Onset of instability due to variable viscosity and dissipation in a plane porous channel

The effect of viscous dissipation is considered on modelling the fully-developed heat transfer in a parallel plane channel filled with a saturated porous medium. The basic Darcy’s flow in a regime of forced convection is analysed insofar as the variability of fluid viscosity with temperature is taken into account. The thermal boundary conditions at the impermeable channel walls are described by assuming external convection with a constant heat transfer coefficient, viz. by imposing Robin conditions for the temperature as parametrised through the Biot number. The emergence of a singular behaviour in the basic velocity and temperature profiles is found when the P ́eclet number and the variable viscosity parameter are large enough as to imply a failure of the linear fluidity model. A linear stability analysis of the basic parallel flow is carried out to detect the reaction of the system to small-amplitude external perturbations. Different odd or even normal modes of the longitudinal type are studied. It is shown that no instability arises until the parametric condition for the emergence of the singularity is approached. An argument to predict the behaviour of normal modes of oblique type is eventually presented.