Numerical Analysis: An accurate integral equation method for simulating multi-phase Stokes flow

We have developed a numerical method based on an integral equation formulation for simulating drops in viscous fluids in the plane. It introduces an interpolatory quadrature approach for handling near-singular integrals. Such integrals typically arise when drop boundaries come close to one another, and are difficult to compute accurately using standard quadrature rules. With the new quadrature method, very general drop configurations can be handled while still maintaining stability and high accuracy. Challenging numerical examples with many drops that are very close to touching can be handled.

This has been further extended to include also solid walls, to allow for very accurate simulation of drops squeezing through channels etc. The accurate treatment of walls with sharp edges and corners is also under way.