In this paper we investigate new boundary conditions for the incompressible, timรจe-dependent Navier-Stokes equation. Especially inflow and outflow conditions are considered. The equations are linearized around a constant flow, so that we can use Laplace-Fourier technique to investigate the strength of boundary layers at open boundaries. Such layers are unphysical, and new boundary conditions are proposed so that these boundary layers are suppressed. We also show that the boundary conditions we propose do not produce divergence. Furthermore, they give solutions that do not grow in time, as long as the forcing function in the system rioes not. We also discuss the numerical treatment of boundaries when fourth-order accurate finite difference operators are used to approximate spatial derivatives. Using higher order methods introduces eigensolutions with boundary layer thickness of the same order of magnitude as the grid size. These eigensolutions have to be suppressed in order to not destroy the fourth-order accuracy of the method. Numerical results for the non-linear Navier-Stokes equation, together with the new boundary conditions, are presented. These calculations confirm that the results for the linearized problem hold for the non-linear problem as well. (c) 1993 Academic Press, Inc.