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Problems of particle dynamics involving unsteady Stokes flows in confinedgeometries are typically harder to solve than their steady counterparts.Approximation techniques are often the only resort. Felderhof (see e.g. 2005,2009b) has developed a point-particle approximation framework to solve suchproblems, especially in the context of Brownian motion. Despite excellentagreement with past experiments, this framework has an inconsistency which weaddress in this work. Upon implementing our modifications, the framework passesconsistency checks that it previously failed. Further, it is not obvious thatsuch an approximation should work for short time-scale motion. We investigateits validity by deriving it from a general formalism based on integralequations through a series of systematic approximations. We also compareresults from the point-particle framework against a calculation performed usingthe method of reflections, for the specific case of a sphere near a full-slipplane boundary. We find from our analysis that the reasons for the success ofthe point-particle approximation are subtle and have to do with the nature ofthe unsteady Oseen tensor. Finally, we provide numerical predictions forBrownian motion near a full-slip and a no-slip plane wall based on thepoint-particle approximation as used by Felderhof, our modified point-particleapproximation, and the method of reflections. We show that our modifications toFelderhof's framework would become significant for systems of metallicnanoparticles in liquids.

Unsteady Stokes flow near boundaries: the point-particle approximation and the method of reflections