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Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids

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Abstract

The problem of sub-shock formation within a shock structure solution of hyperbolic systems of balance laws is investigated for a binary mixture of multi-temperature Eulerian fluids. The main purpose of this work is the analysis of the ranges of Mach numbers characterizing shock-structure solutions with different features, continuous or not, and to show the existence of ranges, below the maximum unperturbed characteristic velocity, for which each constituent of the mixture may develop a sub-shock within a smooth shock structure profile. The theoretical results are supported by numerical calculations.

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References

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

  5. 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, 78, 31–43 (2006)

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

  7. 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 

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

  9. Müller, I., Ruggeri, T.: Rational extended thermodynamics (2nd Ed.). Springer (1998)

  10. 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  MATH  Google Scholar 

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

  12. Ruggeri, T., Simić, S.: Average temperature and Maxwellian iteration in multi-temperature mixtures of fluids. Phys. Rev. E 80, 026317 (2009)

    Article  Google Scholar 

  13. 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 

  14. 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 

  15. Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics beyond the Monatomic Gas. Springer Cham, Heidelberg, New York, Dordrecht, London (2015)

    Book  MATH  Google Scholar 

  16. 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  MATH  Google Scholar 

  17. Bisi, M., Conforto, F., Martalò, G.: Sub-shock formation in Grad 10 moment equations for a binary gas mixture. Continuum Mech. Thermodyn. (2015). doi:10.1007/s00161-015-0476-8

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  22. Currò, C., Fusco, D.: Discontinuous travelling wave solutions for a class of dissipative hyperbolic models. Rend. Mat. Acc. Linceis. 916(1), 61–71 (2005)

  23. Bisi, M., Martalò, G., Spiga, G.: Multi-temperature fluid-dynamic model equations from kinetic theory in a reactive gas: the steady shock problem. Comp. Math. Appl. 66, 1403–1417 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Conforto, F., Monaco, R., Ricciardello, A.: Discontinuous shock structure in a reacting mixture modelled by grad 13 moment approximation. Acta Appl. Math. 132(1), 225–236 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bisi, M., Conforto, F., Martalò, G.: Subshock formation in reacting gas mixtures. Proceedings of the conference Particle systems and Partial Differential Equations III, University of Minho, 2014. P. Gonalves, A.J. Soares (eds.), Springer, (2016 in press)

  26. 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 (2015)

  27. Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory and wave velocities. Cont. Mech. Thermodyn. 9, 205–212 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mentrelli, A., Ruggeri, T., Sugiyama, M., Zhao, N.: Interaction between a shock and an acceleration wave in a perfect gas for increasing shock strength. Wave Motion 45, 498–517 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhao, N., Mentrelli, A., Ruggeri, T., Sugiyama, M.: Admissible shock waves and shock-induced phase transitions in a van der waals fluid. Phys. Fluids 23, 086101 (2011)

    Article  MATH  Google Scholar 

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Acknowledgments

This work has been supported by GNFM/INdAM Young Researchers Project 2014 ‘Subshock formation in gas mixtures’ (PI: F. Conforto); GNFM/INdAM Young Researchers Project 2015 ‘An Eulerian/Lagrangian model for combustion fronts’ (PI: A. Mentrelli) and by University of Bologna FARB 2012 Project ‘Extended Thermodynamics of Non-Equilibrium Processes from Macro- to Nano-Scale’ (PI: T. Ruggeri).

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, University of Messina, V.le F. Stagno d’Alcontres 31, Messina, Italy

    Fiammetta Conforto

  2. Department of Mathematics and Alma Mater Research Center on Applied Mathematics (AM²), University of Bologna, via Saragozza, 8, Bologna, Italy

    Andrea Mentrelli & Tommaso Ruggeri

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  1. Fiammetta Conforto
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  2. Andrea Mentrelli
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  3. Tommaso Ruggeri
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Correspondence to Fiammetta Conforto.

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Conforto, F., Mentrelli, A. & Ruggeri, T. Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids. Ricerche mat 66, 221–231 (2017). https://doi.org/10.1007/s11587-016-0299-3

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