The small thickness of membranes, shells and capsules enables their efficient approximation as hyper-surfaces. Phase-field modeling provides a versatile tool to capture the motion of such elastic hyper-surfaces in fluid flow under bending and surface tension forces. However, the in-plane stretching of the surface has been widely neglected or approximated by inextensibility of the material. In this talk, we develop a novel phase-field model for elastic hyper-surfaces in Navier–Stokes fluids, which includes bending, tension, and in-plane stretching. The model is based on a coupling of a phase-field model for two-phase flow to a fully Eulerian description of the surface deformation tensor. We apply the method to microfluidic experiments on lipid vesicles and illustrate how the results can be used to better understand the mechanical properties of cell membranes.