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


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.


Shock 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.


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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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