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Journal ArticleDOI

Hydrodynamic Theory of the Velocity Correlation Function

01 Nov 1970-Physical Review A (American Physical Society)-Vol. 2, Iss: 5, pp 2005-2012
TL;DR: In this article, the velocity correlation function of an atom in a simple liquid is calculated using a frequency-dependent version of the Stokes-Einstein formula, and good agreement is obtained with the velocities determined by Rahman using computer experiments.
Abstract: The velocity correlation function of an atom in a simple liquid is calculated using a frequency-dependent version of the Stokes-Einstein formula. Stokes's law for the frictional force on a moving sphere is generalized to arbitrary frequency, compressibility, and visco-elasticity, with arbitrary slip of the fluid on the surface of the sphere. This frequency-dependent friction coefficient is then used in a generalized Stokes-Einstein formula, and the velocity correlation function is found by Fourier inversion. By using physically reasonable values for viscoelastic parameters, good agreement is obtained with the velocity correlation function determined by Rahman using computer experiments.
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Journal ArticleDOI
TL;DR: In this paper, the system-size dependence of translational diffusion coefficients and viscosities in molecular dynamics simulations under periodic boundary conditions was studied. But the authors focused on the effect of the number of particles in the simulation box.
Abstract: We study the system-size dependence of translational diffusion coefficients and viscosities in molecular dynamics simulations under periodic boundary conditions. Simulations of water under ambient conditions and a Lennard-Jones (LJ) fluid show that the diffusion coefficients increase strongly as the system size increases. We test a simple analytic correction for the system-size effects that is based on hydrodynamic arguments. This correction scales as N-1/3, where N is the number of particles. For a cubic simulation box of length L, the diffusion coefficient corrected for system-size effects is D0 = DPBC + 2.837297kBT/(6πηL), where DPBC is the diffusion coefficient calculated in the simulation, kB the Boltzmann constant, T the absolute temperature, and η the shear viscosity of the solvent. For water, LJ fluids, and hard-sphere fluids, this correction quantitatively accounts for the system-size dependence of the calculated self-diffusion coefficients. In contrast to diffusion coefficients, the shear viscos...

1,110 citations

Posted Content
TL;DR: In this article, the authors review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics.
Abstract: We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis induces us to introduce a hydrodynamic model suitable to revisit in Sect. 3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In Sect. 4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order $0< \beta <2$. Led by our analysis we express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).

1,064 citations


Cites background from "Hydrodynamic Theory of the Velocity..."

  • ...[75-97]; in most cases hydrodynamic models are adopted....

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Book
05 Oct 2014

756 citations

Journal ArticleDOI
TL;DR: In this article, the friction coefficients for the uniform rotation of spheroids in a viscous fluid with the slipping boundary condition were computed numerically and reported in tabular form.
Abstract: Friction coefficients for the uniform rotation of prolate and oblate spheroids in a viscous fluid, with the slipping boundary condition, are computed numerically and reported in tabular form.

674 citations

Journal ArticleDOI
TL;DR: In this paper, a review of the development, present state, and future directions of the generalized Stokes-Einstein relation (GSER) in microrheology is presented.
Abstract: In microrheology, the local and bulk mechanical properties of a complex fluid are extracted from the motion of probe particles embedded within it. In passive microrheology, particles are forced by thermal fluctuations and probe linear viscoelasticity, whereas active microrheology involves forcing probes externally and can be extended out of equilibrium to the nonlinear regime. Here we review the development, present state, and future directions of this field. We organize our review around the generalized Stokes-Einstein relation (GSER), which plays a central role in the interpretation of microrheology. By discussing the Stokes and Einstein components of the GSER individually, we identify the key assumptions that underpin each, and the consequences that occur when they are violated. We conclude with a discussion of two techniques—multiple particle-tracking and nonlinear microrheology— that have arisen to handle systems in which the GSER breaks down.

591 citations


Additional excerpts

  • ...Following Zwanzig & Bixon (1970), the hydrodynamic mobility M and resistance ζ are generalized for LVE materials according to V(t) = ∫ t −∞ M(t − t′)FH(t′)dt′ and FH(t) = ∫ t −∞ ζ (t − t′)V(t′)dt′, (6) where FH is the hydrodynamic force on the particle....

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