Flows of fluids with free surfaces show complex dynamical behavior. Examples include effects like capillary surface waves, topological transitions such as droplet breakup and coalescence, or pattern formation in wetting and de-wetting dynamics. These complex phenomena result from a highly nonlinear evolution that is driven by the interplay of surface forces and the changing surface geometry. Droplet-based microfluidics both utilizes the free-surface dynamics in a wide range of applications in science and engineering, and, due to the precise control of flows at small scales, allows to study the dynamics experimentally.

An analytical description of the dynamics is made difficult by the high degree of nonlinearity. Numerical tools complement experiments, as they give access to quantities of interest such as the pressure fields inside a fluid or local stresses on the interface, and allow for a precise control of parameters and models of physical effects. We use numerical tools to study the complex dynamics of free surface flows.

In the first part of this thesis, we develop a fully-resolved 3D boundary element method for simulating droplet dynamics in complex geometries. The developed numerical tool allows us to follow the dynamic deformation of droplets with variable viscosity ratio between droplet and continuous phase, under the effect of Young-Laplace surface tension, gravity, and dielectric stresses due to electric fields. Free interfaces are represented by a novel smooth surface representation that gives an accurate description for the surface shape and curvature.

In the second part, we address two practically relevant problems. First, we study the breakup of droplets as concentrated emulsions are injected into a narrow constriction, and describe the underlying physical mechanism that drives the breakup. Second, we analyze the efficiency of droplet sorting with dielectrophoresis, and propose a new sorting device that operates at lower voltage and reduces stress on the droplets. In careful quantitative comparisons between numerics and experiments, we find that in-plane surface stresses due to nonequilibrium surfactant distributions have a major impact on free interface dynamics, and merit further study.