On derivations of evolving surface Navier–Stokes equations

In recent literature several derivations of incompressible Navier–Stokes-type equations that model the dynamics of an evolving fluidic surface have been presented. These derivations differ in the physical principles used in the modeling approach and in the coordinate systems in which the resulting equations are represented. This is an overview paper in the sense that we put five different derivations of surface Navier–Stokes equations into one framework. This then allows a systematic comparison of the resulting surface Navier–Stokes equations and shows that some, but not all, of the resulting models are the same. Furthermore, based on a natural splitting approach in tangential and normal components of the velocity, we show that all five derivations that we consider yield the same tangential surface Navier–Stokes equations.