Pressure boundary condition for the time-dependent incompressible Navier-Stokes equations

In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111-1145; Incompressible Flow and the Finite Element Method, vol. 2. Wiley: New York, 2000) was proposed an important hypothesis regarding the pressure Poisson equation (PPE) for incompressible flow: Stated there but not proven was a so-called equivalence theorem (assertion) that stated/asserted that if the Navier-Stokes momentum equation is solved simultaneously with the PPE whose boundary condition (BC) is the Neumann condition obtained by applying the normal component of the momentum equation on the boundary on which the normal component of velocity is specified as a Dirichlet BC, the solution (u, p) Would be exactly the same as if the 'primitive' equations, in which the PPE Plus Neumann BC is replaced by the usual divergence-free constraint (del (.) u=0), were solved instead. This issue is explored ill sufficient detail in this paper so as to actually prove the theorem for at least some situations. Additionally, like the original/primitive equations that require no BC for the pressure, the new results establish the same Ming when the PPE approach is employed. Copyright (c) 2005 John Wiley & Sons, Ltd.