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Shock structure and sub-shocks formation in a mixture of polyatomic gases with large bulk viscosity

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Abstract

We continue our previous study on the sub-shock formation in a binary mixture of gases (Ruggeri in Phys. Fluids 34: 066116, 2022) by considering the dissipation due to the bulk viscosity in both constituents. We use the model of a mixture of gases proposed by T. Arima, T. Ruggeri, M. Sugiyama, and S. Taniguchi (Arima in Rend. Lincei Mat. Appl. 28: 495, 2017). For prescribed values of the mass ratio of the constituents and the degrees of freedom of molecules, we classify the regions depending on the concentration and the Mach number for which the sub-shock may exist inside the shock profile of one or both constituents or not. Furthermore, we perform for some mixtures of gas numerical calculations of the profile, showing the role of the dissipation with respect to the Eulerian gases. As we expect, the shock profile is more regularized, and in the case that it exists sub-shocks, the amplitude of the sub-shock is reduced compared to those of non-dissipative gases.

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Notes

  1. In [2] the production terms \({\hat{e}}_1\) and \({\hat{\omega }}_1\) differ from the present one by a factor 2. The present choice is better to compare with previous results in the case of Eulerian gases.

References

  1. Ruggeri, T., Taniguchi, S.: A complete classification of sub-shocks in the shock structure of a binary mixture of eulerian gases with different degrees of freedom. Phys. Fluids 34(6), 066116 (2022)

    Article  CAS  ADS  Google Scholar 

  2. Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Galilean invariance and entropy principle for a system of balance laws of mixture type. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28, 66–75 (2017)

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

    Book  Google Scholar 

  4. 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(44), 2799–2803 (2012)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  5. Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Non-linear extended thermodynamics of real gases with 6 fields. Int. J. Non-Linear Mech. 72, 6–15 (2015)

    Article  ADS  Google Scholar 

  6. Ruggeri, T.: Non-linear maximum entropy principle for a polyatomic gas subject to the dynamic pressure. Bull. Inst. Math. Acad. Sin. 11, 1–22 (2016)

    MathSciNet  Google Scholar 

  7. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Continuum Mech. Thermodyn. 24(4), 271–292 (2012)

    Article  CAS  ADS  Google Scholar 

  8. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics. Continuum Mech. Thermodyn. 25(6), 727–737 (2013)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  9. 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(1), 016103 (2014)

    Article  ADS  Google Scholar 

  10. 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, 66–75 (2016)

    Article  ADS  Google Scholar 

  11. Ruggeri, T., Sugiyama, M.: Classical and Relativistic Rational Extended Thermodynamics of Gases. Springer, Cham (2021)

    Book  Google Scholar 

  12. Ruggeri, T., Simić, S.: On the hyperbolic system of a mixture of eulerian fluids: a comparison between single- and multi-temperature models. Math. Methods Appl. Sci. 30(7), 827–849 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  13. Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Ration. Mech. Anal. 137(4), 305–320 (1997)

    Article  MathSciNet  Google Scholar 

  14. Arima, T., Mentrelli, A., Ruggeri, T.: Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments. Ann. Phys. 345, 111–140 (2014)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  15. Lax, P.D.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10(4), 537–566 (1957)

    Article  MathSciNet  Google Scholar 

  16. Boillat, G., Ruggeri, T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Continuum Mech. Thermodyn. 10(5), 285–292 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  17. Ruggeri, T.: Breakdown of shock-wave-structure solutions. Phys. Rev. E 47, 4135–4140 (1993)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  18. Liu, T.-P.: Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Commun. Pure Appl. Math. 30(6), 767–796 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  19. Liu, T.-P.: Large-time behavior of solutions of initial and initial-boundary value problems of a general system of hyperbolic conservation laws. Commun. Math. Phys. 55(2), 163–177 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  20. Brini, F., Ruggeri, T.: On the Riemann problem in extended thermodynamics. In: Proceedings of the 10th International Conference on Hyperbolic Problems (HYP2004), pp. 319–326. Yokohama Publisher Inc., Yokohama (2006)

  21. Brini, F., Ruggeri, T.: On the Riemann problem with structure in extended thermodynamics. Suppl. Rend. Circ. Mat. Palermo II(78), 31–43 (2006)

    MathSciNet  Google Scholar 

  22. Mentrelli, A., Ruggeri, T.: Asymptotic behavior of Riemann and riemann with structure problems for a 2\(\times \)2 hyperbolic dissipative system. Suppl. Rend. Circ. Mat. Palermo II(78), 201–225 (2006)

    MathSciNet  Google Scholar 

  23. Liu, T.-P.: In: Ruggeri, T. (ed.) Nonlinear hyperbolic-dissipative partial differential equations, pp. 103–136. Springer, Berlin, Heidelberg (1996)

  24. Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (2009)

    Book  Google Scholar 

  25. Brini, F., Ruggeri, T.: The Riemann problem for a binary non-reacting mixture of euler fluids. In: Monaco, R., Pennisi, S., Rionero, S., Ruggeri, T. (eds.) Proceedings XII Int. Conference on Waves and Stability in Continuous Media, pp. 102–108. World Scientific, Singapore (2004)

  26. Taniguchi, S., Ruggeri, T.: On the sub-shock formation in extended thermodynamics. Int. J. Non-Linear Mech. 99, 69–78 (2018)

    Article  ADS  Google Scholar 

  27. Taniguchi, S., Ruggeri, T.: A 2 \(\times \) 2 simple model in which the sub-shock exists when the shock velocity is slower than the maximum characteristic velocity. Ricerche mat. 68(1), 119–129 (2019)

    Article  MathSciNet  Google Scholar 

  28. Liotta, S.F., Romano, V., Russo, G.: Central schemes for balance laws of relaxation type. SIAM J. Numer. Anal. 38(4), 1337–1356 (2001)

    Article  MathSciNet  Google Scholar 

  29. Ruggeri, T., Simić, S.: Mixture of gases with multi-temperature: Identification of a macroscopic average temperature. In: Memorie dell’Accademia delle Scienze, Lettere ed Arti di Napoli, Proceedings Mathematical Physics Models and Engineering Sciences, pp. 455–465 (2008). http://www.societanazionalescienzeletterearti.it/pdf/Memorie%20SFM%20-%20Mathematical%20Physics%20Model%20(2008).pdf

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Acknowledgements

This paper is dedicated to the memory of Salvatore Rionero. In particular, Tommaso Ruggeri wants to recall that Salvatore was not only a great mathematician but also a fraternal unforgettable friend. This work is partially supported by GNFM- INdAM (TR) and by JSPS KAKENHI Grant Number JP19K04204 (ST).

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Authors and Affiliations

  1. Department of Mathematics and Alma Mater Research Center on Applied Mathematics, University of Bologna, Via Saragozza, 8, 40123, Bologna, Italy

    Tommaso Ruggeri

  2. Department of Creative Engineering, National Institute of Technology, Kitakyushu College, 5-20-1 Shii, Kitakyushu, 802-0985, Fukuoka, Japan

    Shigeru Taniguchi

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  1. Tommaso Ruggeri
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  2. Shigeru Taniguchi
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Correspondence to Tommaso Ruggeri.

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Ruggeri, T., Taniguchi, S. Shock structure and sub-shocks formation in a mixture of polyatomic gases with large bulk viscosity. Ricerche mat 73 (Suppl 1), 261–274 (2024). https://doi.org/10.1007/s11587-023-00788-8

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