An Explicit Formula for Two-Dimensional Singly-Periodic Regularized Stokeslets Flow Bounded by a Plane Wall (Communications in Computational Physics, 2018)
We derive a closed form expression for the regularized Stokeslet in twospace dimensions with periodic boundary conditions in the x-direction and a solidplane wall at y = 0. To accommodate the no-slip condition on the wall, a system ofimages for the regularized Stokeslets was used. The periodicity is enforced by writingall elements of the image system in terms of a Green's function whose periodicextension is known. Although the formulation is derived in the context of regularizedStokeslets, the expression for the traditional (singular) Stokeslet is easily found by takingthe limit as the regularization parameter approaches zero. The new formulation isvalidated by comparing results of two test problems: the Taylor infinite waving sheetand the motion of a cylinder moving near a wall. As an example of an application,we use our formulation to compute the motion and flow generated by cilia using amodel that does not prescribe the motion so that the beat period and synchronizationof neighboring cilia are a result of the forces developed along the cilia.