This paper aims to focus on the hydrodynamic stability of steady-state natural convection inside a vertical flat porous microchannel. The main objective is to investigate the transition conditions from stable to unstable flow under the Boussinesq approximation, explicitly incorporating velocity slip and temperature jump boundary conditions at the channel walls.
For the first time for this specific channel geometry, the small perturbation method is applied to derive a modified Orr–Sommerfeld equation that accounts for microscale effects and porous media properties. The resulting eigenvalue problem is solved numerically to determine the neutral stability thresholds and analyze the comprehensive effects of the Knudsen, Darcy and Prandtl numbers on the flow behavior.
Critical Grashof numbers, where the imaginary part of the eigenvalues changes sign, were successfully determined. The slip effect acts as a stabilizing factor because reduced fluid-wall interaction lowers friction, requiring a higher temperature difference to destabilize the flow. Conversely, increasing the Darcy number reduces the critical Grashof number toward an asymptotic constant, as less porous material intensifies flow velocity and disturbances due to decreased internal pressure. Furthermore, a higher Prandtl number thins the thermal boundary layer, shifting disturbances away from the hot wall and accelerating flow instability.
The paper delivers a pioneering stability analysis by simultaneously capturing both rarefaction effects (Knudsen number) and porous media characteristics (Darcy number) via a modified Orr–Sommerfeld framework. These physical insights provide valuable guidelines for controlling and optimizing heat transfer and fluid flow in advanced microfluidic devices and porous heat exchangers.
