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

The effects of the vertical throughflow of a non-Newtonian power-law fluid on the onset of convective instability in a horizontal porous layer are investigated. The extended Darcy's model of momentum diffusion is employed together with the Oberbeck-Boussinesq approximation. A stationary basic solution for the vertical throughflow is determined analytically. The basic velocity and temperature fields turn out to be independent of the non-Newtonian rheology. A linear stability analysis is carried out, leading to a fourth-order eigenvalue problem. A numerical solution of the eigenvalue problem is employed to obtain the neutral stability curves and the critical Rayleigh number for the onset of instability. The governing parameters of the transition to instability are the Péclet number associated with the throughflow, and the power-law index of the fluid. These parameters influence the position of the neutral stability curve and also the critical Rayleigh number. The asymptotic cases of absolute pseudoplasticity, absolute dilatancy, and small Péclet number are discussed in detail. The latter case leads to a simple analytical solution that approximates fairly well the numerical data when the Péclet number is smaller than unity.