Advertisement

Continuum Mechanics and Thermodynamics

, Volume 20, Issue 8, pp 489–508 | Cite as

On the ellipsoidal statistical model for polyatomic gases

  • Stéphane BrullEmail author
  • Jacques Schneider
Original Article

Abstract

The aim of this article is to construct a BGK-type model for polyatomic gases which gives in the hydrodynamic limit the proper transport coefficient. Its construction relies upon a systematic procedure: minimizing Boltzmann entropy under suitable moments constraints (Levermore in J Stat Phys 83:1021–1065, 1996; Brull and Schneider in Cont Mech Thermodyn 20(2):63–74, 2008). The obtained model corresponds to the ellipsoidal statistical model introduced in Andries et al. (Eur J Mech B Fluids 19:813–830, 2000). We also study the return to equilibrium of its solutions in the homogeneous case.

Keywords

Kinetic theory BGK operator Polyatomic gases 

PACS

05.20.Dd 02.60.Cb 02.30.Xx 02.60.Nm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andries P., Le Tallec P., Perlat J.P., Perthame B.: Entropy condition for the ES BGK model of Boltzmann equation for mono and polyatomic gases. Eur. J. Mech. B Fluids 19, 813–830 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arkeryd L.: Stability in L 1 for the spatially homogeneous equation. Arch. Rat. Mech. Anal. 103(2), 151–167 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bahi, Y.: Contribution la simulation numrique des coulements en gaz rarfis. Phd Thesis, University Pierre et Marie Curie, Paris (1997)Google Scholar
  4. 4.
    Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. Phys. Rev. 94, 511 (1954)zbMATHCrossRefGoogle Scholar
  5. 5.
    Borgnakke C., Larsen P.S.: Statistical collision model for Monte–Carlo simulation of polyatomic gas mixtures. J. Comp. Phys. 18, 405–420 (1975)CrossRefGoogle Scholar
  6. 6.
    Bourgat J.F., Desvillettes L., Le Tallec P., Perthame B.: Microreversible collisions for polyatomic gases and Boltzmann’s theorem. Eur. J. Mech. B Fluids 13(2), 237–254 (1994)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Brull S., Schneider J.: A new approach of the ellipsoidal statistical model. Cont. Mech. Thermodyn. 20(2), 63–74 (2008)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Cercignani, C.: The Boltzmann Equation and Its Applications, pp. 40–103. Scottish Academic Press, Edinburgh (1988)Google Scholar
  9. 9.
    Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases, 3rd Edn. Cambridge Mathematical Library (1970)Google Scholar
  10. 10.
    Csiszár I.: I-divergence geometry of probability distributions and minimization problems Sanov property. Ann. Probab. 3, 146–158 (1975)zbMATHCrossRefGoogle Scholar
  11. 11.
    Collet J.F.: Extensive Lyapounov functionals for moment-preserving evolution equations. C. R. A. S. Ser. I 334, 429–434 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Desvillettes L.: Sur un Modèle de type Borgnakke-Larsen Conduisant des lois d’Energie Non-linéaires en Température pour les Gaz Parfaits Polyatomiques. Ann. Fac. Sci. Toulouse Sér. 6(2), 257–262 (1997)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Goldstein, D., Sturtevant, B., Broadwell, J.E.: Investigation of the motion of discrete-velocity gases, In: Muntz, E.P., Weaver, D.P., Campbell, D.H. (Eds.) Rarefied gas dynamics: theoretical and computational techniques, progress in astronautics and aeronautics, vol. 118, pp. 100–117 (1989)Google Scholar
  14. 14.
    Holway, L.H.: Kinetic theory of shock structure using an ellipsoidal distribution function. In: Rarefied Gas Dynamics [Proceedings Fourth International Symposium, University Toronto (1964)], vol. I. Academie Press, New York, pp. 193–215 (1966)Google Scholar
  15. 15.
    Kusker I.: A model for rotational energy exchange in polyatomic gases. Physica A 158, 784–800 (1989)CrossRefGoogle Scholar
  16. 16.
    Le Tallec P.: A hierarchy of hyperbolic models linking Boltzmann to Navier Stokes equations for polyatomic gases. ZAMM 80(11–12), 779–790 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Levermore C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Michel P., Schneider J.: Approximation simultanée de réels par des nombres rationnels et noyau de collision de l’équation de Boltzmann. CRAS-série 1 Maths. 330(9), 857–862 (2000)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Mieussens, L.: Modèles à vitesses discrètes et méthodes numériques pour l’équation de Boltzmann BGK. Phd Thesis (1999)Google Scholar
  20. 20.
    Mieussens L.: Discrete velocity models and implicit scheme for the BGK equation rarefied gas dynamic. M3AS 10(8), 1121–1149 (2000)MathSciNetGoogle Scholar
  21. 21.
    Junk M.: Domain of definition of Levermore’s five-moment system. J. Stat. Phys. 93, 1143–1167 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Junk M.: Maximum entropy for reduced moment problems. M3AS 10, 1121–1149 (2000)Google Scholar
  23. 23.
    Rogier F., Schneider J.: A direct method for solving the Boltzmann equation. TTSP 23(1–3), 313–338 (1994)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Pullin D.I.: Kinetic models for polyatomic molecules with phenomenological energy exchange. Phys. Fluids 21, 209–216 (1978)CrossRefGoogle Scholar
  25. 25.
    Schneider J.: Entropic approximation in kinetic theory. M2AN 38(3), 541–561 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Palczewski A., Schneider J., Bobylev A.: A consistency result for a discrete-velocity model of the Boltzmann equation. SIAM. J. Numer. Anal. 34(5), 1865–1883 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Toscani G.: Remarks on entropy and equilibrium states. Appl. Math. Lett. 12(7), 19–25 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Villani C.: Fisher information estimates for Boltzmann’s collision operator. J. Maths Pures Appl. 77, 821–837 (1998)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Wennberg, B.: Stability and Exponential Convergence for the Boltzmann Equation. Phd thesis, Chalmers University Tech. (1993)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institut des mathématiques de Toulouse, UMR 5219Université Paul SabatierToulouse cedex 9France
  2. 2.ANLAUniversity of ToulonLa GardeFrance

Personalised recommendations