Microscopic determination of macroscopic boundary conditions in Newtonian liquids

Abstract

We study boundary conditions applied to the macroscopic dynamics of Newtonian liquids from the view of microscopic particle systems. We assume the existence of microscopic boundary conditions that are uniquely determined from a microscopic description of the fluid and the wall. By using molecular dynamical simulations, we examine a possible form of the microscopic boundary conditions. In the macroscopic limit, we may introduce a scaled velocity field by ignoring the higher order terms in the velocity field that is calculated from the microscopic boundary condition and standard fluid mechanics. We define macroscopic boundary conditions as the boundary conditions that are imposed on the scaled velocity field. The macroscopic boundary conditions contain a few phenomenological parameters for an amount of slip, which are related to a functional form of the given microscopic boundary condition. By considering two macroscopic limits of the non-equilibrium steady state, we propose two different frameworks for determining macroscopic boundary conditions.

Statistical Mechanical Expressions of Slip Length

“…In Ref. [55], we discussed deriving such boundary conditions based on several phenomenological assumptions.…”

“…These conditions are essential to the derivation of the partial slip boundary condition. Previously [55], we discussed what kind of walls satisfies (68) and ( 69) when the fluid is in a steady state. In this section, we derive the linear response formula focusing on the first order approximation of (42) in .…”

Full characterization of the hydrodynamic boundary condition at the atomic scale using an oscillating channel: Identification of the viscoelastic interfacial friction and the hydrodynamic boundary position

“…It is of critical importance to understand and optimize liquid-solid slip, which can enhance dramatically the performance of micro-and nanofluidic systems [14][15][16][17]. Accordingly, the hydrodynamic BC has been an active field of research during the past decades, both on the experimental side [18][19][20][21] and on the modeling side [3,4,6,[22][23][24][25][26][27][28][29][30][31][32][33], where molecular dynamics (MD) simulation proved to be a powerful tool. However, determining both λ and z s that make up the Navier BC is delicate even in a MD simulation, and it is a common practice to assume that z s lies at some prescribed distance from the wall surface [32] or coincides with the position of the wall surface [23][24][25] and only the friction coefficient (or equivalently the slip length) is measured.…”

Theoretical framework for the atomistic modeling of frequency-dependent liquid-solid friction

“…Español and coworkers [22,23] developed a new theory of non-equilibrium statistical mechanics, which led to a corrected form of the GK formula under the assumption that the system is Markovian. In another recent work, Nakano and Sasa [24,25] introduced explicit assumptions on the scale separation between the microscopic motion of molecules and the macroscopic motion of fluid and proposed a new way to estimate λ based on linearized fluctuating hydrodynamics. Both works from the two groups involved elaborate mathematical manipulations, and only reported the pure viscous (Markovian) behavior of the Navier FC, for a Lennard-Jones (LJ) liquid on a simple model wall.…”