A Unified Approach to Two-Dimensional Brinkman-Bénard Convection of Newtonian Liquids in Cylindrical and Rectangular Enclosures

A unified model for the analysis of two-dimensional Brinkman–Bénard/Rayleigh–Bénard/ Darcy–Bénard convection in cylindrical and rectangular enclosures ((Formula presented.)) saturated by a Newtonian liquid is presented by adopting the local thermal non-equilibrium ((Formula presented.)) model for the heat transfer between fluid and solid phases. The actual thermophysical properties of water and porous media are used. The range of permissible values for all the parameters is calculated and used in the analysis. The result of the local thermal equilibrium ((Formula presented.)) model is obtained as a particular case of the (Formula presented.) model through the use of asymptotic analyses. The critical value of the Rayleigh number at which the entropy generates in the system is reported in the study. The analytical expression for the number of Bénard cells formed in the system at the onset of convection as a function of the aspect ratio, (Formula presented.), and parameters appearing in the problem is obtained. For a given value of (Formula presented.) it was found that in comparison with the case of (Formula presented.), more number of cells manifest in the case of (Formula presented.). Likewise, smaller cells form in the (Formula presented.) problem when compared with the corresponding problem of (Formula presented.). In the case of (Formula presented.), fewer cells form when compared to that in the case of (Formula presented.) and (Formula presented.). The above findings are true in both (Formula presented.) and (Formula presented.). In other words, the presence of a porous medium results in the production of less entropy in the system, or a more significant number of cells represents the case of less entropy production in the system. For small and finite (Formula presented.), the appearance of the first cell differs in the (Formula presented.) and (Formula presented.) problems.