threshold.
It is found that the critical value of Ra above which convection arises depends linearly on the value of Rb, which may be described as a convective porous Bingham number. It is also found that, as Ra increases, the proportion of the cavity which is not stagnant also increases, but stagnant regions always exist and are found within the middle of the two circulations, near the corners of the cavity and on the horizontal boundaries near the middle of the cavity.

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