Non-local viscosity from the Green-Kubo formula.

We study through MD simulations the correlation matrix of the discrete transverse momentum density field in real space for an unconfined Lennard-Jones fluid at equilibrium. Mori theory predicts this correlation under the Markovian approximation from the knowledge of the non-local shear viscosity matrix, which is given in terms of a Green-Kubo formula. However, the running Green-Kubo integral for the non-local shear viscosity does not have a plateau. By using a recently proposed correction for the Green-Kubo formula that eliminates the plateau problem [Español et al., Phys. Rev. E 99, 022126 (2019)], we unambiguously obtain the actual non-local shear viscosity. The resulting Markovian equation, being local in time, is not valid for very short times. We observe that the Markovian equation with non-local viscosity gives excellent predictions for the correlation matrix from a time at which the correlation is around 80% of its initial value. A local in space approximation for the viscosity gives accurate results only after the correlation has decayed to 40% of its initial value.

Discrete hydrodynamics near solid planar walls.

We derive, with the projection operator technique, the equations of motion for the time-dependent average of the discrete mass and momentum densities of a fluid confined by planar walls under the assumption that the flow field is translationally invariant along the directions tangent to the walls. Shear flow and sound propagation perpendicular to the walls can be described with the discrete hydrodynamic equations. The interaction with the walls is not given through boundary conditions but rather in terms of impenetrability and friction forces appearing in the discrete hydrodynamic equations. Microscopic expressions for the transport coefficients entering the discrete equations are provided. We further show that the obtained discrete equations can be interpreted as a Petrov-Galerkin finite-element discretization of the continuum equations presented by Camargo et al. [J. Chem. Phys. 148, 064107 (2018)JCPSA60021-960610.1063/1.5010401] when restricted to planar geometries and flows.

Solution to the plateau problem in the Green-Kubo formula.

Transport coefficients appearing in Markovian dynamic equations for coarse-grained variables have microscopic expressions given by Green-Kubo formulas. These formulas may suffer from the well-known plateau problem. The problem arises because the Green-Kubo running integrals decay as the correlation of the coarse-grained variables themselves. The usual solution is to resort to an extreme timescale separation, for which the plateau problem is minor. Within the context of Mori projection operator formulation, we offer an alternative expression for the transport coefficients that is given by a corrected Green-Kubo expression that has no plateau problem by construction. The only assumption is that the Markovian approximation is valid in such a way that transport coefficients can be defined, even in the case that the separation of timescales is not extreme.

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.

Microscopic Slip Boundary Conditions in Unsteady Fluid Flows.

An algebraic tail in the Green-Kubo integral for the solid-fluid friction coefficient hampers its use in the determination of the slip length. A simple theory for discrete nonlocal hydrodynamics near parallel solid walls with extended friction forces is given. We explain the origin of the algebraic tail and give a solution of the plateau problem in the Green-Kubo expressions. We derive the slip boundary condition with a microscopic expression for the slip length and the hydrodynamic wall position, and assess it through simulations of an unsteady plug flow.

Nanoscale hydrodynamics near solids.

Density Functional Theory (DFT) is a successful and well-established theory for the study of the structure of simple and complex fluids at equilibrium. The theory has been generalized to dynamical situations when the underlying dynamics is diffusive as in, for example, colloidal systems. However, there is no such a clear foundation for Dynamic DFT (DDFT) for the case of simple fluids in contact with solid walls. In this work, we derive DDFT for simple fluids by including not only the mass density field but also the momentum density field of the fluid. The standard projection operator method based on the Kawasaki-Gunton operator is used for deriving the equations for the average value of these fields. 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 (solid elasticity is irrelevant). 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 of the solid, 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 resulting hydrodynamic equations should allow one to study dynamical regimes of simple fluids in contact with solid objects in isothermal situations.