Nonlinear steady Darcy-Bénard convection problem: Revisit using the heatlines approach

Abstract

The classical problem of Darcy-Bénard convection($DBC$) in enclosures is revisited using the method of heatlines to have a better perspective of the problem. General aspect ratio is chosen in the analysis which helps in obtaining the results of four different types of enclosures, viz., tall, square, shallow and very shallow. Three different water saturated porous media$\left(WSPM\right)$ and their actual thermophysical properties are used in the computation of the results. The method of heatlines facilitates the observation of fluid and heat flow lines in order to have a good understanding of the dynamics. The neo-classical approach not only accurately predicts the critical Darcy-Rayleigh and wave numbers but also picturizes the heat flow of the problem in the most natural way. The Galerkin method is used in the paper for the normal and convective modes of convection yields accurate analytical results in the heatlines formulation. Theoretical expression to calculate the number of Bénard cells that form in the system at onset is obtained by linear theory, and ranges of aspect ratio at which unicellular, two-cellular and multicellular convection are possible are determined and documented. The weakly non-linear stability analysis is performed to determine the heat transport. Among four considered enclosures, maximum heat transport is achieved in the case of a square enclosure. Out of three chosen $WSPM$, the water-saturated glass balls porous medium and the water-saturated aluminium-foam porous medium show most stable and least stable behaviours. Results obtained from the heatlines approach are validated by comparing with the results of the classical $DBC$ problem in the case of a very shallow enclosure. From the study, we conclude that the square enclosure with water-saturated aluminium-foam porous medium has possible application in heat removal systems.