Determining hydrodynamic boundary conditions from equilibrium fluctuations

Shuyu Chen, Han Wang, Tiezheng Qian, and Ping Sheng
Phys. Rev. E 92, 043007 – Published 7 October 2015

Abstract

The lack of a first-principles derivation has made the hydrodynamic boundary condition a classical issue for the past century. The fact that the fluid can have interfacial structures adds additional complications and ambiguities to the problem. Here we report the use of molecular dynamics to identify from equilibrium thermal fluctuations the hydrodynamic modes in a fluid confined by solid walls, thereby extending the application of the fluctuation-dissipation theorem to yield not only the accurate location of the hydrodynamic boundary at the molecular scale, but also the relevant parameter value(s) for the description of the macroscopic boundary condition. We present molecular dynamics results on two examples to illustrate the application of this approach—one on the hydrophilic case and one on the hydrophobic case. It is shown that the use of the orthogonality condition of the modes can uniquely locate the hydrodynamic boundary to be inside the fluid in both cases, separated from the molecular solid-liquid interface by a small distance Δ that is a few molecules in size. The eigenvalue equation of the hydrodynamic modes directly yields the slip length, which is about equal to Δ in the hydrophilic case but is larger than Δ in the hydrophobic case. From the decay time we also obtain the bulk viscosity which is in good agreement with the value obtained from dynamic simulations. To complete the picture, we derive the Green-Kubo relation for a finite fluid system and show that the boundary fluctuations decouple from the bulk only in the infinite-fluid-channel limit; and in that limit we recover the interfacial fluctuation-dissipation theorem first presented by Bocquet and Barrat. The coupling between the bulk and the boundary fluctuations provides both the justification and the reason for the effectiveness of the present approach, which promises broad utility for probing the hydrodynamic boundary conditions relevant to structured or elastic interfaces, as well as two-phase immiscible flows.

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  • Received 24 November 2014
  • Revised 4 September 2015

DOI:https://doi.org/10.1103/PhysRevE.92.043007

©2015 American Physical Society

Authors & Affiliations

Shuyu Chen1, Han Wang2,3, Tiezheng Qian4, and Ping Sheng1,5,*

  • 1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
  • 2CAEP Software Center for High Performance Numerical Simulation, Beijing, China
  • 3Zuse Institute Berlin, Takustrasse 7, 14195 Berlin, Germany
  • 4Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
  • 5Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

  • *Corresponding author: sheng@ust.hk

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Issue

Vol. 92, Iss. 4 — October 2015

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