Shock Waves

, Volume 29, Issue 3, pp 427–439 | Cite as

Exact solution of shock wave structure in a non-ideal gas under constant and variable coefficient of viscosity and heat conductivity

  • A. PatelEmail author
  • M. Singh
Original Article


This paper investigates the structure of a normal shock wave using the continuum model for steady one-dimensional flow of a viscous non-ideal gas under heat conduction. The coefficients of viscosity and heat conductivity are assumed to be directly proportional to a power of the temperature. The simplified van der Waals equation of state for the non-ideal gas has been assumed in this work. The smooth and rough sphere models of the gas molecules in the kinetic theory of gases are used for the viscosity of a non-ideal gas. Assuming the Prandtl number to be 3 / 4, the complete integral of the energy equation, exact velocity, density, pressure, Mach number, change in entropy, viscous stress, and heat flux across the shock transition zone have been obtained in a perfect and a non-ideal gas under both constant and variable properties of the medium. The validity of the continuum hypothesis with respect to Mach number is examined for the study of shock wave structure in both the smooth and rough sphere models of ideal and non-ideal gas molecules. It has been shown that the continuum theory gives reasonably valid results for flows with higher Mach numbers in the case of a non-ideal gas in comparison with an ideal gas. The inverse thickness of the shock wave is calculated and compared for constant and variable properties of the gases. The shock wave thickness is also discussed as a function of mean free path of the gas molecules computed at different points between the boundary states. It is found that the inverse shock thickness decreases with the increase in non-idealness of the gas. In the rough sphere model of gas molecules, the increase in the non-idealness of the gas and the temperature exponent in the coefficients of viscosity and heat conductivity significantly increases the validity limit of the continuum model.


Shock wave Navier–Stokes equations Non-ideal gas Viscosity Heat conduction 



Arvind Patel expresses thanks to the University of Delhi, Delhi, India, for the R&D Grant vide letter no. RC/2015/9677 dated October 15, 2015. The research of the Manoj Singh is supported by UGC, New Delhi, India, vide letter no. Sch. No. /JRF/AA/139/F-297/2012-13 dated January 22, 2013. The authors would like to present their sincere thanks to the referees and editor for their valuable comments on the original manuscript, which were quite helpful to revise the paper in the present form.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Rankine, W.J.M.: On the thermodynamic theory of waves of finite longitudinal disturbances. Philos. Trans. R. Soc. Lond. 160, 277–288 (1870). CrossRefGoogle Scholar
  2. 2.
    Rayleigh, L.: Aerial plane waves of finite amplitude. Proc. R. Soc. Lond. A 84, 247–284 (1910). CrossRefzbMATHGoogle Scholar
  3. 3.
    Taylor, G.I.: The conditions necessary for discontinuous motion in gases. Proc. R. Soc. Lond. A 84, 371–377 (1910). CrossRefzbMATHGoogle Scholar
  4. 4.
    Becker, R.: Impact waves and detonation. Z. Phys. 8, 321 (1922) translation, NACA-TM-505 (1929).
  5. 5.
    Thomas, L.H.: Note on Becker’s theory of the shock front. J. Chem. Phys. 12, 449–453 (1944). CrossRefGoogle Scholar
  6. 6.
    von Mises, R.: On the thickness of a steady shock wave. J. Aeronaut. Sci. 17, 551–554 (1950). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Morduchow, M., Libby, P.A.: On a complete solution of the one-dimensional flow equations of viscous, heat conducting, compressible gas. J. Aeronaut. Sci. 16, 674–684 (1949). MathSciNetCrossRefGoogle Scholar
  8. 8.
    Meyerhoff, L.: An extension of the theory of the one-dimensional shock waves structure. J. Aeronaut. Sci. 17, 775–786 (1950). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gilbarg, D., Paolucci, D.: Structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2(4), 617–642 (1953).
  10. 10.
    Chapman, S., Cowling, T.: The Mathematical Theory of Non-uniform Gases. Cambridge University Press, Cambringe (1970)zbMATHGoogle Scholar
  11. 11.
    Cercignani, C.: Theory and Application of the Boltzmann Equation. Scottish Academic, Edinburgh (1975)zbMATHGoogle Scholar
  12. 12.
    Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972)Google Scholar
  13. 13.
    Bobylev, A.V.: The Chapman–Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27, 29 (1982)Google Scholar
  14. 14.
    Struchtrup, H.: Failures of the Burnett and super-Burnett equations in steady state processes. Contin. Mech. Thermodyn. 17, 43–50 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Torrilhon, M., Struchtrup, H.: Regularized 13-moment-equations: shock structure calculations and comparison to Burnett models. J. Fluid Mech. 513, 171–198 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sakurai, A.: A note on Mott-Smith’s solution of the Boltzmann equation for a shock wave. Fluid Mech. 3, 255–260 (1957). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sakurai, A.: A note on Mott-Smith’s solution of the Boltzmann equation for a shock wave, II. Research Report, vol. 6, p. 49. Tokyo Electrical Engineering College, Tokyo (1958)Google Scholar
  18. 18.
    Mott-Smith, H.M.: Solution of the Boltzmann equation of a shock wave. Phys. Rev. 82(6), 885–892 (1951). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Talbot, L., Sherman, F.S.: Experiment versus kinetic theory for rarefied gases. In: Devienne, F.M. (ed.) Proceedings of the 1st International Symposium on Rarefied Gas Dynamics. Pergamon, New York (1960)Google Scholar
  20. 20.
    Bird, G.A.: Proceedings of the 7th International Symposium on Rarefied Gas Dynamics, vol. 2, p. 693 (1971)Google Scholar
  21. 21.
    Hicks, B.L., Yen, S.M., Reilly, B.J.: The internal structure of shock wave. J. Fluid Mech. 53, 85–111 (1972). CrossRefzbMATHGoogle Scholar
  22. 22.
    Wang Chang, C.S.: On the Theory of the Thickness of Weak Shock Waves. Department of Engineering Research, University of Michigan, APL/JHU CM-503, UMH-3-F (1948)Google Scholar
  23. 23.
    Zoller, K.: Zur struktur des verdiehrungsstobes. Z. Angew. Phys. 130, 1–38 (1951)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Grad, H.: Profile of a steady plane shock wave. Commun. Pure Appl. Math. 5(3), 257–300 (1952). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Struchtrup, H., Torrilhon, M.: Regularized of Grad’s 13-moment equation: Derivation and linear analysis. Phys. Fluids 15, 2668–2680 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Khidr, M.A., Mahmoud, M.A.A.: The shock-wave structure for arbitrary Prandtl number and high Mach numbers. Astrophys. Space Sci. 113, 289–301 (1985). CrossRefGoogle Scholar
  27. 27.
    Johnson, B.M.: Analytical shock solutions at large and small Prandtl number. J. Fluid Mech. 726(R4), 1–12 (2013). MathSciNetzbMATHGoogle Scholar
  28. 28.
    Myong, R.S.: Analytical solutions of shock structure thickness and asymmetry in Navier -Stokes/Fourier framework. AIAA J. 52(5), 1075–1080 (2014). CrossRefGoogle Scholar
  29. 29.
    Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 2nd edn. Clarendon Press, Oxford (1994)Google Scholar
  30. 30.
    Anand, A.K., Yadav, H.C.: On the structure of MHD shock waves in a viscous non-ideal gas. Theor. Comput. Fluid Dyn. 28, 369–376 (2014). CrossRefGoogle Scholar
  31. 31.
    Anand, A.K., Yadav, H.C.: The effect of viscosity on the structure of shock waves in a non-ideal gas. Acta Phys. Pol. A 129, 28–34 (2016). CrossRefGoogle Scholar
  32. 32.
    Wu, C.C., Roberts, P.H.: Shock-wave propagation in a sonoluminescing gas bubble. Phys. Rev. Lett. 70(22), 3424–3427 (1993). CrossRefGoogle Scholar
  33. 33.
    Roberts, P.H., Wu, C.C.: Structure and stability of a spherical implosion. Phys. Lett. 213, 59–64 (1996). CrossRefGoogle Scholar
  34. 34.
    Vishwakarma, J.P., Chaube, V., Ptel, A.: Self-similar solution of a shock propagation in a non-ideal gas. Int. J. Appl. Mech. Eng. 12, 813–829 (2007)Google Scholar
  35. 35.
    Vishwakarma, J.P., Mahendra, S.: Self-similar cylindrical ionizing shock waves in a non-ideal gas with radiation heat-flux. Appl. Math. 2(1), 1–7 (2012). Google Scholar
  36. 36.
    Nath, G., Vishwakarma, J.P.: Similarity solution for the flow behind a shock wave in a non-ideal gas with heat conduction and radiation heat-flux in magnetogasdynamics. Commun. Nonlinear Sci. Numer. Simul. 19, 1347–1365 (2014). MathSciNetCrossRefGoogle Scholar
  37. 37.
    Landau, L.D., Lifshitz, E.M.: Statistical Physics, Course of Theoretical Physics, vol. 5. Pergamon, Oxford (1958)Google Scholar
  38. 38.
    Bird, G.A.: Definition of mean free path for real gases. Phys. Fluids 26(11), 3222–3223 (1983). CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia

Personalised recommendations