Instability of mixed convection in a vertical porous channel with uniform wall heat flux

Mixed convection in a vertical plane channel filled with a saturated porous medium is investigated. The boundary planes are considered as subject to symmetric uniform heat fluxes, resulting into a net fluid heating or cooling. Either upflow or downflow conditions are considered, thus exhibiting two distinct regimes: buoyancy-assisted and buoyancy-opposed. A basic stationary and parallel flow directed vertically along the channel is examined. The linear stability of this basic solution is developed through the standard normal-mode analysis. The solution of the eigenvalue problem for neutral stability is carried out numerically for the general oblique modes. An analytical solution is provided for the longitudinal modes, with a horizontal wave vector having a direction parallel to the boundary planes. An asymptotic analytical solution is also allowed for oblique modes with either a vanishingly small Peclet number or a vanishingly small wave number. The longitudinal modes are the most unstable, displaying their parametric domain of instability under buoyancy-opposed regime. In this regime, the longitudinal modes with sufficiently small wave numbers are always unstable. This conclusion suggests that conditions of flow reversal or crossing of parametric singularities, characteristic of the parallel flow solution under buoyancy-opposed regime, are unlikely to be observed in an experiment.