  Certain unresolved ambiguities surround pressure determinations for
incompressible flows, both Navier-Stokes and magnetohydrodynamic. For
uniform-density fluids with standard Newtonian viscous terms, taking the
divergence of the equation of motion leaves a Poisson equation for the pressure
to be solved. But Poisson equations require boundary conditions. For the case
of rectangular periodic boundary conditions, pressures determined in this way
are unambiguous. But in the presence of "no-slip" rigid walls, the equation of
motion can be used to infer both Dirichlet and Neumann boundary conditions on
the pressure P, and thus amounts to an over-determination. This has
occasionally been recognized as a problem, and numerical treatments of
wall-bounded shear flows usually have built in some relatively ad hoc dynamical
recipe for dealing with it, often one which appears to "work" satisfactorily.
Here we consider a class of solenoidal velocity fields which vanish at no-slip
walls, have all spatial derivatives, but are simple enough that explicit
analytical solutions for P can be given. Satisfying the two boundary conditions
separately gives two pressures, a "Neumann pressure" and a "Dirichlet pressure"
which differ non-trivially at the initial instant, even before any dynamics are
implemented. We compare the two pressures, and find that in particular, they
lead to different volume forces near the walls. This suggests a reconsideration
of no-slip boundary conditions, in which the vanishing of the tangential
velocity at a no-slip wall is replaced by a local wall-friction term in the
equation of motion.
