Skip to main content
SpringerLink
Log in

Exact Solution for an Unsteady Isothermal Flow Behind a Cylindrical Shock Wave in a Rotating Perfect Gas with an Axial Magnetic Field and Variable Density

  • Published:
Journal of Engineering Physics and Thermophysics Aims and scope

An exact solution of the problem on the propagation of a cylindrical shock wave in a rotating perfect gas with an axial magnetic field in the case of an isothermal flow is obtained. The initial density, magnetic field strength, and the initial angular velocity in the ambient medium are assumed to vary according to the power law. An exact similarity solution obtained by the McVittie method for an isothermal flow in a rotating gas is first reported. Similarity transformations are used to transform a system of partial differential equations into a system of ordinary differential equations, and then the product solution of McVittie is used to obtain the exact solution. The effects of the values of the gas specific heat ratio, rotational parameter, and of the strength of the initial magnetic field are discussed. It is shown that the shock velocity increases and the shock strength decreases with increase in the values of these parameters. The effect of variation in the value of the initial density index is also studied. The obtained solutions show that the radial fluid velocity, density, pressure, and the magnetic field strength tend to zero as the axis of symmetry is approached.

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

Similar content being viewed by others

References

  1. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, Vol. II, Academic Press, New York (1967).

  2. T. S. Lee and T. Chen, Hydrodynamic interplanetary shock waves, Planet. Space Sci., 16, 1483–1502 (1968).

    Article  Google Scholar 

  3. D. Summers, An idealized model of a magnetohydrodynamic spherical blast wave applied to a flare produced shock in the solar wind, Astron. Astophys., 45, 151–158 (1972).

    Google Scholar 

  4. L. I. Sedov, Similarity and Dimensional Methods in Mechanics, Mir, Moscow (1982).

  5. J. P. Vishwakarma and Nanhey Patel, Magnetogasdynamic cylindrical shock waves in a rotating nonideal gas with radiation heat flux, J. Eng. Phys. Thermophys., 88, 521–530 (2015).

    Article  Google Scholar 

  6. P. Chaturani, Strong cylindrical shocks in a rotating gas, Appl. Sci. Res., 23, 197–211 (1970).

    Article  MathSciNet  Google Scholar 

  7. A. Sakurai, Propagation of spherical shock waves in stars, J. Fluid Mech., 1, 436–453 (1956).

    Article  MathSciNet  Google Scholar 

  8. O. Nath, S. Ojha, and H. S. Takhar, Propagation of a shock wave in a rotating interplanetary atmosphere with increasing energy, J. MHD Plasma Res., 8, 269–282 (1999).

    Google Scholar 

  9. J. P. Vishwakarma and G. Nath, Magnetogasdynamic shock waves in a rotating gas with exponentially varying density, ISRN Math. Phys., 2012, Article ID 168315, https://doi.org/10.5402/2012/168315 (2012).

  10. L. Hartmann, Accretion Processes in Star Formation, Cambridge University Press, Cambridge (1998).

    Google Scholar 

  11. B. Balick and A. Frank, Shapes and shaping of planetary nebulae, Annu. Rev. Astron. Astrophys., 40, 439–486 (2002).

    Article  Google Scholar 

  12. J. S. Shang, Recent research in magnetoaerodynamics, Prog. Aerosp. Sci., 37, 1–20 (2001).

    Article  Google Scholar 

  13. R. M. Lock and A. J. Mestel, Annular self-similar solutions in ideal magnetogasdynamics, J. Plasma Phys., 74, 531–554 (2008).

    Article  Google Scholar 

  14. S. Ashraf and P. L. Sachdev, An exact similarity solution in radiation-gas-dynamics, Proc. Indian Acad. Sci. A, 71, No. 6, 275–281 (1970).

    Article  Google Scholar 

  15. J. P. Vishwakarma and S. N. Pandey, Magnetogasdynamic cylindrical shock waves in a non-ideal gas with radiation heat flux, Model. Measure. Control B, 73, No. 1, 23–37 (2004).

    Google Scholar 

  16. G. C. McVittie, Spherically symmetric solutions of the equations of gas dynamics, Proc. R. Soc. Lond., A220, No. 1142, 339–355 (1953).

  17. S. K. Srivastava and R. K. Singh, An exact similarity solution for a spherical shock wave in a self-gravitating system, Astrophys. Space Sci., 92, 365–372 (1983).

    Article  Google Scholar 

  18. G. Nath, M. Dutta, and R. P. Pathak, An exact solution for the propagation of shock waves in self-gravitating medium in the presence of magnetic field and radiative heat flux, Model. Measure. Control B, 86, No. 4, 907–927 (2017).

    Article  Google Scholar 

  19. J. P. Vishwakarma, R. C. Srivastava, and Arun Kumar, An exact similarity solution in radiation magneto-gas-dynamics for the flows behind a spherical shock wave, Astrophys. Space Sci., 129, 45–52 (1987).

    Article  Google Scholar 

  20. G. Nath, Sumeeta Singh, and Pankaj Srivastava, An exact solution for magnetogasdynamic cylindrical shock wave in a self-gravitating rotational axisymmetric perfect gas with radiation heat flux and variable density, J. Eng. Phys. Thermophys., 91, 1301–1312 (2018).

    Article  Google Scholar 

  21. Vinay Chaubey and G. Nath, Magnetogasdynamic shock waves in a non-ideal gas with radiation heat flux, Nat. Acad. Math., India, 15, 45–57 (2001).

    MathSciNet  Google Scholar 

  22. G. Nath, Mrityunjoy Dutta, and S. Chaurasia, Exact solution for isothermal flow behind a shock wave in a self-gravitating gas of variable density in an azimuthal magnetic field, J. Eng. Phys. Thermophys., 93, No. 5, 1292–1299 (2020).

  23. G. Nath, Magnetogasdynamic shock wave generated by a moving piston in a rotational axisymmetric isothermal flow of perfect gas with variable density, Adv. Space Res., 47, 1463–1471 (2011).

  24. G. Nath, Propagation of a cylindrical shock wave in a rotational axisymmetric isothermal flow of a non-ideal gas in magnetogasdynamics, Ain Shams Eng. J., 3, 393–401 (2012).

    Article  Google Scholar 

  25. S. Ashraf and Z. Ahmad, Approximate analytic solution of a strong shock with radiation near the surface of the star, Indian J. Pure Appl. Math., 6, 1090–1098 (1975).

    MATH  Google Scholar 

  26. V. P. Korobeinikov, Unidimensional automodel motions of a conducting gas in a magnetic field, Dokl. Akad. Nauk SSSR, 121, 613–616 (1958).

    MathSciNet  Google Scholar 

  27. D. D. Laumbach and R. F. Probstein, Self-similar strong shocks with radiation in a decreasing exponential atmosphere, Phys. Fluids, 13, 1178–1183 (1970).

    Article  Google Scholar 

  28. P. L. Sachdev and S. Ashraf, Converging spherical and cylindrical shocks with zero temperature gradient in the rear flow field, J. Appl. Math. Phys., 22, 1095–1102 (1971).

    MATH  Google Scholar 

  29. T. A. Zhuravskaya and V. A. Levin, The propagation of converging and diverging shock waves under intense heat exchange conditions, J. Appl. Math. Mech., 60, 745–752 (1996).

    Article  MathSciNet  Google Scholar 

  30. P. Rosenau and S. Frankenthal, Equatorial propagation of axisymmetric magnetohydrodynamic shocks, Phys. Fluids, 19, 1889–1899 (1976).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Allahabad, 211004, India

    G. Nath

Authors
  1. G. Nath
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Nath.

Additional information

Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 93, No. 6, pp. 1593–1602, November–December, 2020.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nath, G. Exact Solution for an Unsteady Isothermal Flow Behind a Cylindrical Shock Wave in a Rotating Perfect Gas with an Axial Magnetic Field and Variable Density. J Eng Phys Thermophy 93, 1538–1547 (2020). https://doi.org/10.1007/s10891-020-02258-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10891-020-02258-6

Keywords

Search

Navigation