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

We derive a general system of images for regularized sources, Stokeslets, and other related elements starting from an arbitrary regularization kernel (blob) used in the simulation of Stokes flows in three dimensions bounded by a plane. This generalizes previous work in which the image system for a Stokeslet had been derived for one specific blob. The significance of this generalization is that recent work on regularization methods requires the use of blobs designed to satisfy certain properties, such as zero moment conditions and fast decay, and thus it is absolutely necessary to have the system of images starting from an arbitrary blob. The system of images for a regularized element consists of a set of several elements, usually of higher order, that produce a flow that is zero at the bounding plane. In order for the resultant flow to vanish analytically at the wall, two different but related blobs must be used. For any given blob, we provide the formula for the companion blob that accomplishes the cancellation and we derive a systematic way to compute the image system of regularized Stokeslets, sources and dipoles. Other elements can be derived from these. By taking the limit as the regularization parameter approaches zero, the system of images for the corresponding singular elements is found.