Mesoscale model of mass and momentum transport in nanopores (2016)
In this work, we derive from first principles the equations of hydrodynamics near a solid wall, valid for the study of the nanoscale. We generalize Dynamic Density Functional Theory (DDFT) by including not only the mass density field as in usual approaches to DDFT, but also the momentum density field of the fluid. The solid is described as featureless under the assumption that all the internal degrees of freedom of the solid relax much faster than those of the fluid. In this new theory, the fluid moves according to a set of non-local hydrodynamic equations that include explicitly the forces due to the solid. These forces are of two types, reversible forces emerging from the free energy density functional, and accounting for impenetrability, and irreversible forces that involve the velocity of both the fluid and the solid. These forces are localized in the vicinity of the solid surface. The non-locality of the equations is due to the non-locality of the transport coefficients, which are given explicitly in terms of Green-Kubo formulae. We particularize this general hydrodynamic DDFT for simple fluids to the case of slit nanopores with planar flow configurations. In this simple geometry, only a reduced number of non-local transport coefficients (wall friction, slip friction, and viscous friction) are needed in this planar configuration. In the planar geometry, the continuum hydrodynamic equations for a fluid in a slit nanopore are discretized into bins. This allows us both, to compute explicitly the Green-Kubo expressions for the non-local transport coefficients, and to solve numerically the continuum hydrodynamic equations. The Green-Kubo formulae are computed from the time correlations of the force density on each bin that the solid exerts on the fluid, and the stress tensor of each bin . The phase functions trajectories are obtained from extensive Equilibrium Molecular Simulations by time span of 54ns. Thes Green-Kubo transport coefficients are subsequently used for the explicit numerical solution of the discrete hydrodynamic equations. Two initial non-equilibrium profiles of the plug and Poiseuille flow form are allowed to decay towards equilibrium. Non-Equilibrium Molecular Dynamics simulations and the predicted flow from the discrete hydrodynamic equation are then compared, with excellent agreement.Finally, we show that the present theory leads, in the limit of macroscopic flows, to a microscopic derivation of the Navier slip boundary conditions for the usual Navier-Stokes hydrodynamics, in which the slip length is given in microscopic terms.