(untitled) (Communications in Computational Physics, 2018)
We derive a closed form expression for the regularized Stokeslet in two space dimensions with periodic boundary conditions in the x-direction and a solid plane wall at y = 0. To accommodate the no-slip condition on the wall, a system of images for the regularized Stokeslets was used. The periodicity is enforced by writing all elements of the image system in terms of a Green's function whose periodic extension is known. Although the formulation is derived in the context of regularized Stokeslets, the expression for the traditional (singular) Stokeslet is easily found by taking the limit as the regularization parameter approaches zero. The new formulation is validated by comparing results of two test problems: the Taylor infinite waving sheet and 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 a model that does not prescribe the motion so that the beat period and synchronization of neighboring cilia are a result of the forces developed along the cilia.