Onset of convection in a non-Newtonian viscous flow through a horizontal porous channel

An impermeable horizontal fluid saturated porous layer is studied with respect to the onset of thermal convection. The saturating fluid is a Ostwald-de Waele type of non-Newtonian fluid. The viscous dissipation is the only heat source present inside the layer. A basic throughflow whose direction is arbitrary in the horizontal plane occurs. The basic flow produces, by means of viscous dissipation, a basic non-uniform vertical temperature gradient. The l​inear stability analysis of the basic state is carried out. The critical values are obtained as functions of the magnitude and of the inclination of the basic throughflow and as functions of the power-law index. The identification of the most unstable mode is performed: while for pseudoplastic fluids the most unstable modes are transverse rolls, for dilatant fluids the most unstable modes can be either longitudinal or transverse rolls.