Continuum Mechanics and Thermodynamics

, Volume 25, Issue 6, pp 727–737 | Cite as

Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics

  • Takashi ArimaEmail author
  • Shigeru Taniguchi
  • Tommaso Ruggeri
  • Masaru Sugiyama
Original Article


We study the dispersion relation for sound in rarefied polyatomic gases (hydrogen, deuterium and hydrogen deuteride gases) basing on the recently developed theory of extended thermodynamics (ET) of dense gases. We compare the relation with those obtained in experiments and by the classical Navier–Stokes Fourier (NSF) theory. The applicable frequency range of the ET theory is proved to be much wider than that of the NSF theory. We evaluate the values of the bulk viscosity and the relaxation times involved in nonequilibrium processes. The relaxation time related to the dynamic pressure has a possibility to become much larger than the other relaxation times related to the shear stress and the heat flux.


Extended thermodynamics Rarefied polyatomic gases Dispersion relation for sound Phase velocity and absorption Bulk viscosity Relaxation time 


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  1. 1.
    Müller I., Ruggeri T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Continuum Mech. Thermodyn. (2011). doi: 10.1007/s00161-011-0213-x
  3. 3.
    Liu I.-S.: Extended thermodynamics of fluids and virial equations of state. Arch. Ration. Mech. Anal. 88, 1–23 (1985)ADSzbMATHGoogle Scholar
  4. 4.
    Kremer G.M.: Extended thermodynamics of non-ideal gases. Physica 144A, 156–178 (1987)ADSGoogle Scholar
  5. 5.
    Liu I.-S., Kremer G.M.: Hyperbolic system of field equations for viscous fluids. Mat. Aplic. Comp 9(2), 123–135 (1990)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Liu I.-S., Salvador J.A.: Hyperbolic system for viscous fluids and simulation of shock tube flows. Continuum Mech. Thermodyn. 2, 179–197 (1990)MathSciNetADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Kremer G.M: On extended thermodynamics of ideal and real gases. In: Sieniutycz, S., Salamon, P. (eds.) Extended Thermodynamics Systems, pp. 140–182. Taylor and Francis, New York (1992)Google Scholar
  8. 8.
    Carrisi M.C., Mele M.A., Pennisi S.: On some remarkable properties of an extended thermodynamics model for dense gases and macromolecular fluids. Proc. R. Soc. A 466, 1645–1666 (2010)MathSciNetADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Pavić, M., Ruggeri, T., Simić, S.: Maximum entropy principle for rarefied polyatomic gases. Physica A (submitted)Google Scholar
  10. 10.
    De Groot S.R., Mazur P.: Non-Equilibrium Thermodynamics. North-Holland, Amsterdam (1963)Google Scholar
  11. 11.
    Landau L.D., Lifshitz E.M.: Fluid Mechanics. Pergamon, London (1958)Google Scholar
  12. 12.
    Ikenberry E., Truesdell C.: On the pressure and the flux of energy in a gas according to Maxwell’s kinetic theory. J. Ration. Mech. Anal. 5, 1–54 (1956)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Muracchini A., Ruggeri T., Seccia L.: Dispersion relation in the high frequency limit and non linear wave stability for hyperbolic dissipative systems. Wave Motion. 15(2), 143–158 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rhodes E.J. Jr.: The velocity of sound in hydrogen when rotational degrees of freedom fail to be excited. Phys. Rev. 70(11), 932–938 (1946)ADSCrossRefGoogle Scholar
  15. 15.
    Sluijter C.G., Knaap H.F.P., Beenakker J.J.M.: Determination of rotational relaxation times of hydrogen isotopes by sound absorption measurements at low temperatures. I. Physica. 30, 745–762 (1964)ADSCrossRefGoogle Scholar
  16. 16.
    Landau L.D., Lifshitz E.M.: Statistical Physics. Pergamon, Oxford (1980)Google Scholar
  17. 17.
    Landau L.D., Lifshitz E.M.: Quantum Mechanics, Non-Relativistic Theory. Pergamon, Oxford (1977)Google Scholar
  18. 18.
    Radzig A.A., Smirnov B.M.: Reference Data on Atoms, Molecules, and Ions. Springer, Berlin (1985)CrossRefGoogle Scholar
  19. 19.
    Mason, W.P. (ed.): Physical Acoustics, Principles and Methods, Vol. II-Part A. Academic Press, New York (1965)Google Scholar
  20. 20.
    Vincenti W.G., Kruger C.H. Jr: Introduction to Physical Gas Dynamics. Wiley, New York (1965)Google Scholar
  21. 21.
    Chapman S., Cowling T.G.: The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge (1991)zbMATHGoogle Scholar
  22. 22.
    Eu B.C., Ohr Y.G.: Generalized hydrodynamics, bulk viscosity, and sound wave absorption and dispersion in dilute rigid molecular gases. Phys. Fluids. 13(3), 744–753 (2001)ADSCrossRefGoogle Scholar
  23. 23.
    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)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Emanuel G.: Bulk viscosity of a dilute polyatomic gas. Phys. Fluids. A 2(12), 2252–2254 (1990)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Meador W.E., Miner G.A., Townsend L.W.: Bulk viscosity as a relaxation parameter: fact or fiction?. Phys. Fluids. A 8(1), 258 (1996)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Emanuel G.: “Bulk viscosity as a relaxation parameter: fact or fiction?” [Phys. Fluids 8, 258 (1996)]. Phys. Fluids. A 8(7), 1984 (1996)ADSCrossRefGoogle Scholar
  27. 27.
    Emanuel G.: Bulk viscosity in the Navier–Stokes equations. Int. J. Eng. Sci. 36, 1313–1323 (1998)CrossRefGoogle Scholar
  28. 28.
    Graves R.E., Argrow B.M.: Bulk viscosity: past to present. J. Thermophys. Heat Transf. 13(3), 337–342 (1999)CrossRefGoogle Scholar
  29. 29.
    Bauer H.J.: Phenomenological theory of the relaxation phenomena in gases. In: Mason, W.P. (ed.) Physical Acoustics II Part A, pp. 47–131. Academic Press, New York (1965)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Takashi Arima
    • 1
    Email author
  • Shigeru Taniguchi
    • 2
  • Tommaso Ruggeri
    • 3
  • Masaru Sugiyama
    • 1
  1. 1.Graduate School of EngineeringNagoya Institute of TechnologyNagoyaJapan
  2. 2.Center for Social Contribution and CollaborationNagoya Institute of TechnologyNagoyaJapan
  3. 3.Department of Mathematics and Research Center of Applied Mathematics (CIRAM)University of BolognaBolognaItaly

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