Linear instability of the horizontal throughflow in a plane porous layer saturated by a power-law fluid

The onset of the convective instability in the horizontal throughflow of a power-law fluid saturating a horizontal porous layer heated from below is studied. A linear stability analysis of the basic flow is carried out and the disturbance equations are solved analytically. The problem examined here is an extension of the classical Prats problem for Newtonian fluids. It is shown that the marginal stability condition, as well as the critical values of the wave number and of the Darcy–Rayleigh number, is affected by the value of the Péclet number associated with the basic flow, except for the special case of a Newtonian fluid. The limit of a vanishingly small Péclet number is considered leading to the special case of the Horton–Rogers–Lapwood (HRL) problem for a power-law fluid, i.e., the Prats problem with a vanishing basic throughflow. It is shown that the generalized HRL problem is always linearly stable for pseudoplastic fluids and always linearly unstable for dilatant fluids.