Comparison of the Results of Steady Darcy-Be ́nard Convection Problems of the Classical and the Barletta Types

The linear stability analysis of the Barletta-Darcy-Be ́nard convection problem in a horizontal fluid-saturated porous layer is extended to a weakly nonlinear stability analysis considering local thermal equilibrium (LTE) between the fluid and solid phases. The minimal Fourier-Galerkin expansion is used for the case of a free upper surface (Neumann boundary condition on the stream function) along with isothermal boundary condition for which heat transport is quantified in terms of the Nusselt number. The present article aims to fill the literature gap between the linear and non-linear stability analyses of classical Darcy-Be ́nard convection and of Barletta-Darcy-Be ́nard convection. Weakly non-linear stability analysis has not been performed in the case of the non-classical Darcy-Be ́nard convection problem. A comparison of results of the present problem with those of the classical Darcy-Be ́nard convection problem is made. It is found that the cell size is larger in the case of the former problem compared to the latter. The critical Darcy-Rayleigh number, however is smaller in the former one. The Nusselt number varies inversely as the Rayleigh number, R and hence the Nusselt number increases with decrease in R which implies that more heat is transported in Barletta-Darcy-Be ́nard convection compared to classical Darcy-Be ́nard convection.