A general system of images for regularized Stokeslets and other elements near a plane wall

Computation of the singular and regularized image systems for doubly-periodic Stokes flow in the presence of a wall

“…In this section, we develop a fast summation method based on Beenakker's approach [3,4] and the method of images for regularized Stokeslets [17,25].…”

“…Note that the 'global' term (x, y; ξ) can be realized as the regularized Stokeslet corresponding to the blob (8) with blob parameter δ = ξ −1 . As a result, the 'local' term which is now realized as the difference of a singular Stokeslet and a regularized Stokeslet has decay rate comparable to that of the blob (8) since it has second moment zero [26]. Since the splitting parameter ξ is the reciprocal of the blob parameter δ, from now on, in this paper, we also refer to the blob parameter δ as the splitting parameter when working with the Fourier space sum.…”

“…In summary, the Stokes flow due to a doubly-periodic array of three dimensional point forces subject to the no-slip condition on the wall can be computed as (25) Note that, when the observation point moves to infinity in the z-direction, the velocity goes to a constant velocity…”

“…As detailed in [17] for a specific blob and [25] for a general blob, we need two blobs to build the image system for a regularized Stokeslet: one for the potential dipole, which we denote by φ δ d (r), and one for the Stokeslet and Stokeslet doublet, which we denote by φ δ s (r).…”

“…The image system for the regularized force of strength f located at x 0 corresponding to the pair (φ δ s , φ δ d ) is then given by [25] U φ δ…”