Skip to main content
SpringerLink
Log in

Extended thermodynamics of dense gases in the presence of dynamic pressure

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

Extended thermodynamics (ET) developed up to now fails when a gas is very dense and is composed of molecules with small internal degrees of freedom because the condition of convexity (stability) is violated. The aim of this paper is to explore a possible approach to construct an ET theory that is valid for any dense gas with the condition that it reduces to the usual ET theory when a gas is sufficiently rarefied. We restrict our study, for simplicity, within the simplest case in which the dissipation is only due to the dynamic pressure. Therefore the basic system of equations is the simplest variant of the Euler system, that is, the system composed of the equations for the conservation laws and an equation for the dynamic pressure (6-field theory).

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

Similar content being viewed by others

References

  1. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)

    Book  MATH  Google Scholar 

  2. Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer, New York (2015)

    Book  Google Scholar 

  3. Liu, I.-S., Müller, I.: Extended thermodynamics of classical and degenerate ideal gases. Arch. Rational Mech. Anal. 83(4), 285 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Liu, I.-S., Müller, I., Ruggeri, T.: Relativistic thermodynamics of gases. Ann. Phys. 169, 191 (1986)

  5. Engholm Jr, H., Kremer, G.M.: Thermodynamics of a diatomic gas with rotational and vibrational degrees of freedom. Int. J. Eng. Sci. 32(8), 1241 (1984)

    Article  Google Scholar 

  6. Kremer, G.M.: Extended thermodynamics and statistical mechanics of a polyatomic ideal gas. J. Non-Equilib. Thermodyn. 14, 363 (1989)

    MATH  Google Scholar 

  7. Kremer, G.M.: Extended thermodynamics of molecular ideal gases. Contin. Mech. Thermodyn. 1, 21 (1989)

    Article  MathSciNet  Google Scholar 

  8. Liu, I.-S.: Extended thermodynamics of fluids and virial equations of state. Arch. Rational Mech. Anal. 88, 1 (1985)

    MATH  MathSciNet  Google Scholar 

  9. Kremer, G.M.: Extended thermodynamics of non-ideal gases. Physica 144A, 156 (1987)

    Article  Google Scholar 

  10. Liu, I.-S., Kremer, G.M.: Hyperbolic system of field equations for viscous fluids. Math. Appl. Comput. 9(2), 123 (1990)

    MATH  MathSciNet  Google Scholar 

  11. Liu, I.-S., Salvador, J.A.: Hyperbolic system for viscous fluids and simulation of shock tube flows. Contin. Mech. Thermodyn. 2, 179 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kremer, G.M.: On extended thermodynamics of ideal and real gases. In: Sieniutycz, S., Salamon, P. (eds.) Extended Thermodynamics Systems, p. 140. Taylor and Francis, New York (1992)

    Google Scholar 

  13. Carrisi, M.C., Pennisi, S.: Some open problems in non-linear extended thermodynamics and their possible solutions. Ric. Mat. 60, 45 (2012)

    Article  MathSciNet  Google Scholar 

  14. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Contin. Mech. Thermodyn. 24, 271 (2012)

    Article  MATH  Google Scholar 

  15. Arima, T., Sugiyama, M.: Characteristic features of extended thermodynamics of dense gases. Atti Accad. Pelorit. Pericol. 91(Suppl. No. 1), A1 (2013)

  16. Ruggeri, T., Sugiyama, M.: Recent developments in extended thermodynamics of dense and rarefied polyatomic gases. Acta Appl. Math. 132, 527 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Monatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics. Phys. Lett. A 377, 2136 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pavić, M., Ruggeri, T., Simić, S.: Maximum entropy principle for polyatomic gases. Physica A 392, 1302 (2013)

    Article  MathSciNet  Google Scholar 

  19. 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 (2014)

    Article  MathSciNet  Google Scholar 

  20. Arima, T., Mentrelli, A., Ruggeri, T.: Extended thermodynamics of rarefied polyatomic gases and characteristic velocities. Rend. Lincei Mat. Appl. 25, 275 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Arima, T., Taniguchi, S., Sugiyama, M.: Light scattering in rarefied polyatomic gases based on extended thermodynamics. In: Proceedings of the 34th Symposium on Ultrasonic Electronics, vol. 15 (2013)

  24. Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: beyond the Bethe–Teller theory. Phys. Rev. E. 89, 013025 (2014)

    Article  Google Scholar 

  25. 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, 016103 (2014)

    Article  Google Scholar 

  26. Arima, T., Barbera, E., Brini, F., Sugiyama, M.: The role of the dynamic pressure in stationary heat conduction of a rarefied polyatomic gas. Phys. Lett. A 378, 2695 (2014)

    Article  MATH  Google Scholar 

  27. Carrisi, M.C., Pennisi, S.: An 18 moments model for dense gases, entropy and Galilean relativity principles without expansions. Entropy 17, 214–230 (2015)

    Article  MathSciNet  Google Scholar 

  28. Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: On the six-field model of fluids based on extended thermodynamics. Meccanica 49, 2181 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  29. 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 (2015)

    Article  Google Scholar 

  30. Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. The structure of the extended thermodynamics. Contin. Mech. Thermodyn. 1, 3 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ikenberry, E., Truesdell, C.: On the pressure and the flux of energy in a gas according to Maxwell’s kinetic theory. J. Rational Mech. Anal. 5, 1 (1956)

    MATH  MathSciNet  Google Scholar 

  32. Münster, A.: Statistical Thermodynamics, vol. 2. Springer, Berlin (1974)

    Google Scholar 

  33. Barker, J.A., Henderson, D.: What is “liquid”? Understanding the states of matter. Rev. Mod. Phys. 48, 587–671 (1976)

    Article  MathSciNet  Google Scholar 

  34. Hansen, J.P., McDonald, J.R.: Theory of Simple Liquids. Academic Press, London (1986)

    Google Scholar 

  35. Taniguchi, S., Mentrelli, A., Zhao, N., Ruggeri, T., Sugiyama, M.: Shock-induced phase transition in systems of hard spheres with internal degrees of freedom. Phys. Rev. E 81, 066307 (2010)

    Article  Google Scholar 

  36. Carnahan, N.F., Starling, K.E.: Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51, 635 (1969)

    Article  Google Scholar 

  37. Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. Contin. Mech. Thermodyn. 1, 3 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  38. Liu, I.-S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rational Mech. Anal. 46, 131 (1972)

    MATH  MathSciNet  Google Scholar 

  39. Pennisi, S., Ruggeri, T.: A new method to exploit the entropy prnciple and Galilean invariance in the macroscopic approach of extended thermodynamics. Richerche Mat. 55, 159–179 (2006)

    Article  MathSciNet  Google Scholar 

  40. Meixner, J.: Absorption und Dispersion des Schalles in Gasen mit chemisch Reagierenden und Anregbaren komponenten. I. Teil. Ann. Phys. 43, 470 (1943)

    Article  MathSciNet  Google Scholar 

  41. Meixner, J.: Allgemeine Theorie der Schallabsorption in Gasen und Flussigkeiten unter Berucksichtigung der Transporterscheinungen. Acoustica 2, 101 (1952)

    MathSciNet  Google Scholar 

  42. de Groot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. Dover, New York (1984)

    Google Scholar 

Download references

Acknowledgments

This work was supported by National Group of Mathematical Physics GNFM-INdAM (M.C.C., S.P. and T.R.) and by University of Bologna: FARB 2012 Project Extended Thermodynamics of Non-Equilibrium Processes from Macro- to Nano-Scale (T. R.) and by Japan Society of Promotion of Science (JSPS) No. 25390150 (M.S.).

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Cagliari, Cagliari, Italy

    Maria Cristina Carrisi & Sebastiano Pennisi

  2. Department of Mathematics, Alma Mater Research Center on Applied Mathematics AM 2, University of Bologna, Bologna, Italy

    Tommaso Ruggeri

  3. Graduate School of Engineering, Nagoya Institute of Technology, Nagoya, Japan

    Masaru Sugiyama

Authors
  1. Maria Cristina Carrisi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sebastiano Pennisi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tommaso Ruggeri
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Masaru Sugiyama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tommaso Ruggeri.

Additional information

Communicated by Salvatore Rionero.

To the memory of Prof. Carlo Ciliberto.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carrisi, M.C., Pennisi, S., Ruggeri, T. et al. Extended thermodynamics of dense gases in the presence of dynamic pressure. Ricerche mat. 64, 403–419 (2015). https://doi.org/10.1007/s11587-015-0247-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-015-0247-7

Keywords

Mathematics Subject Classification

Search

Navigation