Microscopic statistical description of incompressible Navier-Stokes granular fluids

Regular Article

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.

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Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Massimo Tessarotto
    • 1
    • 2
  • Michael Mond
    • 3
  • Claudio Asci
    • 1
  1. 1.Department of Mathematics and GeosciencesUniversity of TriesteTriesteItaly
  2. 2.Institute of Physics, Faculty of Philosophy and ScienceSilesian University in OpavaOpavaCzech Republic
  3. 3.Department of Mechanical EngineeringBen Gurion University of the NegevBe’er ShevaIsrael

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