Skip to main content
SpringerLink
Log in

Recent results on nonlinear extended thermodynamics of real gases with six fields Part II: shock wave structure

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

On the basis of the nonlinear extended thermodynamics theory discussed in Part I, the shock wave structure in a rarefied non-polytropic gas is analyzed. It is found that the effect of nonlinearity in the constitutive equation on the shock wave structure becomes significant only when the Mach number is large. The deviation from the exponential decay in the relaxation profile of the mass density for large Mach numbers is also discussed.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Vincenti, W.G., Kruger Jr., C.H.: Introduction to Physical Gas Dynamics. Wiley, New York (1965)

    Google Scholar 

  2. Zel’dovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover Publications, Mineola (2002)

    Google Scholar 

  3. Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Recent results on nonlinear extended thermodynamics of real gases with six fields. Part I: general theory. Ricerche di Matematica (2016)

  4. Smiley, E.F., Winkler, E.H., Slawsky, Z.I.: Measurement of the vibrational relaxation effect in CO\(_2\) by means of shock tube interferograms. J. Chem. Phys. 20, 923 (1952)

    Article  Google Scholar 

  5. Smiley, E.F., Winkler, E.H.: Shock-tube measurements of vibrational relaxation. J. Chem. Phys. 22, 2018 (1954)

    Article  Google Scholar 

  6. Griffith, W.C., Bleakney, W.: Shock waves in gases. Am. J. Phys. 22, 597 (1954)

    Article  Google Scholar 

  7. Griffith, W., Brickl, D., Blackman, V.: Structure of shock waves in polyatomic gases. Phys. Rev. 102, 1209 (1956)

    Article  Google Scholar 

  8. 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 (1962)

    Article  Google Scholar 

  9. Griffith, W.C., Kenny, A.: On fully-dispersed shock waves in carbon dioxide. J. Fluid Mech. 3, 286 (1957)

    Article  Google Scholar 

  10. Gilbarg, D., Paolucci, D.: The structure of shock waves in the continuum theory of fluids. J. Rat. Mech. Anal. 2, 617 (1953)

    MathSciNet  MATH  Google Scholar 

  11. Bethe, H.A., Teller, E.: Deviations from Thermal Equilibrium in Shock Waves. University of Michigan, Ann Arbor (1941). (Reprinted by Engineering Research Institute)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  14. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press, Oxford (1994)

    Google Scholar 

  15. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  16. Zhdanov, V.M.: The kinetic theory of a polyatomic gas. Soviet Phys. JETP 26, 1187 (1968)

    Google Scholar 

  17. MacCormack, F.J.: Kinetic equations for polyatomic gases: the 17-moment approximation. Phys. Fluids 11, 2533 (1968)

    Article  Google Scholar 

  18. MacCormack, F.J.: Kinetic moment equations for a gas of polyatomic molecules with many internal degrees of freedom. Phys. Fluids 13, 1446 (1970)

    Article  Google Scholar 

  19. Mallinger, F.: Generalization of the Grad theory to polyatomic gases. Research Report RR-3581, INRIA (1998)

  20. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  22. Arima, T., Sugiyama, M.: Characteristic features of extended thermodynamics of dense gases. Atti della Accademia Peloritana dei Pericolanti 91(Suppl. 1), A1 (2013)

    MathSciNet  Google Scholar 

  23. Ruggeri, T., Sugiyama, M.: Recent developments in extended thermodynamics of dense and rarefied polyatomic gases. Acta Appl. Math. 132, 527 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Monatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics. Phys. Lett. A 377, 2136 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer, Heidelberg (2015)

    Book  MATH  Google Scholar 

  26. 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)

    Article  Google Scholar 

  27. Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Shock wave structure in a rarefied polyatomic gas based on extended thermodynamics. Acta Appl. Math. 132, 583 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Effect of the dynamic pressure on the shock wave structure in a rarefied polyatomic gas. Phys. Fluids 26, 016103 (2014)

    Article  Google Scholar 

  29. 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 (2012)

    Article  MathSciNet  Google Scholar 

  30. Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: On the six-field model of fluids based on extended thermodynamics. Meccanica 49, 2181 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Nonlinear extended thermodynamics of real gases with 6 fields. Int. J. Non Linear Mech. 72, 6 (2015)

    Article  Google Scholar 

  33. Ruggeri, T.: Non-linear maximum entropy principle for a polyatomic gas subject to the dynamic pressure. Bull. Inst. Math. Acad. Sinica (N. Ser.) 11(1), 1 (2016)

    MATH  Google Scholar 

  34. Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure. Int. J. Non Linear Mech. 79, 66 (2016)

    Article  Google Scholar 

  35. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics. Cont. Mech. Thermodyn. 25, 727 (2013)

    Article  MathSciNet  Google Scholar 

  36. Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E 52, R5760 (1995)

    Article  Google Scholar 

  37. JSME Data Book.: Thermophysical Properties of Fluids. Japan Society of Mechanical Engineers, Tokyo (1983)

  38. Cramer, M.S.: Numerical estimates for the bulk viscosity of ideal gases. Phys. Fluids 24, 066102 (2012)

    Article  Google Scholar 

  39. Boillat, G., Ruggeri, T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Cont. Mech. Thermodyn. 10, 285 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was partially supported by Japan Society of Promotion of Science (JSPS) No. 15K21452 (TA) No. 25390150 (MS), and by National Group of Mathematical Physics GNFM-INdAM and by University of Bologna: FARB 2012 Project Extended Thermodynamics of Non-Equilibrium Processes from Macro- to Nano-Scale (TR).

Author information

Authors and Affiliations

  1. Department of Creative Engineering, National Institute of Technology, Kitakyushu College, Kitakyushu, Japan

    Shigeru Taniguchi

  2. Department of Mechanical Engineering, Faculty of Engineering, Kanagawa University, Yokohama, Japan

    Takashi Arima

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

    Tommaso Ruggeri

  4. Graduate School of Engineering, Nagoya Institute of Technology, Nagoya, Japan

    Masaru Sugiyama

Authors
  1. Shigeru Taniguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takashi Arima
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tommaso Ruggeri
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Masaru Sugiyama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tommaso Ruggeri.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Taniguchi, S., Arima, T., Ruggeri, T. et al. Recent results on nonlinear extended thermodynamics of real gases with six fields Part II: shock wave structure. Ricerche mat. 65, 279–288 (2016). https://doi.org/10.1007/s11587-016-0280-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-016-0280-1

Keywords

Mathematics Subject Classification

Search

Navigation