Spectrally-Accurate Close Evaluation Schemes for Stokes Boundary Integral Operators

Abstract

Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this thesis, two spectrally-accurate integration schemes for close evaluation of 2D Stokes layer potentials are developed -- a global quadrature for the moving particles (e.g., blood cells, vesicles) represented as smooth closed curves, and an adaptive panel quadrature for the stationary boundaries (e.g., vessel walls, microfluidic channels) which are more complex curves that can be non-smooth. Both schemes rely on expressing Stokes layer potentials in terms of Laplace potentials and related complex contour integrals, which are then evaluated accurately either through a singularity cancellation technique or using analytic expressions. Numerical examples are presented to demonstrate the robustness and super-algebraic convergence of both schemes. Finally, as an application of the integration schemes, we investigate the electrohydrodynamic interactions between (possibly deflated) vesicles, where interesting behaviors unique to vesicles, such as circulatory and oscillatory motions, are observed and analyzed.