Starting from a Navier-Stokes-Cahn-Hilliard equation for a two-phase flow problem we discuss efficient numerical approaches based on adaptive finite element methods. Various extensions of the model are discussed: a) we consider the model on implicitly described geometries, which is used to simulate the sliding of droplets over nano-patterned surfaces, b) we consider the effect of soluble surfactants and show its influence on tip splitting of droplets under shear flow, and c) we consider bijels as a new class of soft matter materials, in which colloidal particles are jammed on the fluid-fluid interface and effect the motion of the interface due to an elastic force.
The work is based on joint work with Sebastian Aland (TU Dresden), John Lowengrub (UC Irvine) and Knut Erik Teigen (U Trondheim).