The linear stability of a basic forced and free convection flow in an inclined porous channel is analysed by using the Darcy law and the Oberbeck–Boussinesq approximation. The basic velocity and temperature distributions are influenced by the effect of viscous dissipation, as well as by the boundary conditions. The boundary planes are assumed to be impermeable and isothermal, with a temperature of the lower boundary higher than that of the upper boundary. The instability against longitudinal rolls is studied by employing a second-order weighted residual solution and an accurate sixth-order Runge–Kutta solution of the disturbance equations. The instability against transverse rolls is also investigated. It is shown that these disturbances are in every case less unstable than the longitudinal rolls.

The thermal instability of the plane Poiseuille flow as a consequence of the effect of viscous dissipation is investigated. No external temperature difference is assumed in the environment; the lower boundary is considered adiabatic, while the upper boundary is isothermal. Thus, the sole cause of the unstable thermal stratification is the flow rate, through the volumetric heating induced by the viscous dissipation. A linear stability analysis is carried out numerically by the analysis of normal modes perturbing the basic flow with different inclinations. The study of cases with different Prandtl numbers and Gebhart numbers suggests that the most unstable perturbations are the longitudinal rolls, namely the normal modes with a wave vector perpendicular to the basic flow direction. A possible comparison with the hydrodynamic instability of the plane Poiseuille flow, described by the Orr–Sommerfeld eigenvalue problem is proposed. This comparison, when referred to high values of the Prandtl number, reveals that the dissipation instability may be effective at a Reynolds number smaller than that needed for the onset of the hydrodynamic instability.