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Ricerche di Matematica

, Volume 65, Issue 1, pp 279–288 | Cite as

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

  • Shigeru Taniguchi
  • Takashi Arima
  • Tommaso RuggeriEmail author
  • Masaru Sugiyama
Article

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.

Keywords

Shock wave structure Rarefied polyatomic gas  Extended thermodynamics Non-polytropic gas 

Mathematics Subject Classification

76L05 76N15 82C35 35L60 82D05 

Notes

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).

References

  1. 1.
    Vincenti, W.G., Kruger Jr., C.H.: Introduction to Physical Gas Dynamics. Wiley, New York (1965)Google Scholar
  2. 2.
    Zel’dovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover Publications, Mineola (2002)Google Scholar
  3. 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)Google Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Smiley, E.F., Winkler, E.H.: Shock-tube measurements of vibrational relaxation. J. Chem. Phys. 22, 2018 (1954)CrossRefGoogle Scholar
  6. 6.
    Griffith, W.C., Bleakney, W.: Shock waves in gases. Am. J. Phys. 22, 597 (1954)CrossRefGoogle Scholar
  7. 7.
    Griffith, W., Brickl, D., Blackman, V.: Structure of shock waves in polyatomic gases. Phys. Rev. 102, 1209 (1956)CrossRefGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 9.
    Griffith, W.C., Kenny, A.: On fully-dispersed shock waves in carbon dioxide. J. Fluid Mech. 3, 286 (1957)CrossRefGoogle Scholar
  10. 10.
    Gilbarg, D., Paolucci, D.: The structure of shock waves in the continuum theory of fluids. J. Rat. Mech. Anal. 2, 617 (1953)MathSciNetzbMATHGoogle Scholar
  11. 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. 12.
    Meixner, J.: Absorption und dispersion des schalles in gasen mit chemisch reagierenden und anregbaren komponenten. I. Teil. Ann. Physik 43, 470 (1943)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Meixner, J.: Allgemeine theorie der schallabsorption in gasen und flussigkeiten unter berucksichtigung der transporterscheinungen. Acoustica 2, 101 (1952)MathSciNetGoogle Scholar
  14. 14.
    Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press, Oxford (1994)Google Scholar
  15. 15.
    Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge (1991)zbMATHGoogle Scholar
  16. 16.
    Zhdanov, V.M.: The kinetic theory of a polyatomic gas. Soviet Phys. JETP 26, 1187 (1968)Google Scholar
  17. 17.
    MacCormack, F.J.: Kinetic equations for polyatomic gases: the 17-moment approximation. Phys. Fluids 11, 2533 (1968)CrossRefGoogle Scholar
  18. 18.
    MacCormack, F.J.: Kinetic moment equations for a gas of polyatomic molecules with many internal degrees of freedom. Phys. Fluids 13, 1446 (1970)CrossRefGoogle Scholar
  19. 19.
    Mallinger, F.: Generalization of the Grad theory to polyatomic gases. Research Report RR-3581, INRIA (1998)Google Scholar
  20. 20.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  21. 21.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Cont. Mech. Thermodyn. 24, 271 (2012)CrossRefzbMATHGoogle Scholar
  22. 22.
    Arima, T., Sugiyama, M.: Characteristic features of extended thermodynamics of dense gases. Atti della Accademia Peloritana dei Pericolanti 91(Suppl. 1), A1 (2013)MathSciNetGoogle Scholar
  23. 23.
    Ruggeri, T., Sugiyama, M.: Recent developments in extended thermodynamics of dense and rarefied polyatomic gases. Acta Appl. Math. 132, 527 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer, Heidelberg (2015)CrossRefzbMATHGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 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)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: On the six-field model of fluids based on extended thermodynamics. Meccanica 49, 2181 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rat. Mech. Anal. 137, 305 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)CrossRefGoogle Scholar
  33. 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)zbMATHGoogle Scholar
  34. 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)CrossRefGoogle Scholar
  35. 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)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E 52, R5760 (1995)CrossRefGoogle Scholar
  37. 37.
    JSME Data Book.: Thermophysical Properties of Fluids. Japan Society of Mechanical Engineers, Tokyo (1983)Google Scholar
  38. 38.
    Cramer, M.S.: Numerical estimates for the bulk viscosity of ideal gases. Phys. Fluids 24, 066102 (2012)CrossRefGoogle Scholar
  39. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2016

Authors and Affiliations

  • Shigeru Taniguchi
    • 1
  • Takashi Arima
    • 2
  • Tommaso Ruggeri
    • 3
    Email author
  • Masaru Sugiyama
    • 4
  1. 1.Department of Creative EngineeringNational Institute of Technology, Kitakyushu CollegeKitakyushuJapan
  2. 2.Department of Mechanical Engineering, Faculty of EngineeringKanagawa UniversityYokohamaJapan
  3. 3.Department of Mathematics and Alma Mater Research Center on Applied Mathematics AM2University of BolognaBolognaItaly
  4. 4.Graduate School of EngineeringNagoya Institute of TechnologyNagoyaJapan

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