Convection in a Horizontal Porous Layer with Vertical Pressure Gradient Saturated by a Power-Law Fluid

The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such a way that a vertical throughflow is described. The threshold for a buoyancy-driven cellular flow is investigated by means of a linear stability analysis. This study consists in introducing disturbances with small amplitude. The disturbances are plane waves, i.e. a normal modes stability analysis of the basic stationary solution is performed. The resulting problem is an ordinary differential equation eigenvalue problem which is solved numerically by coupling the Runge–Kutta method with the shooting method. Results are presented in the form of marginal stability curves and their critical points representing the values of the control parameters such that the growth rate of the disturbances is zero. It is found that, among roll disturbances, the most unstable modes are stationary and uniform with infinite wavelength. For this reason, an asymptotic analysis for vanishing wave numbers is carried out. The results of this asymptotic analysis are obtained analytically displaying a very good agreement with the numerical solution. It is found that vertical throughflow plays a destabilising role for pseudoplastic fluids and a stabilising role for dilatant fluids.