Linear stability of pipe flow with magnetic field and internal heat source: A non-Darcian approach

Abstract

This study examines the stability of convective flow in a vertical pipe with a magnetic field, driven by an internal heat source. To formulate governing equations, the non-Darcy Brinkman Forchheimer extended model has been used and solved numerically by the Chebyshev spectral collocation method (CSCM). Stability analysis is conducted for various fluids (mercury and liquids) corresponding to Prandtl numbers ($Pr$) of 0.0248 and 7, showing that increasing the magnetic field enhances the stability region, with varying effects on critical wave number ($k_c$) and critical Grashof number ($G{r}_c$) depending on $Pr$. The analysis reveals that the magnetic field increases the velocity profile in the central region, while decreasing it near the boundary, with the velocity profile near the boundary increasing with the Darcy number ($Da$). Increasing Hartmann number ($Ha$) significantly stabilizes the flow by suppressing convective motion through Lorentz forces. For $Da=10^{-1}$ and $Pr=0.0248$, $G{r}_c$ increases from $6.75\times 10^4$ at $Ha=0$ to $7.03\times 10^5$ at $Ha=10$, indicating requirement of higher buoyancy to trigger convection. Similarly, for $Da=10^{-2}$, $G{r}_c$ rises from $2.15\times 10^5$ to $2.40\times 10^6$. Reducing $Da$ enhances porous resistance, resulting in weaker secondary vortices and diminished convective transport, with secondary flow intensity dropping by over 60% as $Da$ decreases from $10^{-1}$ to $10^{-2}$. Low-$Pr$ fluids exhibit stronger stabilization under magnetic damping compared to high-$Pr$ fluids. Visualizations of the secondary flow in a circular cross-section, showcasing the disturbance velocities (radial, circumferential, and axial) and temperature for various $Da$ in both absence and presence of a magnetic field. Additionally, the stream function, streamwise velocity, and temperature are depicted in the meridional cross-section of the pipe at critical parameters, providing a comprehensive understanding of the flow dynamics.