Theoretical Prediction of the Number of Bénard Cells in Low-Porosity Cylindrical/Rectangular Enclosures Saturated by a Fast Chemically Reacting Fluid

Many applications including chemical engineering and meteorology require the study of a chemically driven convection in cylindrical, as well as rectangular enclosures. The present paper reports a unified analysis of a chemically driven convection in densely packed porous cylindrical/rectangular enclosures saturated by a chemically reactive binary fluid mixture. Employing the degeneracy technique and the single-term Galerkin method involving Bessel functions in a linear stability analysis, an analytical expression for the critical Rayleigh number, (Formula presented.), was obtained. An analytical expression for the number of cells that manifest in a given enclosure, at the onset of convection, was derived from (Formula presented.). The connection between the stabilizing and destabilizing effects of various parameters and the size or the number of Bénard cells that manifest are described in detail. The results depicted that the chemical parameters related to the heat of reaction destabilize and the parameter depending inversely on the rate of the chemical reaction stabilizes the system. In the latter case, a greater number of smaller cells were formed in the system compared to the former case. Hence, we concluded that the chemically reactive fluid advances the onset of convection compared to the chemically non-reactive fluid. The results of a similar problem in rectangular enclosures of infinite horizontal extent and chemically non-reactive liquid-saturated porous medium were recovered as limiting cases. Thus, the present model presents a unified analysis of six individual problems.