On flow simplification occurring in viscous three-dimensional water flows with constant non-vanishing vorticity

Abstract

We show that a viscous steady water flow of constant non-vanishing vorticity situated below a free surface (assumed to have the shape of a two-dimensional time dependent graph) and above a flat bottom has to be two-dimensional; that is, while satisfying the viscous three-dimensional water wave equations, it turns out that the free surface, the velocity field and the pressure present no variation in one of the horizontal directions. Moreover, the vorticity must have only one non-zero component which points in the horizontal direction orthogonal to the direction of the surface wave propagation. Compared with previous works, our study here has the advantage that it considers the Navier–Stokes equations and also takes into account the normal and tangential stress boundary conditions.