Ricerche di Matematica

, Volume 68, Issue 2, pp 485–502 | Cite as

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

  • Valeria Artale
  • Fiammetta ConfortoEmail author
  • Giorgio Martalò
  • Angela Ricciardello


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.


Grad 10 moment approximation Mixtures of gases Sub-shock formation Rankine–Hugoniot conditions Riemann problem 

Mathematics Subject Classification

35Q20 58J45 35L67 



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.


  1. 1.
    Gilbarg, D., Paolucci, D.: The structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2, 617–642 (1953)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ruggeri, T.: Breakdown of shock-wave-structure solutions. Phys. Rev. E 47(6), 4135–4140 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ruggeri, T.: Non existence of shock structure solutions for hyperbolic dissipative systems including characteristic shocks. Appl. Anal. 57, 23–33 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E 52(6), R5760–R5763 (1995)CrossRefGoogle Scholar
  5. 5.
    Boillat, G., Ruggeri, T.: Hyperbolic principal subsystem: entropy convexity and subcharacteristic conditions. Arch. Ration. Mech. Anal. 137, 305–320 (1997)MathSciNetCrossRefGoogle 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(5), 285–292 (1998)MathSciNetCrossRefGoogle Scholar
  7. 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)Google Scholar
  8. 8.
    Simić, S.: Shock structure in continuum models of gas dynamics: stability and bifurcation analysis. Nonlinearity 22, 1337–1366 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Madjarevic, D., Simić, S.: Shock structure in Helium–Argon mixture—a comparison of hyperbolic multi-temperature model with experiment. EPL 102, 44002 (2013)CrossRefGoogle Scholar
  10. 10.
    Madjarević, D., Ruggeri, T., Simić, S.: Shock structure and temperature overshoot in macroscopic multi-temperature model of mixtures. Phys. Fluids 26, 106102 (2014)CrossRefGoogle Scholar
  11. 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)MathSciNetCrossRefGoogle Scholar
  12. 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)MathSciNetCrossRefGoogle Scholar
  13. 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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Taniguchi, S., Ruggeri, T.: On the sub-shock formation in extended thermodynamics. Int. J. Non-Linear Mech. 99, 69–78 (2018)CrossRefGoogle Scholar
  15. 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)Google Scholar
  16. 16.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, Berlin (1998)CrossRefGoogle Scholar
  17. 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)MathSciNetCrossRefGoogle Scholar
  18. 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)MathSciNetCrossRefGoogle Scholar
  19. 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)Google Scholar
  20. 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)MathSciNetzbMATHGoogle Scholar
  21. 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)zbMATHGoogle Scholar
  22. 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)Google Scholar
  23. 23.
    Mentrelli, A., Ruggeri, T.: The Riemann problem for a hyperbolic model of incompressible fluids. Int. J. Nonlinear Mech. 51, 87–96 (2013)CrossRefGoogle Scholar
  24. 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)MathSciNetCrossRefGoogle Scholar
  25. 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)zbMATHGoogle Scholar
  26. 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)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)MathSciNetCrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.University of CataniaCataniaItaly
  2. 2.University of MessinaMessinaItaly
  3. 3.University of Naples Federico IINaplesItaly
  4. 4.University of Enna KoreEnnaItaly

Personalised recommendations