Skip to main content
SpringerLink
Log in

Propagation of non-planar weak and strong shocks in a non-ideal relaxing gas

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

In this article, the kinematics of one-dimensional motion have been applied to construct evolution equations for non-planar weak and strong shocks propagating into a non-ideal relaxing gas. The approximate value of exponent of shock velocity, at the instant of shock collapse, obtained from systematic approximation method is compared with those obtained from characteristic rule and Guderley’s scheme. Computation of exponent is carried out for different values of van der Waals excluded volume. Effects of non-ideal and relaxation parameters on the wave evolution, governed by the evolution equations, are analyzed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Sharma, V.D.: Quasilinear Hyperbolic Systems, Compressible Flows, and Waves. Chapman and Hall/CRC, Boca Raton (2010)

    Book  Google Scholar 

  2. Scott, W.A., Johannesen, N.H.: Spherical nonlinear wave propagation in a vibrationally relaxing gas. Proc. R. Soc. Lond. A 382, 103–134 (1982)

    Article  Google Scholar 

  3. Jena, J., Sharma, V.D.: Interaction of a characteristic shock with a weak discontinuity in a relaxing gas. J. Eng. Math. 60, 43–53 (2008)

    Article  MathSciNet  Google Scholar 

  4. Singh, R., Jena, J.: One dimensional steepening of waves in non-ideal relaxing gas. Int. J. Non Linear Mech. 77, 158–161 (2015)

    Article  Google Scholar 

  5. Madhumita, G., Sharma, V.D.: Imploding cylindrical and spherical shock waves in a non-ideal medium. J. Hyperbolic Diff. Equ. 1, 521–530 (2004)

    Article  MathSciNet  Google Scholar 

  6. Clarke, J.F., McChesney, M.: Dynamics of Relaxing Gases. Butterworth, London (1976)

    Google Scholar 

  7. Sharma, V.D., Venkatraman, R.: Evolution of weak shocks in one dimensional planar and non-planar gas dynamics flows. Int. J. Nonlinear Mech. 47, 918–926 (2012)

    Article  Google Scholar 

  8. Quintanilla, R., Straughan, B.: A note on the discontinuity waves in type III thermoelasticity. Proc. R. Soc. Lond. A 460, 1169–1175 (2004)

    Article  MathSciNet  Google Scholar 

  9. Sharma, V.D., Radha, C.: On one dimensional planar and non-planar shock waves in a relaxing gas. Phys. Fluids 6, 2177–2190 (1994)

    Article  MathSciNet  Google Scholar 

  10. Zhao, N., Mentrelli, A., Ruggeri, T., Sugiyama, M.: Admissible shock waves and shock induced phase transitions in a van der Waals fluid. Phys. Fluids 23, 086101 (2011)

    Article  Google Scholar 

  11. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  12. Anile, A.M., Hunter, J.K., Pantano, P., Russo, G.: Ray Method for Non-linear Waves in Fluids and Plasmas. Longman, New York (1993)

    MATH  Google Scholar 

  13. Zheng, Y.: Systems of Conservation Laws. Birkhauser, Boston (2001)

    Book  Google Scholar 

  14. Straughan, B.: Heat Waves, Applied Mathematical Sciences, vol. 177. Springer, New York (2011)

    MATH  Google Scholar 

  15. Mentrelli, A., Ruggeri, T.: The Riemann problem for a hyperbolic model of incompressible fluids. Int. J. Nonlinear Mech. 51, 87–96 (2013)

    Article  Google Scholar 

  16. Mentrelli, A., Ruggeri, T.: The propagation of shock waves in incompressible fluids: the case of freshwater. Acta Appl. Math. 132, 427–437 (2014)

    Article  MathSciNet  Google Scholar 

  17. Conforto, F., Mentrelli, A., Ruggeri, T.: Shock structure and multiple sub-shocks in binary mixures of Eulerian fluids. Ric. Mat. 66, 221–231 (2016)

    Article  Google Scholar 

  18. Pandey, M., Sharma, V.D.: Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas. Wave Motion 44, 346–354 (2007)

    Article  MathSciNet  Google Scholar 

  19. Truesdell, C., Toupin, R.A.: The Classical Field Theories, Handbuch der physik. Springer, Berlin (1960)

    Google Scholar 

  20. Guderley, G.: Starke kugelige und zylindrische Verdichtungsstosse in der Nahe des Kugelmittelunktes bzw der Zylinderachse. Luftfahrtforschung 19, 302–312 (1942)

    MathSciNet  MATH  Google Scholar 

  21. Arora, R., Siddiqui, M.J., Singh, V.P.: Similarity method for imploding strong shocks in a non-ideal relaxing gas. Int. J. Nonlinear Mech. 57, 1–9 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

The first author is highly thankful to CSIR India (Ref. No. 09/045(1444)/2016-EMR-I) for fellowship. Research and Development grant from University of Delhi, Delhi (Ref. No. RC/2015/9677) is gratefully acknowledged by the second author.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Sarswati Shah & Randheer Singh

Authors
  1. Sarswati Shah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Randheer Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Randheer Singh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shah, S., Singh, R. Propagation of non-planar weak and strong shocks in a non-ideal relaxing gas. Ricerche mat 70, 371–393 (2021). https://doi.org/10.1007/s11587-019-00472-w

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-019-00472-w

Keywords

Mathematics Subject Classification

Search

Navigation