Maximum density perspectives on the stability of Brinkman porous convection in a vertical channel

Abstract

The stability of parallel buoyant flow is studied in a vertical layer of Brinkman porous medium possessing a density maximum in its interior with quadratic density law. The vertical boundaries are considered to be rigid and maintained at different uniform temperatures. The similarities and differences between the linear and quadratic density-temperature relationships on the base flow, stability features of fluid flow, and the mode of instability are underlined. If the flow is governed by classical Darcy's law, then it is proved analytically that the base flow remains unconditionally stable irrespective of how the density varies with temperature. On the contrary, the change in the density-temperature relationship affects the stability of base flow when the flow is governed by either unsteady Darcy's law or the Brinkman-extended Darcy law and for these cases, the eigenvalue problem is solved numerically. For the unsteady Darcy flow model, instability occurs through the traveling-wave mode in a certain range of Darcy–Prandtl number for the quadratic density law yet the base flow is always stable if the density law is linear. The nature of the density-temperature relationship and the magnitude of governing parameters combine together in influencing the instability of fluid flow in the Brinkman regime. The possibility of otherwise stabilization of the base flow becoming unstable with the change in the density-temperature relationship from linear to quadratic for the same values of governing parameters is identified. These intricacies are discussed in detail by computing the critical stability parameters for different values of the Darcy–Prandtl number and the Darcy number.