Influence of vertical throughflow on the linear and nonlinear stability analyses of Rayleigh–Bénard convection in a biviscous Bingham fluid saturating a porous medium

This article aims to investigate the influence of vertical throughflow on the stability analysis of a biviscous Bingham fluid-saturated horizontal porous layer. Specifically, both linear and nonlinear stability thresholds are examined. The Darcy–Brinkman law is employed to formulate the momentum equation for the system. In this study, all three types of boundary conditions are considered: rigid-rigid, rigid-free, and free-free. The well-known energy method is applied to conduct the nonlinear stability analysis, while the linear stability analysis is carried out using the normal mode approach. The resultant eigenvalue problems are solved using the bvp4c-scheme in MATLAB 2022(a). The critical Rayleigh number and the corresponding wave numbers are obtained numerically by minimizing the neutral stability curves for both theories, and are calculated for the specified values of the flow-governing parameters, with the results presented graphically. It is observed that an increase in the Péclet number (vertical throughflow) delays the onset of convection, whereas an increase in the biviscous Bingham fluid parameter accelerates the onset of convection.