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Acta Applicandae Mathematicae

, Volume 132, Issue 1, pp 583–593 | Cite as

Shock Wave Structure in a Rarefied Polyatomic Gas Based on Extended Thermodynamics

  • Shigeru TaniguchiEmail author
  • Takashi Arima
  • Tommaso Ruggeri
  • Masaru Sugiyama
Article

Abstract

A theory of the shock wave structure in a rarefied polyatomic gas is developed on the basis of the recent new approach to extended thermodynamics. We summarize the following points (i) and (ii) based on the previous study on this subject and also show the new point (iii): (i) The theory can explain the change of types of the shock wave structure observed experimentally with the increase of the Mach number from unity; the nearly symmetric shock wave structure (Type A, small Mach number), the asymmetric structure (Type B, moderate Mach number), and the structure composed of thin and thick layers (Type C, large Mach number). (ii) The theoretical prediction of the mass density profile agrees well with experimental data. (iii) The points (i) and (ii) are not strongly affected by the details of the temperature dependence of the bulk viscosity.

Keywords

Shock wave structure Extended thermodynamics Rarefied polyatomic gas Bethe–Teller theory Navier–Stokes–Fourier theory 

Mathematics Subject Classification (2000)

76L05 82C35 76N15 35L60 76P05 

References

  1. 1.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998) CrossRefzbMATHGoogle Scholar
  2. 2.
    Vincenti, W.G., Kruger, C.H. Jr.: Introduction to Physical Gas Dynamics. Wiley, New York (1965) Google Scholar
  3. 3.
    Zel’dovich, Ya.B., Raizer, Yu.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover, New York (2002) Google Scholar
  4. 4.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Contin. Mech. Thermodyn. 24, 271–292 (2012) CrossRefzbMATHGoogle Scholar
  5. 5.
    Arima, T., Sugiyama, M.: Characteristic features of extended thermodynamics of dense gases. Atti Accad. Pelorit. Pericol., Cl. Sci. Med.-Biol. 91(1), A1–A15 (2013) MathSciNetGoogle Scholar
  6. 6.
    Ruggeri, T., Sugiyama, M.: Recent developments in extended thermodynamics of dense and rarefied polyatomic gases. Acta Appl. Math. (in press) Google Scholar
  7. 7.
    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) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics. Contin. Mech. Thermodyn. 25, 727–737 (2013) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: A study of linear waves based on extended thermodynamics for rarefied polyatomic gases. Acta Appl. Math. (in press). doi: 10.1007/s10440-014-9888-x
  10. 10.
    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 (2014) CrossRefGoogle Scholar
  11. 11.
    Smiley, E.F., Winkler, E.H., Slawsky, Z.I.: Measurement of the vibrational relaxation effect in CO2 by means of shock tube interferograms. J. Chem. Phys. 20, 923–924 (1952) CrossRefGoogle Scholar
  12. 12.
    Smiley, E.F., Winkler, E.H.: Shock-tube measurements of vibrational relaxation. J. Chem. Phys. 22, 2018–2022 (1954) CrossRefGoogle Scholar
  13. 13.
    Griffith, W.C., Bleakney, W.: Shock waves in gases. Am. J. Phys. 22, 597–612 (1954) CrossRefGoogle Scholar
  14. 14.
    Griffith, W., Brickl, D., Blackman, V.: Structure of shock waves in polyatomic gases. Phys. Rev. 102, 1209–1216 (1956) CrossRefGoogle Scholar
  15. 15.
    Johannesen, N.H., Zienkiewicz, H.K., Blythe, P.A., Gerrard, J.H.: Experimental and theoretical analysis of vibrational relaxation regions in carbon dioxide. J. Fluid Mech. 13, 213–224 (1962) CrossRefGoogle Scholar
  16. 16.
    Griffith, W.C., Kenny, A.: On fully-dispersed shock waves in carbon dioxide. J. Fluid Mech. 3, 286–288 (1957) CrossRefGoogle Scholar
  17. 17.
    Gilbarg, D., Paolucci, D.: The structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2, 617–642 (1953) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Bethe, H.A., Teller, E.: Deviations from Thermal Equilibrium in Shock Waves. Reprinted by Engineering Research Institute, University of Michigan Google Scholar
  19. 19.
    Data Book, J.S.M.E.: Thermophysical Properties of Fluids. Japan Society of Mechanical Engineers, Tokyo (1983) Google Scholar
  20. 20.
    Haynes, W.M., Lide, D.R. (eds.): CRC Handbook of Chemistry and Physics, 91st edn. CRC Press, Boca Raton (2010) Google Scholar
  21. 21.
    Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E 52, R5760–R5763 (1995) CrossRefGoogle Scholar
  22. 22.
    Cramer, M.S.: Numerical estimates for the bulk viscosity of ideal gases. Phys. Fluids 24, 066102 (2012), 23 pp. CrossRefGoogle Scholar
  23. 23.
    Boillat, G., Ruggeri, T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin. Mech. Thermodyn. 10, 285–292 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    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 (2014) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Shigeru Taniguchi
    • 1
    • 2
    Email author
  • Takashi Arima
    • 3
    • 4
  • Tommaso Ruggeri
    • 5
  • Masaru Sugiyama
    • 1
  1. 1.Graduate School of EngineeringNagoya Institute of TechnologyNagoyaJapan
  2. 2.Department of Control and Information Systems EngineeringKitakyushu National College of TechnologyKitakyushuJapan
  3. 3.Center for Social Contribution and CollaborationNagoya Institute of TechnologyNagoyaJapan
  4. 4.Department of Mechanical Engineering, Faculty of EngineeringKanagawa UniversityYokohamaJapan
  5. 5.Department of MathematicsUniversity of BolognaBolognaItaly

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