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Molecular Extended Thermodynamics of a Rarefied Polyatomic Gas

  • Tommaso RuggeriEmail author
Chapter
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Part of the Springer INdAM Series book series (SINDAMS, volume 27)

Abstract

Extended Thermodynamics can be considered as a theory of continuum with structure because there are new field variables with respect to the classical approach and they are dictated at mesoscopic level by the kinetic theory. In this survey I present some recent results on the so called Molecular Extended Thermodynamics (MET) in which the macroscopic fields are related to the moments of a distribution function that for polyatomic gas contains an extra variable taking into account the internal degrees of freedom of a molecule. The closure is obtained via the variational procedure of the Maximum Entropy Principle (MEP). Particular attention will be paid on the simple model of MET with six independent fields, i.e., the mass density, the velocity, the temperature and the dynamic pressure, without adopting near-equilibrium approximation. The model obtained is the simplest example of non-linear dissipative fluid after the ideal case of Euler. The system is symmetric hyperbolic with the convex entropy density and the K-condition is satisfied. Therefore, in contrast to the Euler case, there exist global smooth solutions provided that the initial data are sufficiently smooth.

References

  1. 1.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer Tracts in Natural Philosophy, vol. 37. Springer, New York (1998)CrossRefGoogle Scholar
  2. 2.
    Liu, I.-S., Müller, I.: Extended thermodynamics of classical and degenerate ideal gases. Arch. Rat. Mech. Anal. 83, 285–332 (1983)Google Scholar
  3. 3.
    Liu, I.-S., Müller, I., Ruggeri, T.: Relativistic thermodynamics of gases. Ann. Phys. 169, 191–219 (1986)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Müller, I., Ruggeri, T.: Extended Thermodynamics. Springer Tracts in Natural Philosophy, vol. 37. Springer, New York (1993)CrossRefGoogle Scholar
  5. 5.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957); Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev. 108, 171–190 (1957)Google Scholar
  6. 6.
    Kapur, J.N.: Maximum Entropy Models in Science and Engineering. Wiley, New York (1989)Google Scholar
  7. 7.
    Kogan, M.N.: Rarefied Gas Dynamics. Plenum Press, New York (1969)CrossRefGoogle Scholar
  8. 8.
    Grad, H.: On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331–407 (1949)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dreyer, W.: Maximization of the entropy in non-equilibrium. J. Phys. A: Math. Gen. 20, 6505–6517 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Continuum Mech. Thermodyn. 9, 205–212 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Continuum Mech. Thermodyn. 9, 205–212 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics beyond the Monatomic Gas. Springer, Cham-Heidelbergh-New York-Dorderecht-London (2015)CrossRefGoogle Scholar
  13. 13.
    Godunov, S.K.: An interesting class of quasilinear systems. Sov. Math. 2, 947 (1961)Google Scholar
  14. 14.
    Boillat, G.: Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systémes hyperboliques. C. R. Acad. Sci. Paris A 278, 909–912 (1974)Google Scholar
  15. 15.
    Ruggeri, T., Strumia, A.: Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics. Ann. Inst. H. Poincaré Sect. A 34, 65–84 (1981)Google Scholar
  16. 16.
    Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. The structure of extended thermodynamics. Continuum Mech. Thermodyn. 1, 3–20 (1989)Google Scholar
  17. 17.
    Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rat. Mech. Anal. 137, 305–320 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Brini, F., Ruggeri, T.: Entropy principle for the moment systems of degree α associated to the Boltzmann equation. Critical derivatives and non controllable boundary data. Continuum Mech. Thermodyn. 14, 165 (2002)Google Scholar
  20. 20.
    Kremer, G.M.: An Introduction to the Boltzmann Equation and Transport Processes in Gases. Springer, Berlin (2010)CrossRefGoogle Scholar
  21. 21.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Continum Mech. Thermodyn. 24, 271–292 (2012)CrossRefGoogle Scholar
  22. 22.
    Borgnakke, C., Larsen, P.S.: Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. J. Comput. Phys. 18, 405–420 (1975)CrossRefGoogle Scholar
  23. 23.
    Bourgat, J.-F., Desvillettes, L., Le Tallec, P., Perthame, B.: Microreversible collisions for polyatomic gases. Eur. J. Mech. B/Fluids 13, 237–254 (1994)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Pavić, M., Ruggeri, T., Simić, S.: Maximum entropy principle for rarefied polyatomic gases. Physica A 392, 1302–1317 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Arima, T., Mentrelli, A., Ruggeri, T.: Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments. Ann. Phys. 345, 111–140 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Monatomic gas as a singular limit of polyatomic gas in molecular extended thermodynamics with many moments. Ann. Phys. 372, 83–109 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ikenberry, E., Truesdell, C.: On the pressure and the flux of energy in a gas according to Maxwell’s kinetic theory. J. Rat. Mech. Anal. 5, 1–54 (1956)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ruggeri, T.: Can constitutive relations be represented by non-local equations? Quart. Appl. Math. 70, 597–611 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pennisi, S., Ruggeri, T.: Relativistic extended thermodynamics of rarefied polyatomic gas. Ann. Phys. 377, 414–445 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of real gases with dynamic pressure: An extension of Meixner’s theory. Phys. Lett. A 376, 2799–2803 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Nonlinear extended thermodynamics of real gases with 6 fields. Int. J. Non-Linear Mech. 72, 6–15 (2015)CrossRefGoogle Scholar
  32. 32.
    Ruggeri, T.: Non-linear maximum entropy principle for a polyatomic gas subject to the dynamic pressure. Bull. Inst. Math. Acad. Sinica (New Series) 11(1), 1–22 (2016)Google Scholar
  33. 33.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Recent results on nonlinear extended thermodynamics of real gases with six fields Part I: general theory. Ric. Mat. 65, 263–277 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Bisi, M., Ruggeri, T., Spiga, G.: Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamic. Kinetic and related models (KRM) 11, 71–95 (2018)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Meixner, J.: Absorption und dispersion des schalles in gasen mit chemisch reagierenden und anregbaren komponenten. I. Teil. Ann. Physik 43, 470 (1943)CrossRefGoogle Scholar
  36. 36.
    Meixner, J.: Allgemeine theorie der schallabsorption in gasen und flussigkeiten unter berucksichtigung der transporterscheinungen. Acoustica 2, 101 (1952)MathSciNetGoogle Scholar
  37. 37.
    Secchi, P.: Existence theorems for compressible viscous fluid having zero shear viscosity, Rend. Sem. Padova 70, 73–102 (1983)zbMATHGoogle Scholar
  38. 38.
    Frid, H., Shelukhin, V.: Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry. SIAM J. Math. Anal. 31(5), 1144–1156 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften, vol. 325, 3rd edn. Springer, Berlin Heidelberg (2010)CrossRefGoogle Scholar
  40. 40.
    Kawashima, S., Shizuta, Y.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14, 249–275 (1985)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Hanouzet, B., Natalini, R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rat. Mech. Anal. 169, 89–117 (2003)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Yong, W.-A.: Entropy and global existence for hyperbolic balance laws. Arch. Rat. Mech. Anal. 172(2), 247–266 (2004)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Bianchini, S., Hanouzet, B., Natalini, R.: Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Comm. Pure Appl. Math., 60, 1559–1622 (2007)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Ruggeri, T., Serre, D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Quart. Appl. Math 62(1), 163–179 (2004)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Lou, J., Ruggeri, T.: Acceleration waves and weak Shizuta-Kawashima condition. Suppl. Rend. Circ. Mat. Palermo. Non Linear Hyperbolic Fields and Waves. A tribute to Guy Boillat, Series II, Suppl. 78, 187–200 (2006)Google Scholar
  46. 46.
    Ruggeri, T.: Entropy Principle and Global Existence of Smooth Solutions in Extended Thermodynamics. In: Hyperbolic Problems: Theory, Numerics, Applications, Vol. II, pp. 267–274. Yokohama Publishers Inc. (2006)Google Scholar
  47. 47.
    Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: Beyond the Bethe-Teller theory., Phys. Rev. E 89 013025-1–013025-11 (2014)Google Scholar
  48. 48.
    Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Effect of dynamic pressure on the shock wave structure in a rarefied polyatomic gas. Phys. Fluids 26, 016103-1–016103-15 (2014)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Alma Mater Research Center of Applied Mathematics AM2University of BolognaBolognaItaly

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