Skip to main content
SpringerLink
Log in

Molecular Extended Thermodynamics of a Rarefied Polyatomic Gas

  • Chapter
  • First Online:
Trends in Applications of Mathematics to Mechanics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 27))

  • 739 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    There are some typos in the paper [24] that were corrected in the Chapter 12 of the book [12].

References

  1. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer Tracts in Natural Philosophy, vol. 37. Springer, New York (1998)

    Book  Google Scholar 

  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. Liu, I.-S., Müller, I., Ruggeri, T.: Relativistic thermodynamics of gases. Ann. Phys. 169, 191–219 (1986)

    Article  MathSciNet  Google Scholar 

  4. Müller, I., Ruggeri, T.: Extended Thermodynamics. Springer Tracts in Natural Philosophy, vol. 37. Springer, New York (1993)

    Book  Google Scholar 

  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. Kapur, J.N.: Maximum Entropy Models in Science and Engineering. Wiley, New York (1989)

    Google Scholar 

  7. Kogan, M.N.: Rarefied Gas Dynamics. Plenum Press, New York (1969)

    Book  Google Scholar 

  8. Grad, H.: On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331–407 (1949)

    Article  MathSciNet  Google Scholar 

  9. Dreyer, W.: Maximization of the entropy in non-equilibrium. J. Phys. A: Math. Gen. 20, 6505–6517 (1987)

    Article  MathSciNet  Google Scholar 

  10. Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Continuum Mech. Thermodyn. 9, 205–212 (1997)

    Article  MathSciNet  Google Scholar 

  11. Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Continuum Mech. Thermodyn. 9, 205–212 (1997)

    Article  MathSciNet  Google Scholar 

  12. Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics beyond the Monatomic Gas. Springer, Cham-Heidelbergh-New York-Dorderecht-London (2015)

    Chapter  Google Scholar 

  13. Godunov, S.K.: An interesting class of quasilinear systems. Sov. Math. 2, 947 (1961)

    Google Scholar 

  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. 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. 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. Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rat. Mech. Anal. 137, 305–320 (1997)

    Article  MathSciNet  Google Scholar 

  18. Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021 (1996)

    Article  MathSciNet  Google Scholar 

  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. Kremer, G.M.: An Introduction to the Boltzmann Equation and Transport Processes in Gases. Springer, Berlin (2010)

    Book  Google Scholar 

  21. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Continum Mech. Thermodyn. 24, 271–292 (2012)

    Article  Google Scholar 

  22. Borgnakke, C., Larsen, P.S.: Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. J. Comput. Phys. 18, 405–420 (1975)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  24. Pavić, M., Ruggeri, T., Simić, S.: Maximum entropy principle for rarefied polyatomic gases. Physica A 392, 1302–1317 (2013)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  28. Ruggeri, T.: Can constitutive relations be represented by non-local equations? Quart. Appl. Math. 70, 597–611 (2012)

    Article  MathSciNet  Google Scholar 

  29. Pennisi, S., Ruggeri, T.: Relativistic extended thermodynamics of rarefied polyatomic gas. Ann. Phys. 377, 414–445 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  35. Meixner, J.: Absorption und dispersion des schalles in gasen mit chemisch reagierenden und anregbaren komponenten. I. Teil. Ann. Physik 43, 470 (1943)

    Article  Google Scholar 

  36. Meixner, J.: Allgemeine theorie der schallabsorption in gasen und flussigkeiten unter berucksichtigung der transporterscheinungen. Acoustica 2, 101 (1952)

    MathSciNet  Google Scholar 

  37. Secchi, P.: Existence theorems for compressible viscous fluid having zero shear viscosity, Rend. Sem. Padova 70, 73–102 (1983)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  39. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften, vol. 325, 3rd edn. Springer, Berlin Heidelberg (2010)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  42. Yong, W.-A.: Entropy and global existence for hyperbolic balance laws. Arch. Rat. Mech. Anal. 172(2), 247–266 (2004)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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. 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. 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 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Alma Mater Research Center of Applied Mathematics AM2, University of Bologna, Bologna, Italy

    Tommaso Ruggeri

Authors
  1. Tommaso Ruggeri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tommaso Ruggeri .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Pavia, Italy

    Elisabetta Rocca

  2. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Ulisse Stefanelli

  3. CNRS, Paris, Paris, France

    Lev Truskinovsky

  4. Dipartimento di Matematica, Università degli Studi di Trento, Povo di Trento, Trento, Italy

    Augusto Visintin

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ruggeri, T. (2018). Molecular Extended Thermodynamics of a Rarefied Polyatomic Gas. In: Rocca, E., Stefanelli, U., Truskinovsky, L., Visintin, A. (eds) Trends in Applications of Mathematics to Mechanics. Springer INdAM Series, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-75940-1_13

Download citation

Publish with us

Policies and ethics

Search

Navigation