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Theoretical and Computational Fluid Dynamics

, Volume 33, Issue 6, pp 537–559 | Cite as

Shock wave structure in a non-ideal gas under temperature and density-dependent viscosity and heat conduction

  • Manoj Singh
  • Arvind PatelEmail author
Original Article

Abstract

The structure of a shock wave is investigated using the continuum hypothesis 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 and density of the gas. The simplified van der Waals equation of state for the non-ideal gas has been assumed in this work. Qualitative analysis of shock wave structure has been done in terms of singularity analysis, isoclines, and integral curves. The exact and numerical solutions of shock structure equations are obtained under the quantitative analysis. The validation of solution is established by comparing the results in the literature (Iannelli in Int J Numer Methods Fluids 72(2):157–176, 2013). The variation of normalized gas velocity, viscous stress, heat flux, and shock thickness have been investigated across shock transition zone with the non-idealness of the gas, temperature, and density exponents in the viscosity and heat conductivity of the gas and initial Mach number. It is found that gas velocity decreases significantly with the increase in non-idealness parameter, temperature, and density exponent in the viscosity of the gas. Shock wave thickness decreases with the increase in the non-idealness of the gas under constant viscosity and heat conductivity but increase under variable gas properties. The thickness of a shock wave decreases with the increase in the temperature exponent and increases with the increase in the density exponent.

Keywords

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

Notes

Acknowledgements

The research of the First author (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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  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.: Arial plane waves of finite amplitude. Proc. R. Soc. Lond. A 84, 247–284 (1910)CrossRefGoogle Scholar
  3. 3.
    Taylor, G.I.: The conditions necessary for discontinuous motion in gases. Proc. R. Soc. Lond. A 84, 371–377 (1910)CrossRefGoogle Scholar
  4. 4.
    Becker, R.: Impact waves and detonation. Z. Phys. 8, 321 (1922) transtation, N.A.C.A.- T.M. No. 505 (1929)Google Scholar
  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.
    Reissner, H.J., Meyerhoff., L.: A contribution to the exact solutions of the problem of a one-dimensional shock wave in a viscous, heat conduction fluid. PIBAL Report, vol. 138 (1948)Google Scholar
  7. 7.
    Meyerhoff, L., Reissner, H.J.: The standing, one-dimensional shock wave under the influence of temperature-dependent viscosity, heat conduction and specific heat. PIBAL Report, vol. 150 (1949)Google Scholar
  8. 8.
    Puckett, A.E., Stewart, H.J.: The thickness of a shock wave in air. Quart. Appl. Math. 7, 457–463 (1950)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Meyerhoff, L.: An extension of the theory of the one-dimensional shock waves structure. J. Aeronaut. Sci. 17, 775–786 (1950)MathSciNetCrossRefGoogle Scholar
  10. 10.
    von Mises, R.: On the thickness of a steady shock wave. J. Aeronaut. Sci. 17, 551–554 (1950)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gilbarg, D., Paolucci, D.: structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2(4), 617–642 (1953)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chapman, S., Cowling, T.: The Mathematical Theory of Non-uniform Gases. Cambringe University Press, Cambringe (1970)zbMATHGoogle Scholar
  13. 13.
    Cercignani, C.: Theory and Application of the Boltzmann Equation. Scottish Academic, Edinburgh (1975)zbMATHGoogle Scholar
  14. 14.
    Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972)Google Scholar
  15. 15.
    Bobylev, A.V.: The Chapman–Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27, 29 (1982)Google Scholar
  16. 16.
    Struchtrup, H.: Failures of the Burnett and Super-Burnett equations in steady state processes. Contin. Mech. Thermodyn. 17, 43–50 (2005).  https://doi.org/10.1007/s00161-004-0186-0 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Torrilhon, M., Struchtrup, H.: Regularized 13-moment-equations: shock structure calculations and comparison to Burnett models. J. Fluid Mech. 513, 171 (2004).  https://doi.org/10.1017/S0022112004009917 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Talbot, L.,Sherman, F.S.: Experiment versus kinetic theory for rarefied gases. In rarefied gas dynamics: Proc. 1st Intl Symp (ed. F.M. Devienne). Pergamon, New YorkGoogle Scholar
  19. 19.
    Bird, G. A.: Proceedings of the 7th International Symposium on Rarefied Gas Dynamics, vol. 2, p. 693 (1971)Google Scholar
  20. 20.
    Hicks, B.L., Yen, S.M., Reilly, B.J.: The internal structure of shock wave. J. Fluid Mech. 53, 85 (1972).  https://doi.org/10.1017/S0022112072000059 CrossRefzbMATHGoogle Scholar
  21. 21.
    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. August (1948)Google Scholar
  22. 22.
    Zoller, K.: Zur struktur des verdiehrungsstobes. Zeitschrift fur Physik 130, 1 (1951)CrossRefGoogle Scholar
  23. 23.
    Grad, H.: The profile of a steady plane shock wave. Commun. Pure Appl. Math. 5, Wiley, New York, p. 257–300 (1952).  https://doi.org/10.1002/cpa.3160050304
  24. 24.
    Struchtrup, H., Torrilhon, M.: Regularized of Grad’s 13-moment equation: derivation and linear analysis. Phys. Fluids. 15, 2668–2680 (2003).  https://doi.org/10.1063/1.1597472 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Alsmeyer, H.: Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluids Mech. 74(3), 497–613 (1976)CrossRefGoogle Scholar
  26. 26.
    Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E Part A 52, R5760 (1995)CrossRefGoogle Scholar
  27. 27.
    Ruggeri, T.: Breakdown of shock-wave-structure solutions. Phys. Rev. E 47, 4135–4140 (1993)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Muller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn, pp. 277–308. Springer, New York (1998)CrossRefGoogle Scholar
  29. 29.
    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
  30. 30.
    Ludford, G.S.S.: The classification of one-dimensional flows and the general problem of a compressible, viscous, heat-conducting fluid. J. Aeronaut. Sci. 18(12), 830–834 (1951)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Griffith, W.C., Bleakney, W.: Shock waves in gases. A. J. Phys. 22(9), 597–612 (1954)CrossRefGoogle Scholar
  32. 32.
    Thompson, P.A., Lambrakis, K.C.: Negative shock waves. J. Fluid Mech. 60, 187–208 (1973)CrossRefGoogle Scholar
  33. 33.
    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
  34. 34.
    Cramer, M.S., Crickenberger, A.B.: The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223, 325–355 (1991)CrossRefGoogle Scholar
  35. 35.
    Iannelli, J.: An implicit Galerkin finite element Runge–Kutta algorithm for shock-structure investigations. J. Compt. Phys. 230, 260–286 (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Johnson, B.M.: Analytical shock solutions at large and small Prandtl number. J. Fluid Mech. 726(R4), 1–12 (2013)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Johnson, B.M.: Closed-form shock solutions. J. Fluid Mech. 745(R1), 1–11 (2014)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Iannelli, J.: An exact non-linear Navier–Stokes compressible-flow solution for CFD code verification. Int. J. Numer. Meth. Fluids 72(2), 157–176 (2013)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Myong, R.S.: Technical note for analytical solutions of shock structure thickness and asymmetry in Navier–Stokes/Fourier framework. AIAA J. 52(5), 1075–1080 (2014)CrossRefGoogle Scholar
  40. 40.
    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
  41. 41.
    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
  42. 42.
    Patel, A., Manoj, S.: Exact solution of shock wave structure in a non-ideal gas under constant and variable coefficient of viscosity and heat conductivity. Shock Waves 29, 427–439 (2019)CrossRefGoogle Scholar
  43. 43.
    Roberts, P.H., Wu, C.C.: Structure and stability of a spherical implosion. Phys. Lett. 213, 59–64 (1996)CrossRefGoogle Scholar
  44. 44.
    Roberts, P.H., Wu, C.C.: The shock wave theory of sonoluminescene. In: Srivastava, R.C., Leutloff, D., Takayama, K., Groning, H. (eds.) Shock Focussing Effect in Medical Science and Sonoluminescene. Springer, Berlin (2003)Google Scholar
  45. 45.
    Vishwakarma, J.P., Chaube, V., Patel, A.: Self-similar solution of a shock propagation in a non-ideal gas. Int. J. Appl. Mech. Eng 12, 813–829 (2007)Google Scholar
  46. 46.
    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).  https://doi.org/10.5923/j.am.20120201.01 CrossRefGoogle Scholar
  47. 47.
    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. Number. Simul. 19, 1347–1365 (2014).  https://doi.org/10.1016/j.cnsns.2013.09.009 MathSciNetCrossRefGoogle Scholar
  48. 48.
    Ghoniem, A.F., Kamel, M.M., Berger, S.A., Oppenheim, A.K.: Effects of internal heat transfer on the structure of self-similar blast waves. J. Fluid Mech 117, 473–491 (1982)CrossRefGoogle Scholar
  49. 49.
    Landau, L.D., Lifshitz, E.M.: Statistical Physics, Course of Theoretical Physics, vol. 5. Pergamon, Oxford (1958)Google Scholar
  50. 50.
    Anisimov, S.I., Spiner, O.M.: Motion of an almost ideal gas in the presence of a strong point explosion. J. Appl. Math. Mech. 36, 883–887 (1972)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiNew DelhiIndia

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