Asymptotic limits of the axisymmetric solution of the Brinkman equation for a point force near a no-slip wall

Abstract

We derive the far-field and near-field solutions for the Green's function of a point force acting perpendicular to a no-slip wall in a Brinkman fluid, focusing on the regime where the distance between the force and the wall is much smaller than the screening length. The general solution is obtained in closed form up to a single integral and can be systematically expanded in a Taylor series in both the far-field and near-field limits. The flow can then be expressed as a series of source-multipole singularities with an additional, analytically known, correction in the proximity of the wall. Comparisons with numerical integration demonstrate the accuracy and reliability of the asymptotic expansions. The results are also applicable to the unsteady Stokes flow driven by a localized assembly of forces, such as a beating cilium protruding from a flat surface.

9 pages, 2 figures, accepted for publication in JFM

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