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