A linear stability investigation of non-Darcian MHD flow in a vertical pipe via numerical methods

This study investigates the linear stability of buoyancy-assisted Poiseuille flow of an electrically conducting fluid through a vertical porous pipe subjected to a transverse magnetic field. The flow behavior is modeled using the Brinkman-extended non-Darcy formulation to capture the influence of both viscous and inertial effects in a porous medium. A linear stability analysis is performed, and the consequential eigenvalue problem is solved numerically using the Chebyshev spectral collocation method. The impact of key dimensionless parameters, including the Prandtl number ($Pr=0.01$ to 100), Darcy number ($Da=10^{-1}$ to $10^{-4}$), and Hartmann number ($Ha$), are systematically examined to understand their roles in flow stability. The results reveal that the base velocity profile exhibits an inflection point, and the applied magnetic field significantly alters both velocity and temperature distributions. For water ($Pr=7$), the flow exhibits least stability at higher magnetic influence ($Ha=2$), indicating the potential for enhanced heat transfer, particle dispersion, and flow manipulation. Conversely, for heavy oil ($Pr=50$), the flow is least stable without a magnetic field ($Ha=0$), highlighting magnetic field-based control strategies for applications such as thermal management, flow control, and smart fluidic devices. These findings offer important insights for optimizing magnetohydrodynamic flows in porous systems for engineering and industrial applications.