Viscosity dissipation and Brinkman–Bénard convection with thermal anisotropy: stability studies in both linear and nonlinear

This study presents both linear and nonlinear stability analyses of Brinkman–Bénard convection in a porous medium, considering the effects of thermal anisotropy. The flow occurs between two walls maintained at uniform but different temperatures. The critical Rayleigh number is examined, including variations in the Darcy number, porosity, Prandtl number, and anisotropic thermal conductivity, with both linear and nonlinear stability regimes analyzed. Contour plots of streamlines and isotherms are provided to visualize fluid and heat flow directions. The results demonstrate that the presence of the porous medium inhibits convection and reduces the cell size at the onset of instability. Additionally, thermal anisotropy stabilizes the system, with the region of subcritical instability shrinking as the anisotropy parameter increases. While the linear stability analysis does not reveal any significant impact of viscous dissipation, the nonlinear stability analysis shows that viscous dissipation destabilizes the system. These findings contribute to a deeper understanding of the interplay between thermal anisotropy, porosity, and convection behavior in porous media, with implications for various engineering and geophysical applications.