The Far-field Green's Integral in Stokes Flow from the Boundary Integral Formulation

摘要

In boundary integral methods for Stokes flow, the far-field Green's integral is usually taken to be zero without proof. However, this is not obviously the case, the reason being that Stokes flow is a near-field approximation and breaks down in the far-field. Here, we show that it is zero as expected by matching it to a far-field Green's integral in Oseen flow. Hence, there are similarities to the matched asymptotic procedure matching a near-field Stokes flow to a far-field Oseen flow, except in this case a different and new procedure is required to deal with the Green's integrals. In particular, the velocity is represented in the near-field by an integral distribution of stokeslets, and in the far-field by an integral distribution of oseenlets, and the two integral distributions are matched together by equating the stokeslets with the oseenlets in the matching region. A boundary integral representation is then obtained which holds throughout the whole flow region, enabling the velocity in the boundary integral scheme to be determined everywhere in the flow region.