Unstable mixed convection in a heated inclined porous channel

This paper presents a stability analysis of a mixed convection problem in an inclined parallel-plate channel with uniform heating (or cooling) from the top and bottom. The channel is filled with a saturated homogeneous porous medium and the momentum equation is given by Darcy's model. A forced through-flow is prescribed across the channel. Linear stability analysis is thus employed to determine the onset of thermoconvective instability. The channel inclination is shown to play an important role in the stability of the problem, where two different regimes can be present: a buoyancy-assisted regime and a buoyancy-opposed regime. The analysis of the problem leads to a differential eigenvalue problem composed of a system of four complex-valued equations that are used to determine the critical values of the Rayleigh number leading to an instability under different problem configurations. This eigenproblem is solved by employing the generalised integral transform technique (GITT), in which simpler real eigenfunction bases are used to expand the complex eigenproblem. The results indicate that the longitudinal rolls are always more unstable than oblique and transverse rolls. For a buoyancy-opposed regime, even with a very small channel inclination angle, the basic through-flow is always unstable. This result has an important implication for experimental research, as it shows that a perfect alignment must be employed for horizontal mixed-convection experiments to avoid instabilities that arise in the buoyancy-opposed regime.