We consider the impact of surface hydrodynamics on the interplay between curvature and composition in coarsening processes on model systems for biomembranes. This includes scaling laws and equilibrium configurations, which are investigated by computational studies of a surface two-phase flow problem with additional phase-depending bending terms. These additional terms geometrically favor specific configurations. We find that as in 2D the effect of hydrodynamics strongly depends on the composition. In situations where the composition allows a realization of a geometrically favored configuration, the hydrodynamics enhances the evolution into this configuration. We restrict our model and numerics to stationary surfaces and validate the numerical approach with various benchmark problems and convergence studies.