Certain unresolved ambiguities surround pressure determinations for ; incompressible flows: both Navier–Stokes and magnetohydrodynamic (MHD). ; 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 wallbounded shear flows ; usually have built in some relatively ad hoc dynamical ; recipe for dealing with it – often one that appears to ‘work’ satisfactorily. Here ; we consider a class of solenoidal velocity fields that 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 nontrivially 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.
