Discrete hydrodynamics near solid walls: Non-Markovian effects and the slip boundary condition.

A simple Markovian theory for the prediction of averages and correlations of discrete hydrodynamics near parallel solid walls is presented. The discrete momentum of bins is defined through a finite element basis function. The effect of the walls on the fluid is through irreversible extended friction forces appearing in the very equations of hydrodynamics. The Markovian assumption is critically assessed from the exponential decay of the eigenvalues of the correlation matrix of the discrete transverse momentum. We observe that for bins smaller than molecular dimensions, allowing one to resolve the density layering near the wall, the dynamics near the wall is non-Markovian. Bins larger than the molecular size do behave in a Markovian way. We measure the nonlocal viscosity and frictions kernels that appear in the discrete hydrodynamic equations, which are given in terms of Green-Kubo formulas. They suffer dramatically from the plateau problem. We use a recent procedure for reliably extracting the transport kernels out of the plateau-problematic Green-Kubo formula. With the so-measured transport kernels the nonlocal theory predicts very well the decay of the average of the transverse momentum when the initial velocity profile is a plug flow. The theory allows us to derive the slip boundary condition with microscopic expressions for the slip length and the hydrodynamic position of the wall. The slip boundary condition is not satisfied at the initial stages of the discontinous plug flow, but good agreement is obtained at later stages.