Zero-mean normal velocity in periodic half-space Stokes flow (${\partial }_y⟨v{⟩}_x=0$)

Incompressibility and periodicity imply that the spatial mean of the normal velocity is constant with height. Zero normal flow at infinity therefore requires zero mean normal boundary velocity. In the bounded biharmonic stream function for planar Stokes flow solution, the mean streamfunction is a constant plus a term linear in height.