Eur. Phys. J. Plus (2017) 132: 213
https://doi.org/10.1140/epjp/i2017-11472-2

Regular Article

Microscopic statistical description of incompressible Navier-Stokes granular fluids

¹ Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12/1, 34127, Trieste, Italy
² Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám.13, CZ-74601, Opava, Czech Republic
³ Department of Mechanical Engineering, Ben Gurion University of the Negev, Be’er Sheva, Israel

^* e-mail: maxtextss@gmail.com

Received: 18 November 2016
Accepted: 29 March 2017
Published online: 10 May 2017

Abstract

Based on the recently established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the validity of the Navier-Stokes fluid equation. In addition, it is shown that the conservation of the Boltzmann-Shannon entropy can similarly be warranted. Applications to the plane Couette and Poiseuille flows are considered showing that they can be regarded as final decaying states for suitable non-stationary flows. As a result, it is shown that an arbitrary initial stochastic 1-body PDF evolving in time by means of MMKE necessarily exhibits the phenomenon of Decay to Kinetic Equilibrium (DKE), whereby the same 1-body PDF asymptotically relaxes to a stationary and spatially uniform Maxwellian PDF.