Advertisement

Differential constraints and exact solutions for the ET6 model

  • Carmela Curró
  • Natale Manganaro
Article

Abstract

In this paper a first order quasilinear hyperbolic system describing a polyatomic gas far from the equilibrium is considered. After giving a classification of all the possible first order differential constraints admitted by the governing equations under interest, classes of exact solutions parameterized in terms of arbitrary functions are determined. That can help in solving initial or boundary value problems. Finally the consistency of the exact solutions characterized during the reduction procedure with the entropy principle is studied.

Keywords

Differential constraints Exact solutions Rational extended thermodynamics 

Mathematics Subject Classification

35L40 35L45 35N10 

Notes

Acknowledgements

This work was supported by Italian National Group of Mathematical Physics (GNFM-INdAM). The results contained in the present paper have been partially presented in Wascom 2017.

References

  1. 1.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer, Cham (2015)CrossRefzbMATHGoogle Scholar
  3. 3.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Contin. Mech. Thermodyn. 24, 271–292 (2012)CrossRefzbMATHGoogle Scholar
  4. 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, 2799–2803 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: On the six-field model of fluids based on extended thermodynamics. Meccanica 49, 2181–2187 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Nonlinear extended thermodynamics of real gases with 6 fields. Int. J. Non Linear Mech. 72, 6–15 (2015)CrossRefzbMATHGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    Janenko, N.N.: Compatibility theory and methods of integration of systems of nonlinear partial differential equation. In: Proceedings of the Fourth All-Union Mathematicians Congress (Leningrad) (Leningrad: Nauka), pp. 247–252 (1964)Google Scholar
  9. 9.
    Rozhdestvenski, B.L., Janenko, N.N.: Systems of Quasilinear Equations and Their applications to Gas Dynamics. (Translation of Mathematical Monographs vol. 55). Americal Mathematical Society, Providence Rhode Island (1983)Google Scholar
  10. 10.
    Sidorov, A.F., Shapeev, V.P., Janenko, N.N.: The Method of Differential Constraints and its Applications. Nauka, Novosibirsk (1984)Google Scholar
  11. 11.
    Raspopov, V.E., Shapeev, V.P., Yanenko, N.N.: The application of the method of differential constraints to one-dimensional gas dynamics equations. Izv. V. U. Z. Mat. 11, 69–74 (1974)Google Scholar
  12. 12.
    Gzigzin, A.E., Shapeev, V.P.: To the problem about continuous adjoining of particular solutions of systems of partial differential equations. Chisl. Metody Mech. Splosh. Sredy 6, 44–52 (1975)Google Scholar
  13. 13.
    Meleshko, S.V., Shapeev, V.P.: An application of DP-solutions to the problem on the decay of an arbitrary discontinuity. Chisl. Metody Mech. Splosh. Sredy 10, 85–96 (1979)Google Scholar
  14. 14.
    Meleshko, S.V., Shapeev, V.P.: The Goursat’s problem for inhomogeneous systems of differential equations. Chisl. Metody Mech. Splosh. Sredy 11, 796–798 (1980)Google Scholar
  15. 15.
    Meleshko, S.V., Shapeev, V.P., Janenko, N.N.: The method of differential constraints and the problem of decay of an arbitrary discontinuity. Sov. Math. Dokl. 22, 447–449 (1980)zbMATHGoogle Scholar
  16. 16.
    Meleshko, S.V.: On the class of solutions of the systems of quasilinear differential equations with many independent variables. Chisl. Metody Mech. Splosh. Sredy 12, 87–100 (1981)MathSciNetGoogle Scholar
  17. 17.
    Manganaro, N., Meleshko, S.V.: Reduction procedure and generalized simple waves for systems written in the Riemann variables. Nonlinear Dyn. 30(1), 87–102 (2002).  https://doi.org/10.1023/A:1020341610639 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Meleshko, S.V., Shapeev, V.P.: An application of the differential constraints method for the two-dimensional equations of gas dynamics. Prikl. Matem. Mech. 63(6), 909–916 (1999). (English transl. in J. Appl. Maths. Mechs. 63 (6), 885-891)Google Scholar
  19. 19.
    Fusco, D., Manganaro, N.: A reduction approach for determining generalized simple waves. ZAMP 59, 63–75 (2008).  https://doi.org/10.1007/s00033-006-5128-1 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Manganaro, N., Pavlov, M.V.: The constant astigmatism equation. New exact solution. J. Phys. A Math. Theor. 47(7), 075203 (2014).  https://doi.org/10.1088/1751-8113/47/7/075203 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Curró, C., Fusco, D., Manganaro, N.: Exact solutions in ideal chromatography via differential constraints method. AAPP Atti Della Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat. 93(1), A2 (2015).  https://doi.org/10.1478/AAPP.931A2 MathSciNetGoogle Scholar
  22. 22.
    Curró, C., Manganaro, N.: Double-wave solutions to quasilinear hyperbolic systems of first-order PDEs. ZAMP 68(5), 103 (2017).  https://doi.org/10.1007/s00033-017-0850-4 MathSciNetzbMATHGoogle Scholar
  23. 23.
    Curró, C., Manganaro, N.: Exact solutions and wave interactions for a viscoelastic medium. AAPP Atti Della Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat. 96(1), A1 (2018).  https://doi.org/10.1478/AAPP.961A1 Google Scholar
  24. 24.
    Curró, C., Fusco, D., Manganaro, N.: Hodograph transformation and differential constraints for wave solutions to \(2 \times 2\) quasilinear hyperbolic nonhomogeneous systems. J. Phys. A Math. Theor. 45(19), 195207 (2012).  https://doi.org/10.1088/1751-8113/45/19/195207 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Curró, C., Fusco, D., Manganaro, N.: An exact description of nonlinear wave interaction processes ruled by \(2 \times 2\) hyperbolic systems. ZAMP 64, 1227–1248 (2013).  https://doi.org/10.1007/s00033-012-0282-0 MathSciNetzbMATHGoogle Scholar
  26. 26.
    Curró, C., Fusco, D., Manganaro, N.: Exact description of simple wave interactions in multicomponent chromatography. J. Phys A Math. Theor. 48(1), 015201 (2015).  https://doi.org/10.1088/1751-8113/48/1/015201 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Curró, C., Fusco, D., Manganaro, N.: A reduction procedure for generalized Riemann problems with application to nonlinear transmission lines. J. Phys. A Math. Theor. 44(33), 335205 (2011).  https://doi.org/10.1088/1751-8113/44/33/335205 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Curró, C., Fusco, D., Manganaro, N.: Differential constraints and exact solution to Riemann problems for a traffic flow model. Acta Appl. Math. 122(1), 167–178 (2012).  https://doi.org/10.1007/s10440-012-9735-x MathSciNetzbMATHGoogle Scholar
  29. 29.
    Curró, C., Manganaro, N.: Riemann problems and exact solutions to a traffic model. J. Math. Phys. 54, 071503 (2013).  https://doi.org/10.1063/1.4813473 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Curró, C., Manganaro, N.: Generalized Riemann problems and exact solutions for \(p\)-systems with relaxation. Ric. Mat. 65(2), 549–562 (2016).  https://doi.org/10.1007/s11587-016-0274-z MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Curró, C., Manganaro, N., Pavlov, M.: Nonlinear wave interaction problems in the three-dimensional case. Nonlinearity 30, 207–224 (2017).  https://doi.org/10.1088/1361-6544/30/1/207 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Manganaro, N.: Riemann problems for viscoelastic media. Rend. Lincei Mat. Appl. 28, 479–494 (2017).  https://doi.org/10.4171/RLM/772 MathSciNetzbMATHGoogle Scholar
  33. 33.
    Meleshko, S.V.: DP-conditions and the problem of adjoinment different DP-solutions to each other. Chisl. Metody Mech. Splosh. Sredy 11, 96–109 (1980)MathSciNetGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.MIFTUniversity of MessinaMessinaItaly

Personalised recommendations