Vorticity‐stream function approaches are inappropriate to solve the surface Navier‐Stokes equation on a torus

Abstract The growing interest in interfacial flows in engineering and biological applications demands to numerically solve the incompressible surface (Navier)‐Stokes equation. A common approach, based on a vorticity‐stream function formulation reformulates the equation into a system of scalar‐valued surface partial differential equations, for which various numerical methods have been developed. However, on surfaces, which are not simply connected, this approach is not appropriate. We here explain the underlying situation and provide details and examples.