Incompressible two-phase flows with an inextensible Newtonian fluid interface

We introduce a diffuse interface approximation for an incompressible two-phase flow problem with an inextensible Newtonian fluid interface. This approach allows to model lipid membranes as viscous fluids. In the present setting the membranes are assumed to be stationary. We validate the model and the numerical approach, which is based on a stream function formulation for the surface flow problem, an operator splitting approach and a semi-implicit adaptive finite element discretization, against observed flow patterns in vesicles, which are adhered to a solid surface and are subjected to shear flow. The influence of the Gaussian curvature on the surface flow pattern is discussed.