Stable Numerical Approximation of Two-Phase Flow with a Boussinesq--Scriven Surface Fluid

We consider two-phase Navier--Stokes flow with a Boussinesq--Scriven surface fluid. In such a fluid the rheological behaviour at the interface includes surface viscosity effects, in addition to the classical surface tension effects. We introduce and analyze parametric finite element approximations, and show, in particular, stability results for semi-discrete versions of the methods, by demonstrating that a free energy inequality also holds on the discrete level. We perform several numerical simulations for various scenarios in two and three dimensions, which illustrate the effects of the surface viscosity.