Buoyancy–induced instability of a power–law fluid saturating a vertical porous slab

Many engineering applications involve porous media and rely on non–Newtonian working fluids. In the present paper, the seepage flow of a non–Newtonian fluid saturating a vertical porous layer is studied. The buoyant flow is thermally driven by the boundaries of the porous layer, which are permeable surfaces kept at different temperatures. In order to model the seepage flow of both shear–thinning (pseudoplastic) and shear–thickening (dilatant) fluids, reference is made to the Ostwald–de Waele rheological model implemented with the power–law extended form of Darcy’s law. The basic stationary flow is parallel to the vertical axis and shows a single–cell pattern, where the cell has infinite height and can display a core–region of enhanced/ inhibited flow according to the fluid’s rheological behaviour. By applying small perturbations, a linear stability analysis of the basic flow is performed to determine the onset conditions for a multicellular pattern. This analysis is carried out numerically by employing the shooting method. The neutral stability curves and the values of the critical Rayleigh number are computed for different pseudoplastic and dilatant fluids. The behaviour of a Newtonian fluid is also obtained as a limiting case.