The slip velocity at the edge of a porous medium: Effects of interior resistance and interface curvature

Singularity methods are used to analyze creeping planar flow in the annulus between concentric cylinders, when a portion of the annulus is filled with an array of regularly spaced rods adjacent to the inner cylinder. The rods are evenly spaced on concentric circles, and the circles are spaced such that the array resembles a square lattice bent into a circle. The rods and inner cylinder are stationary, and steady rotation of the outer cylinder generates the flow. The quantity of interest is the 'slip velocity', the mean velocity at the interface between the array and the unfilled portion of the annulus. The primary part of the study concerns the influence of the interior rods on the interfacial velocity, and to this end the velocity is found as successive circles of rods are removed, starting with the circle closest to the inner cylinder. The calculations are carried out for solid volume fractions from 0.0001 to 0.1, and these show that the slip velocity is virtually unchanged as the interior circles of rods are removed, until only one circle remains and then the velocity is of order 10% larger than that for the full array. Hence the velocity at the edge of a sparse porous medium depends minimally on the hydrodynamic resistance of the obstacles in the interior. In the secondary part of the study, it is found that curvature of the interface does not influence the velocity there.