A Unified Treatment of Benard Convection in Newtonian, Non-Newtonian, Porous Media and Double Diffusive Flows

ABSTRACT

A numerical method based on the finite elements is applied to the study double diffusive convection in porous media. The problem on hand is the onset of cellular motion in a two dimensional rectangular box of aspect ratio four which is subjected to both temperature and concentration gradients vertically. Non Darcian model has been used here for which sufficiently small Darcy numbers (less than 0.0001 in this study), the results are virtually indistinguishable from Darcian flows. The non-Newtonain nature of the fluid is described by the so-called power law model, which is a generalization of the Newtonian fluid using the power law index, n. Numerical results are presented for n varying from 0.4 to 1.5. It is found that the parameter that governs the onset of cellular motion is the sum of thermal and solutal Rayleigh numbers (Ra_s + Ra_T). This value for Newtonian fluids is found to be 40 which is close to the theoretical value of 4π². An interesting feature that is found is that boundary conditions on the horizontal walls have little effect on the results, in the sense that whether it is a slip or free (no shear condition) the results appear to be the same for onset of cellular motion. It is also found that the value of critical Rayleigh number increases with the power law index.

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