Shock wave structure in a non-ideal gas under temperature and density-dependent viscosity and heat conduction
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.
KeywordsShock wave Navier–Stokes equation Non-ideal gas Viscosity Heat conduction
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.
- 4.Becker, R.: Impact waves and detonation. Z. Phys. 8, 321 (1922) transtation, N.A.C.A.- T.M. No. 505 (1929)Google Scholar
- 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.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
- 14.Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972)Google Scholar
- 15.Bobylev, A.V.: The Chapman–Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27, 29 (1982)Google Scholar
- 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.Bird, G. A.: Proceedings of the 7th International Symposium on Rarefied Gas Dynamics, vol. 2, p. 693 (1971)Google Scholar
- 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
- 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
- 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.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
- 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
- 49.Landau, L.D., Lifshitz, E.M.: Statistical Physics, Course of Theoretical Physics, vol. 5. Pergamon, Oxford (1958)Google Scholar