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Microscopic statistical description of incompressible Navier-Stokes granular fluids

The European Physical Journal Plus volume 132, Article number: 213 (2017) Cite this 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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References

  1. S. Chapman, T. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press 1951)

  2. C. Cercignani, The Boltzmann Equation and Its Applications, in Applied Mathematical Sciences, Vol. 67 (Springer-Verlag, 1988)

  3. C. Bardos, F. Golse, D. Levermore, J. Stat. Phys. 63, 323 (1991)

    ADS  Article  Google Scholar 

  4. D. Belardinelli, R. Marra, Nonlinearity 27, 209 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  5. G.R. McNamara, G. Zanetti, Phys. Rev. Lett. 61, 2332 (1988)

    ADS  Article  Google Scholar 

  6. F. Higuera, S. Succi, R. Benzi, Europhys. Lett. 9, 345 (1989)

    ADS  Article  Google Scholar 

  7. S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, in Numerical Mathematics and Scientific Computation (Oxford Science Publications, 2001)

  8. S. Chen, H. Chen, D.O. Martinez, W.H. Matthaeus, Phys. Rev. Lett. 67, 3776 (1991)

    ADS  Article  Google Scholar 

  9. M. Tessarotto, M. Ellero, AIP Conf. Proc. 762, 108 (2005)

    ADS  Article  Google Scholar 

  10. M. Ellero, M. Tessarotto, Physica A 355, 233 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  11. M. Tessarotto, M. Ellero, Physica A 373, 142 (2007)

    ADS  Article  Google Scholar 

  12. M. Tessarotto, M. Ellero, P. Nicolini, Phys. Rev. A 75, 012105 (2007) arXiv:quant-ph/0606091v1

    ADS  Article  Google Scholar 

  13. M. Tessarotto, M. Ellero, P. Nicolini, AIP Conf. Proc. 1084, 33 (2008)

    ADS  Article  Google Scholar 

  14. M. Tessarotto, M. Ellero, N. Aslan, M. Mond, P. Nicolini, AIP Conf. Proc. 1084, 224 (2008)

    ADS  Article  Google Scholar 

  15. M. Tessarotto, C. Asci, C. Cremaschini, A. Soranzo, G. Tironi, Eur. Phys. J. Plus 127, 36 (2012)

    Article  Google Scholar 

  16. M. Tessarotto, M. Mond, D. Batic, Found. Phys. 46, 1127 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  17. M. Grmela, H.C. Ottinger, Phys. Rev. E 56, 6620 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  18. H.C. Ottinger, M. Grmela, Phys. Rev. E 56, 6633 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  19. Massimo Tessarotto, Claudio Cremaschini, Marco Tessarotto, Eur. Phys. J. Plus 128, 32 (2013)

    Article  Google Scholar 

  20. M. Tessarotto, C. Cremaschini, Phys. Lett. A 378, 1760 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  21. M. Tessarotto, C. Cremaschini, Eur. Phys. J. Plus 129, 157 (2014)

    Article  Google Scholar 

  22. M. Tessarotto, C. Cremaschini, Eur. Phys. J. Plus 129, 243 (2014)

    Article  Google Scholar 

  23. M. Tessarotto, C. Cremaschini, Phys. Lett. A 379, 1206 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  24. M. Tessarotto, C. Cremaschini, Eur. Phys. J. Plus 130, 91 (2015)

    Article  Google Scholar 

  25. M. Tessarotto, C. Asci, C. Cremaschini, A. Soranzo, G. Tironi, Eur. Phys. J. Plus 130, 160 (2015)

    Article  Google Scholar 

  26. M. Tessarotto, C. Asci, Phys. Lett. A 381, 1484 (2017)

    ADS  Article  Google Scholar 

  27. L. Boltzmann, Wiener Ber. 66, 275 (in WA I, paper 23 (1872))

  28. L. Boltzmann, Ann. Phys. 57, 773 (1896) (English translation by S.G. Brush, The kinetic theory of gases

    Article  Google Scholar 

  29. L. Boltzmann, Vorlesungen über Gasstheorie, 2 vols. (J.A. Barth, Leipzig, 1896-1898) English translation by H. Brush, Lectures on gas theory (University of California Press, 1964)

  30. L. Boltzmann, Ann. Phys. 60, 392 (1897) (English translation by S.G. Brush, The kinetic theory of gases

    Article  Google Scholar 

  31. D. Enskog, Kungl. Svensk Vet. Akade. 63, 4 (1921) (English translation by S.G. Brush)

    Google Scholar 

  32. C.E. Shannon, Bell Syst. Tech. J. 27, 379 (1948)

    Article  Google Scholar 

  33. H. Grad, Handbook Phys. XII, 205 (1958)

    ADS  Google Scholar 

  34. C. Cercignani, Theory and applications of the Boltzmann equation (Scottish Academic Press, Edinburgh and London, 1975)

  35. C. Cercignani, Mathematical methods in kinetic theory (Plenum Press, New York, 1969)

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Authors and Affiliations

  1. Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12/1, 34127, Trieste, Italy

    Massimo Tessarotto & Claudio Asci

  2. Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám.13, CZ-74601, Opava, Czech Republic

    Massimo Tessarotto

  3. Department of Mechanical Engineering, Ben Gurion University of the Negev, Be’er Sheva, Israel

    Michael Mond

Authors
  1. Massimo Tessarotto
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  2. Michael Mond
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Correspondence to Massimo Tessarotto.

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Tessarotto, M., Mond, M. & Asci, C. Microscopic statistical description of incompressible Navier-Stokes granular fluids. Eur. Phys. J. Plus 132, 213 (2017). https://doi.org/10.1140/epjp/i2017-11472-2

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  • DOI: https://doi.org/10.1140/epjp/i2017-11472-2