  We study well-posedness of a velocity-vorticity formulation of the
Navier--Stokes equations, supplemented with no-slip velocity boundary
conditions, a no-penetration vorticity boundary condition, along with a natural
vorticity boundary condition depending on a pressure functional. In the
stationary case we prove existence and uniqueness of a suitable weak solution
to the system under a small data condition. The topic of the paper is driven by
recent developments of vorticity based numerical methods for the Navier--Stokes
equations.
