BIE method for Stokes Flow in Axisymmetric and Periodic Geometry
Date:
Peristalsis is an important biological process in which tubes in our body contract and relax to move solid or fluid along. One important inverse problem in mimicing this behavior artificially is to find the optimal wall configuration that generates the optimal flow, or minimizes the energy loss. During the forward direction, one solves for the flow generated by a certain wall geometry. This can be done using boundary integral equations (BIEs), by imposing a surface source density on the wall. In this paper, we reformulate the periodic geometry using proxy sources and periodic boundary discrepancies. We then use axisymmetry to reduce the surface BI into a 1D integral by analytically integrating over the angle of revolution. We also discuss the special quadrature needed to address singular and near-singular integrals that appears from BIE. Finally we give convergence and simulation results.