Close interactions of drops and particles in viscous flow
Integral equation based numerical methods are attractive for the simulations of fluid mechanics at the micro scale such as in droplet-based microfluidics, with tiny water drops dispersed in oil, stabilized by surfactants. We have developed highly accurate numerical methods for drops with insoluble surfactants, both in two and three dimensions with the latter recently extended to include also electric fields.
This involves addressing several fundamental challenges that are highly relevant also to other applications: accurate quadrature methods for singular and nearly singular integrals, adaptive time-stepping, and reparameterization of time-dependent surfaces for high quality discretization of the drops throughout the simulations. In this talk, particular emphasis will be on quadrature methods applied to the evaluation of nearly singular layer potentials including error estimates and their use for adaptive parameter selection.