Linear, hydrodynamic flow in a rotating saturated porous medium

Abstract

Linear, steady, axisymmetric flow of a homogeneous fluid in a rigid, bounded, rotating, saturated porous medium is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The dynamics of the interior region and vertical boundary layers are investigated as functions of the Ekman number E(=v/ΩL 2) and rotational Darcy 3 numberN(=kΩ/v) which measures the ratio between the Coriolis force and the Darcy frictional term. IfN≫E −1/2, the permeability is sufficiently high and the flow dynamics are the same as those of the conventional free flow problem with Stewartson'sE 1/3 andE 1/4 double layer structure. For values ofN≤E −1/2 the effect of porous medium is felt by the flow; the Taylor-Proudman constraint is no longer valid. ForN≪E −1/3 the porous medium strongly affects the flow; viscous side wall layer is absent to the lowest order and the fluid pumped by the Ekman layer, returns through a region of thicknessO(N −1). The intermediate rangeE −1/3≪N≪E −1/2 is characterized by double side wall layer structure: (1)E 1/3 layer to return the mass flux and (ii) (NE)1/2 layer to adjust the interior azimuthal velocity to that of the side wall. Spin-up problem is also discussed and it is shown that the steady state is reached quickly in a time scaleO(N).