Stability of thermal convection in couple stress liquid saturated porous slab in the presence of internal heating

In this paper, a numerical study uses linear and weakly nonlinear stability analyses to investigate the impact of internal heating on convection stability in a couple-stress liquid porous slab heated from below and cooled from above. The density variation in the slab is modeled using the Navier-Stokes equations under the Boussinesq approximation. The onset of convective instability is analyzed by linearizing the governing equations for perturbations, considering only two-dimensional perturbation stability equations. The stability of eigenvalue problem is solved to establish the principle of exchange of stability. The critical Rayleigh number at the onset of stationary instability is determined using the weighted residual technique applied to the stability eigenvalue problem. Results indicate that the couple stress parameter stabilizes the flow, whereas internal heating destabilizes the system. For weakly nonlinear theory, a nonlinear Ginzburg–Landau equation is derived using multi-scale analysis to describe the evolution of the amplitude of convection rolls near the onset of instability. This approach captures nonlinear interactions between modes and provides insight into secondary instabilities and pattern formation. Heat transport is quantified via the Nusselt number. The results show that the Darcy number stabilizes, leading to stability. These findings contribute to a deeper understanding of convection dynamics in couple-stress liquid porous systems with internal heating.