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Dynamics of a dilute sheared inelastic fluid. II. The effect of correlations

Kumaran, V (2009) Dynamics of a dilute sheared inelastic fluid. II. The effect of correlations. In: Physical Review E, 79 (1, Par). 011302-1-011302-19.

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The effect of correlations on the viscosity of a dilute sheared inelastic fluid is analyzed using the ring-kinetic equation for the two-particle correlation function. The leading-order contribution to the stress in an expansion in epsilon=(1-e)(1/2) is calculated, and it is shown that the leading-order viscosity is identical to that obtained from the Green-Kubo formula, provided the stress autocorrelation function in a sheared steady state is used in the Green-Kubo formula. A systemmatic extension of this to higher orders is also formulated, and the higher-order contributions to the stress from the ring-kinetic equation are determined in terms of the terms in the Chapman-Enskog solution for the Boltzmann equation. The series is resummed analytically to obtain a renormalized stress equation. The most dominant contributions to the two-particle correlation function are products of the eigenvectors of the conserved hydrodynamic modes of the two correlated particles. In Part I, it was shown that the long-time tails of the velocity autocorrelation function are not present in a sheared fluid. Using those results, we show that correlations do not cause a divergence in the transport coefficients; the viscosity is not divergent in two dimensions, and the Burnett coefficients are not divergent in three dimensions. The equations for three-particle and higher correlations are analyzed diagrammatically. It is found that the contributions due to the three-particle and higher correlation functions to the renormalized viscosity are smaller than those due to the two-particle distribution function in the limit epsilon -> 0. This implies that the most dominant correlation effects are due to the two-particle correlations.

Item Type: Journal Article
Publication: Physical Review E
Publisher: The American Physical Society
Additional Information: Copyright of this article belongs to The American Physical Society.
Keywords: Boltzmann equation;eigenvalues and eigenfunctions;granular flow;hydrodynamics;Navier-Stokes equations;shear flow;viscosity.
Department/Centre: Division of Mechanical Sciences > Chemical Engineering
Date Deposited: 30 Apr 2009 05:11
Last Modified: 19 Sep 2010 05:28
URI: http://eprints.iisc.ac.in/id/eprint/19466

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