Skip to main content
SpringerLink
Log in

Shock structure and multiple sub-shocks in Grad 10-moment binary mixtures of monoatomic gases

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

The problem of sub-shock occurrence within a shock structure solution is investigated for an inert binary mixture of monoatomic gases, modelled by a Grad 10-moment approximation of the Boltzmann equations. The main purpose of this paper is to show by numerical simulations the existence of discontinuous shock structure solutions for values of the shock speed below the maximum unperturbed characteristic velocity. Moreover, for suitable concentrations of the two species, and for shock velocities beyond the maximum unperturbed characteristic velocity, each constituent of the mixture generates a jump discontinuity, and the shock structure solution exhibits two sub-shocks.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Gilbarg, D., Paolucci, D.: The structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2, 617–642 (1953)

    MathSciNet  MATH  Google Scholar 

  2. Ruggeri, T.: Breakdown of shock-wave-structure solutions. Phys. Rev. E 47(6), 4135–4140 (1993)

    Article  MathSciNet  Google Scholar 

  3. Ruggeri, T.: Non existence of shock structure solutions for hyperbolic dissipative systems including characteristic shocks. Appl. Anal. 57, 23–33 (1995)

    Article  MathSciNet  Google Scholar 

  4. Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E 52(6), R5760–R5763 (1995)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Ruggeri, T., Simić, S.: Non linear wave propagation in binary mixtures of Euler fluids. In: Monaco, R., et al. (eds.) Proceedings XII International Conference on Waves and Stability in Continuous Media, World Scientific, Singapore, pp. 455–462 (2004)

  8. Simić, S.: Shock structure in continuum models of gas dynamics: stability and bifurcation analysis. Nonlinearity 22, 1337–1366 (2009)

    Article  MathSciNet  Google Scholar 

  9. Madjarevic, D., Simić, S.: Shock structure in Helium–Argon mixture—a comparison of hyperbolic multi-temperature model with experiment. EPL 102, 44002 (2013)

    Article  Google Scholar 

  10. Madjarević, D., Ruggeri, T., Simić, S.: Shock structure and temperature overshoot in macroscopic multi-temperature model of mixtures. Phys. Fluids 26, 106102 (2014)

    Article  Google Scholar 

  11. Bisi, M., Martalò, G., Spiga, G.: Shock wave structure of multi-temperature Euler equations from kinetic theory for a binary mixture. Acta Appl. Math. 132(1), 95–105 (2014)

    Article  MathSciNet  Google Scholar 

  12. Bisi, M., Conforto, F., Martalò, G.: Sub-shock formation in Grad 10 moment equations for a binary gas mixture. Contin. Mech. Thermodyn. 28(5), 1295–1324 (2016)

    Article  MathSciNet  Google Scholar 

  13. Conforto, F., Mentrelli, A., Ruggeri, T.: Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids. Ric. Mat. 66(1), 221–231 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  15. Conforto, F., Mentrelli, A., Ruggeri, T.: Shock structure and multiple sub-shocks in hyperbolic systems of balance laws: the case of a multi-temperature mixture of Eulerian fluids. Preprint (2018)

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. 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, 163–177 (1977)

    Article  MathSciNet  Google Scholar 

  19. Brini, F., Ruggeri, T.: The Riemann problem for a binary non-reacting mixture of Euler fluids. In: Monaco, R. et al., (eds.) Proceedings XII International Conference on Waves and Stability in Continuous Media. World Scientific, Singapore, pp. 102–108 (2004)

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

    MathSciNet  MATH  Google Scholar 

  21. Mentrelli, A., Ruggeri, T.: Asymptotic behavior of Riemann and Riemann with structure problems for a \(2\times 2\) hyperbolic dissipative system. Rend. Circ. Mat. Palermo Ser. II Suppl. 78, 31–43 (2006)

    MATH  Google Scholar 

  22. Brini, F., Ruggeri, T.: On the Riemann problem in extended thermodynamics. In: Asakura, F., et al., (eds.) Proceedings of the 10th International Conference on Hyperbolic Problems (HYP2004), vol. I, pp. 319–326. Yokohama Publishers Inc., (2006)

  23. Mentrelli, A., Ruggeri, T.: The Riemann problem for a hyperbolic model of incompressible fluids. Int. J. Nonlinear Mech. 51, 87–96 (2013)

    Article  Google Scholar 

  24. Bisi, M., Groppi, M., Spiga, G.: Grads distribution functions in the kinetic equations for a chemical reaction. Contin. Mech. Thermodyn. 14, 207–222 (2002)

    Article  MathSciNet  Google Scholar 

  25. Bisi, M., Groppi, M., Spiga, G.: Kinetic approach to chemically reacting gas mixtures. In: Pareschi, L., Russo, G., Toscani, G. (eds.) Modelling and Numerics of Kinetic Dissipative Systems, pp. 107–126. Nova Science, New York (2005)

    MATH  Google Scholar 

  26. Liotta, S.F., Romano, V., Russo, G.: Central schemes for balance laws of relaxation type. SIAM J. Numer. Anal. 38(4), 1337–1356 (2000)

    Article  MathSciNet  Google Scholar 

  27. Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)

    Article  MathSciNet  Google Scholar 

  28. Harnett, L.N., Muntz, E.P.: Experimental investigation of normal shock wave velocity distribution functions in mixtures of argon and helium. Phys. Fluids 15, 565 (1972)

    Article  Google Scholar 

Download references

Acknowledgements

This work is performed in the frame of activities sponsored by INdAM–GNFM, by Universities of Messina, and of Enna Kore. G. Martalò is a post-doc fellow supported by the National Institute of Advanced Mathematics (INdAM). This work is dedicated to Professor Tommaso Ruggeri on the occasion of his 70th birthday. The authors would like to express their deep and sincere gratitude to Professor Ruggeri for all the extremely useful discussions, the deep analysis and the constructive criticisms, which turned out to be crucial for the development of their research.

Author information

Authors and Affiliations

  1. University of Catania, Catania, Italy

    Valeria Artale

  2. University of Messina, Messina, Italy

    Fiammetta Conforto

  3. University of Naples Federico II, Naples, Italy

    Giorgio Martalò

  4. University of Enna Kore, Enna, Italy

    Angela Ricciardello

Authors
  1. Valeria Artale
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fiammetta Conforto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giorgio Martalò
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Angela Ricciardello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fiammetta Conforto.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Artale, V., Conforto, F., Martalò, G. et al. Shock structure and multiple sub-shocks in Grad 10-moment binary mixtures of monoatomic gases. Ricerche mat 68, 485–502 (2019). https://doi.org/10.1007/s11587-018-0421-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-018-0421-9

Keywords

Mathematics Subject Classification

Search

Navigation