Effect of the dynamic pressure on the similarity solution of cylindrical shock waves in a rarefied polyatomic gas

Abstract

The similarity solution for a strong cylindrical shock wave in a rarefied polyatomic gas is analyzed on the basis of Rational Extended Thermodynamics with six independent fields; the mass density, the velocity, the pressure and the dynamic pressure. A new ODE system for the similarity solution is derived in a systematic way by using the method based on the Lie group theory proposed in the context of the spherical shock wave in a rarefied monoatomic gas in Donato and Ruggeri (J Math Anal Appl 251:395, 2000). The boundary conditions are also specified from the Rankine–Hugoniot conditions for the sub-shock. The derived similarity solution is characterized by only one dimensionless parameter \(\alpha \) related to the relaxation time for the dynamic pressure. The numerical analysis of the similarity solution is also performed. The solution agrees with the well-known Sedov–von Neumann–Taylor (SNT) solution when \(\alpha \) is small. When \(\alpha \) is larger, due to the presence of the dynamic pressure, the deviation from the SNT solution is evident; the strength of a peak near the shock front becomes smaller and the profile becomes broader.

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

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

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

    Google Scholar 

  2. 2.

    Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, 2nd edn. Pergamon Press, London (1987)

    Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

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

    Google Scholar 

  5. 5.

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

    Google Scholar 

  6. 6.

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

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 8.

    Taniguchi, S., Ruggeri, T.: On the sub-shock formation in extended thermodynamics. Int. J. Non Linear Mech. 99, 69 (2018)

    Article  Google Scholar 

  9. 9.

    Bisi, M., Martalò, G., Spiga, G.: Shock wave structure of multi-temperature Euler equations from kinetic theory for a binary mixtures. Acta Appl. Math. 132, 95 (2014)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Conforto, F., Mentrelli, A., Ruggeri, T.: Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids. Ric. Mat. 66, 221 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ruggeri, T., Taniguchi, S.: Shock Waves in Hyperbolic Systems of Non-Equilibrium Thermodynamics. In: Berezovski, A., Soomere, T. (eds.) Applied Wave Mathematics II, pp. 167–186. Springer, Cham (2019)

    Google Scholar 

  12. 12.

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

    MathSciNet  MATH  Google Scholar 

  13. 13.

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

    Google Scholar 

  14. 14.

    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 

  15. 15.

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

    Article  Google Scholar 

  16. 16.

    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)

    MathSciNet  Article  Google Scholar 

  17. 17.

    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 

  18. 18.

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

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Bisi, M., Ruggeri, T., Spiga, G.: Dynamical pressure in a polyatomic gas: interplay between kinetic theory and extended thermodynamics. Kinet. Relat. Models 11, 71 (2018)

    MathSciNet  Article  Google Scholar 

  20. 20.

    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 

  21. 21.

    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 

  22. 22.

    Kosuge, S., Aoki, K., Goto, T.: Shock wave structure in polyatomic gases: numerical analysis using a model Boltzmann equation. In: AIP Conference Proceedings, vol. 1786, pp. 180004 (2016)

  23. 23.

    Kosuge, S., Aoki, K.: Shock-wave structure for a polyatomic gas with large bulk viscosity. Phys. Rev. Fluids 3, 023401 (2018)

    Article  Google Scholar 

  24. 24.

    Kosuge, S., Kuo, H.W., Aoki, K.: A kinetic model for a polyatomic gas with temperature-dependent specific heats and its application to shock-wave structure. J. Stat. Phys. 177, 209 (2019)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Nagaoka, R., Taniguchi, S., Ruggeri, T.: Similarity solution of strong spherical shock waves in a rarefied polyatomic gas based on extended thermodynamics. In: AIP Conference Proceedings, vol. 2153, pp. 020014 (2019)

  26. 26.

    Donato, A., Ruggeri, T.: Similarity solutions and strong shocks in extended thermodynamics of rarefied gas. J. Math. Anal. Appl. 251, 395 (2000)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Donato, A., Oliveri, F.: When nonautonomous equations are equivalent to autonomous ones. Appl. Anal. 58, 313 (1995)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Sedov, L.I.: Similarity and Dimensional Methods in Mechanics, 10th edn. CRC Press, Boca Raton (1993)

    Google Scholar 

  29. 29.

    Ikenberry, E., Truesdell, C.: On the pressure and the flux of energy in a gas according to Maxwell’s kinetic theory, I. J. Ration. Mech. Anal. 5, 1 (1956)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The present paper is dedicated to Professor Masaru Sugiyama who invited the author to the present research field and always encourages him to proceed with the study. The author thanks Professor Tommaso Ruggeri for valuable discussions. The results contained in the present paper have been partially presented in WASCOM 2019. This work was supported by JSPS KAKENHI Grant Number JP19K04204.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shigeru Taniguchi.

Ethics declarations

Conflict of interest

The author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Taniguchi, S. Effect of the dynamic pressure on the similarity solution of cylindrical shock waves in a rarefied polyatomic gas. Ricerche mat (2020). https://doi.org/10.1007/s11587-020-00505-9

Download citation

Keywords

  • Shock wave
  • Extended thermodynamics
  • Similarity solution
  • Cylindrical symmetry
  • Rarefied polyatomic gas
  • Dynamic pressure

Mathematics Subject Classification

  • 76L05
  • 82C35
  • 35L60
  • 76N15
  • 76M60