Onset of Darcy–Brinkman convection with thermal anisotropy in an inclined porous layer

In this article, we study the linear instability and the nonlinear stability (through energy functional) analyses of thermal convection in an inclined Darcy–Brinkman porous layer considering uniformly heated horizontal rigid, impermeable walls from below and above. The effects of a uniform internal heat source and anisotropy in effective thermal diffusivity on heat transfer are also considered. Heating the porous layer from below yields the temperature gradient, influencing the buoyancy and making the convection happen. This temperature gradient also impacts the base state. The basic solution for velocity incorporates both hyperbolic and polynomial functions, significantly increasing the complexity of linear and nonlinear analyses. The Chebyshev-tau method, together with the QZ algorithm, is used to solve the linear and nonlinear perturbed system of equations numerically. The region of subcritical instability is obtained by comparing the linear and nonlinear thresholds for the longitudinal and transverse rolls, respectively. We found that perturbations for longitudinal and transverse rolls do not grow after inclination is more than 30.3° and 31.3°, respectively. It has been noted that in transverse roll scenarios, the flow becomes stabilized when the inclination angle, ϕ, is equal to or exceeds 60°, where ϕ plays a leading role in surpassing the impact of internal heating. However, when the inclination angle is ϕ<60°, then internal heating dominates and destabilizes the flow. For the longitudinal rolls, the internal heating dominates the whole range of ϕ, destabilizing the system. Furthermore, it can be seen that the Darcy number (Da) and the anisotropic thermal diffusivity (ξ) delay the onset of convection.