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Certain unresolved ambiguities surround pressure determinations forincompressible flows, both Navier-Stokes and magnetohydrodynamic. Foruniform-density fluids with standard Newtonian viscous terms, taking thedivergence of the equation of motion leaves a Poisson equation for the pressureto be solved. But Poisson equations require boundary conditions. For the caseof rectangular periodic boundary conditions, pressures determined in this wayare unambiguous. But in the presence of "no-slip" rigid walls, the equation ofmotion can be used to infer both Dirichlet and Neumann boundary conditions onthe pressure P, and thus amounts to an over-determination. This hasoccasionally been recognized as a problem, and numerical treatments ofwall-bounded shear flows usually have built in some relatively ad hoc dynamicalrecipe for dealing with it, often one which appears to "work" satisfactorily.Here we consider a class of solenoidal velocity fields which vanish at no-slipwalls, have all spatial derivatives, but are simple enough that explicitanalytical solutions for P can be given. Satisfying the two boundary conditionsseparately gives two pressures, a "Neumann pressure" and a "Dirichlet pressure"which differ non-trivially at the initial instant, even before any dynamics areimplemented. We compare the two pressures, and find that in particular, theylead to different volume forces near the walls. This suggests a reconsiderationof no-slip boundary conditions, in which the vanishing of the tangentialvelocity at a no-slip wall is replaced by a local wall-friction term in theequation of motion.Comment: 8 pages, 4 figures, to appear in Journal of Plasma Physics, send  e-mail to bkress@darthmouth.ed

Pressure determinations for incompressible fluids and magnetofluids