Hydrodynamic boundary conditions, correlation functions, and Kubo relations for confined fluids

Dynamical correlation functions of a fluid slab confined between two solid walls are computed using a phenomenologial, hydrodynamic appoach that generalizes Onsager's principle of linear regression of fluctuations to inhomogeneous systems. The phenomenological results are compared to exact molecular-dynamics simulations on simple model systems. This comparison permits a determination of the phenomenological parameters that describe the hydrodynamics of the fluid slab and especially of the boundary conditions (BC's) that account for the presence of solid walls. In most cases, the hydrodynamic BC for the tangential velocity field is found to be a no-slip BC and must be applied in a plane that is separated from the solid by about one layer of fluid atoms. A set of formal relations between the parameters that characterize the hydrodynamic BC and the equilibrium correlation functions of the inhomogeneous fluid is also derived. These relations are analogous to the usual Green-Kubo equalities for the transport coefficients of bulk fluids.