Since the pioneering work of Youngren and Acrivos (JFM, 1976) thirty years ago, interfacial dynamics in Stokes flow and/or gravity has been implemented through explicit time integration of boundary integral schemes which require that the time step is sufficiently small to ensure numerical stability. To avoid this difficulty, we have developed an efficient, fully-implicit time integration algorithm based on a mathematically rigorous combination of implicit formulas with a Jacobian-free three-dimensional Newton method. The resulting algorithm preserves the stability of the employed implicit formula and thus it has excellent stability properties, e.g. it is not affected by the Courant condition or by physical stiffness such as that associated with the critical conditions of interfacial deformation.

In this talk, we present our results, and comparisons with previous experimental, analytical and computational studies, in an array of different problems. In particular, we consider the deformation of free-suspended droplets in both subcritical and supercritical flows, interfaces in close contact with other interfaces or with solid walls as well as the dynamics of viscous droplets attached to rough solid substrates in shearing flows.