Generalized reflection method for the stokes equation in confined geometries. Applications to microfluidics and particle transport

A generalized reflection method is proposed for determining the hydrodynamic resistance matrix of a rigid particle translating in a confined Stokesian fluid, by reducing its estimate to the solution of two simpler problems: (i) the singular expansion of the hydrodynamic variables due to the particle motion in the free space, represented by the Faxén operator associated with the particle geometry, and (ii) the determination of the Green function associated with the specific channel geometry. This approach extends the Stokeslet approximation to a Faxén approximation that is reliable to ratios of the particles characteristic length l to the characteristic distance d from the walls higher than the existing literature approximations, the error of which is order of O((l/d)4). Given the solution of the two above mentioned subproblems, the proposed method applies to any particle and channel geometry. The application to prolate spheroidal particles translating parallel to a plane is discussed and compared against numerical simulations.