Convective instability of the Darcy–Bénard problem with through flow in a porous layer saturated by a power-law fluid

The linear stability of the horizontal through flow in a porous layer saturated by a non-Newtonian power-law fluid is analysed. The bounding horizontal walls are assumed to be impermeable and subject to different temperature boundary conditions parametrised by a pair of Biot numbers. The exchange of stabilities in the comoving reference frame is proved for the general case, meaning that the disturbances display a drift motion in the same direction and at the same speed of the through flow independently of the chosen boundary conditions. Several possible models of thermal boundary conditions are studied in detail, corresponding to different combinations of Robin boundary conditions at the top and bottom walls. The different behavior of pseudoplastic fluids and dilatant fluids at the onset of instability is described: the preferred patterns for convection are transverse rolls for pseudoplastic fluids, while longitudinal rolls turn out to be the most unstable for dilatant fluids. In both cases, an increase in either Biot numbers leads to a less unstable flow with smaller rolls.