Exact solution of shock wave structure in a non-ideal gas under constant and variable coefficient of viscosity and heat conductivity
- 16 Downloads
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.
KeywordsShock 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.
- 4.Becker, R.: Impact waves and detonation. Z. Phys. 8, 321 (1922) translation, NACA-TM-505 (1929). https://doi.org/10.1007/BF01329605
- 9.Gilbarg, D., Paolucci, D.: Structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2(4), 617–642 (1953). http://www.jstor.org/stable/24900350
- 12.Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972)Google Scholar
- 13.Bobylev, A.V.: The Chapman–Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27, 29 (1982)Google Scholar
- 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
- 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.Bird, G.A.: Proceedings of the 7th International Symposium on Rarefied Gas Dynamics, vol. 2, p. 693 (1971)Google Scholar
- 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
- 29.Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 2nd edn. Clarendon Press, Oxford (1994)Google Scholar
- 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
- 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). https://doi.org/10.1016/j.cnsns.2013.09.009 MathSciNetCrossRefGoogle Scholar
- 37.Landau, L.D., Lifshitz, E.M.: Statistical Physics, Course of Theoretical Physics, vol. 5. Pergamon, Oxford (1958)Google Scholar