Integral equation methods are ideal for high-order accurate, linear-scaling solvers for viscous flows involving rigid or non-rigid particles, possibly in confined geometries. We focus on computing the effective shear viscosity of a suspension of 2D rigid particles with periodic boundary conditions, whose applications include industrial flows and complex fluids. We overview recent new tools needed for efficient and accurate solution: 1) Boundary integral quadratures for handling close-to-touching curves. 2) Periodization of the fast multipole method in arbitrary lattices (which also applies to other PDEs such as the 3D Maxwell equations). In the context of Stokes flow through channels, we also discuss: 3) Adaptive panel quadratures to handle complex boundary shapes and corners. We highlight the beautiful complex analysis needed to develop these tools.
Joint work with Jun Wang (Flatiron Institute), Ehssan Nazockdast (UNC), Shravan Veerapaneni, Bowei Wu, and Hai Zhu (Michigan).