Pressure determinations for incompressible fluids and magnetofluids

Certain unresolved amiguities surround pressure determinations for incompre ssible flows: both Navier-Stokes and magnetohydrodynamic (MHD). For uniform -density fluids with standard Newtonian viscous terms, taking the divergenc e of the equation of motion leaves a Poisson equation for the pressure to b e solved. But Poisson equations require boundary conditions. For the case o f rectangular periodic boundary conditions, pressures determined in this wa y are unambiguous. But in the presence of 'no-slip' rigid walls, the equati on of motion can be used to infer both Dirichlet and Neumann boundary condi tions on the pressure P, and thus amounts to an over-determination. This ha s occasionally been recognized as a problem, and numerical treatments of wa ll-bounded shear flows usually have built in some relatively ad hoc dynamic al recipe for dealing with it often one that appears to 'work' satisfactori ly. Here we consider a class of solenoidal velocity fields that vanish at n o-slip walls, have all spatial derivatives, but are simple enough that expl icit analytical solutions for P can be given. Satisfying the two boundary c onditions separately gives two pressures, a 'Neumann pressure" and a 'Diric hlet pressure", which differ non-trivially at the initial instant, even bef ore any dynamics are implemented. We compare the two pressures, and find th at, in particular, they lead to different volume forces near the walls. Thi s suggests a reconsideration of no-slip boundary conditions, in which the v anishing of the tangential velocity at a no-slip wall is replaced by a loca l wall-friction term in the equation of motion.