Convection Dynamics in a Brinkman Bidisperse Porous Medium Under Internal Heating

Abstract This study explores the onset of thermal instability within a bidisperse porous medium saturated with a homogeneous, incompressible fluid, subjected to a non-uniform internal heat generation and constant temperature gradient due to heating from below. The fluid motion is modelled with the Darcy’s law in micropores, while the Brinkman’s law is employed in macropores to ensure a more accurate representation of momentum transfer across different scales. The system is modelled under the Oberbeck–Boussinesq approximation, where density variations are incorporated solely in the buoyancy term, with the fluid density being temperature-dependent. Linear and nonlinear stability analyses are performed and different profiles of depth-dependent heat source are considered to investigate its effect in various physical scenarios. Both analyses lead to a generalized eigenvalue problem that is solved numerically by means of the Chebyshev- $$\tau$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> method. The nonlinear stability analysis is carried out in the context of the energy theory by means of the differential constraints method. A golden section algorithm is implemented to determine the critical thresholds for linear and nonlinear stability analyses and discuss their proximity.