Skip to main content

Similarity solutions for strong magnetogasdynamic cylindrical shock wave in rotating axisymmetric ideal gas with radiation heat flux using Lie group theoretic method

Abstract

Mathematical model for propagation of cylindrical shock wave under the influence of azimuthal magnetic field in rotating medium with radiation heat flux for adiabatic flow condition, using Lie group of transformation method is formulated and similarity solutions are obtained. The medium ahead of the shock is assumed to be at rest. The density, magnetic field, azimuthal and axial fluid velocities are presumed to be varying in the undisturbed medium. We have obtained two different cases of potential solutions considering different cases for the arbitrary constants appearing in the expressions of infinitesimals. Numerical solutions are obtained in the case of power law shock path. Distributions of magnetogasdynamical quantities are discussed through figures. The effects of increase in ambient azimuthal fluid velocity variation index, strength of magnetic field and ambient density exponent are examined on shock strength and on the flow variables. It is observed that shock strength decreases due to increase in strength of magnetic field. Whereas there is increase in strength of shock due to increase in ambient density or ambient azimuthal fluid velocity variation index. In general, density, azimuthal fluid velocity, pressure, radial fluid velocity, temperature and radiation heat flux decrease as we move inwards from the shock to the axis of symmetry. Magnetic field, axial fluid velocity and non-dimensional azimuthal component of vorticity vector \(l_{\theta }\) increase as we move inwards from the shock to the axis of symmetry. Non dimensional axial component of vorticity vector \(l_{z}\) increases, attains the maximum and then decreases as we move inwards from the shock to the axis of symmetry. Numerical calculations are done and graphs are being plot using software Mathematica.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 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(5), 1347–1365 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  2. Marshak, R.E.: Effect of radiation on shock wave behavior. Phys. Fluids 1(1), 24–29 (1958)

    MathSciNet  MATH  Article  Google Scholar 

  3. Elliott, L.A.: Similarity methods in radiation hydrodynamics. Proc. R. Soc. Lond. A 258(1294), 287–301 (1960)

    MathSciNet  Article  Google Scholar 

  4. Wang, K.C.: The ‘piston problem’ with thermal radiation. J. Fluid Mech. 20(3), 447–455 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  5. Ashraf, S., Sachdev, P.L.: An exact similarity solution in radiation-gas-dynamics. In: Proceedings of the Indian Academy of Sciences-Section A (Vol. 71, No. 6, pp. 275-281). Springer India (1970)

  6. Laumbach, D.D., Probstein, R.F.: Self-similar strong shocks with radiation in a decreasing exponential atmosphere. Phys. Fluids 13(5), 1178–1183 (1970)

    Article  Google Scholar 

  7. Ojha, S.N.: A solution to the radiative blast wave in stellar interiors. Acta Phys. Hung. 31(4), 375–383 (1972)

    Article  Google Scholar 

  8. NiCastro, J.R.A.J.: Similarity analysis of the radiative gas dynamic equations with spherical symmetry. Phys. Fluids 13(8), 2000–2006 (1970)

    Article  Google Scholar 

  9. Sharma, V.D., Ch, R.: Strong converging shock waves in a radiating gas. ZAMM-J. Appl. Math. Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 75(12), 847–859 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  10. Nath, G.: Propagation of a spherical shock wave in mixture of non-ideal gas and small solid particles under the influence of gravitational field with conductive and radiative heat fluxes. Astrophys. Space Sci. 361(1), 31 (2016)

    MathSciNet  Article  Google Scholar 

  11. Lin, S.C.: Cylindrical shock waves produced by instantaneous energy release. J. Appl. Phys. 25(1), 54–57 (1954)

    MATH  Article  Google Scholar 

  12. Taylor, G.I.: The formation of a blast wave by a very intense explosion I. Theoretical discussion. Proc. R. Soc. Lond. A 201(1065), 159–174 (1950)

    MATH  Article  Google Scholar 

  13. Taylor, G.I.: The formation of a blast wave by a very intense explosion.-II. The atomic explosion of 1945. Proc. R. Soc. Lond. A 201(1065), 175–186 (1950)

    MATH  Article  Google Scholar 

  14. Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic Press, New York (1959)

    MATH  Google Scholar 

  15. Zel’Dovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and High-temperature Hydrodynamic Phenomena. Courier Corporation, Cambridge (2002)

    Google Scholar 

  16. Chaturani, P.: Strong cylindrical shocks in a rotating gas. Appl. Sci. Res. 23(1), 197–211 (1971)

    MathSciNet  MATH  Article  Google Scholar 

  17. Vishwakarama, J.P., Vishwakarama, S.: Magnetogasdinamic cylindrical shock wave in a rotating gas with variable density. Int. J. Appl. Mech. Eng. 12(1), 283–297 (2007)

    Google Scholar 

  18. Vishwakarma, J.P., Maurya, A.K., Singh, K.K.: Self-similar adiabatic flow headed by a magnetogasdynamic cylindrical shock wave in a rotating non-ideal gas. Geophys. Astrophys. Fluid Dyn. 101(2), 155–168 (2007)

    MathSciNet  Article  Google Scholar 

  19. Hishida, M., Fujiwara, T., Wolanski, P.: Fundamentals of rotating detonations. Shock Waves 19(1), 1–10 (2009)

    MATH  Article  Google Scholar 

  20. Nath, G.: Magnetogasdynamic shock wave generated by a moving piston in a rotational axisymmetric isothermal flow of perfect gas with variable density. Adv. Space Res. 47(9), 1463–1471 (2011)

    Article  Google Scholar 

  21. Nath, O., Ojha, S.N., Takhar, H.S.: Propagation of a shock wave in a rotating interplanetary atmosphere with increasing energy. Theoret. Chim. Acta 44(1), 87–98 (1999)

    Google Scholar 

  22. Vishwakarma, J.P., Patel, N.: Magnetogasdynamic cylindrical shock waves in a rotating nonideal gas with radiation heat flux. J. Eng. Phys. Thermophys. 88(2), 521–530 (2015)

    Article  Google Scholar 

  23. Nagasawa, M.: Gravitational instability of the isothermal gas cylinder with an axial magnetic field. Progress Theoret. Phys. 77(3), 635–652 (1987)

    Article  Google Scholar 

  24. Summers, D.: An idealised model of a magnetohydrodynamic spherical blast wave applied to a flare produced shock in the solar wind. Astron. Astrophys. 45, 151–158 (1975)

    Google Scholar 

  25. Lee, T.S., Chen, T.: Hydromagnetic interplanetary shock waves. Planet. Space Sci. 16(12), 1483–1502 (1968)

    Article  Google Scholar 

  26. Christer, A.H., Helliwell, J.B.: Cylindrical shock and detonation waves in magnetogasdynamics. J. Fluid Mech. 39(4), 705–725 (1969)

    MATH  Article  Google Scholar 

  27. Korobeĭnikov, V.P.: Problems in the theory of point explosion in gases (Vol. 119). American Mathematical Soc. (1976)

  28. Shang, J.S.: Recent research in magneto-aerodynamics. Prog. Aerosp. Sci. 37(1), 1–20 (2001)

    Article  Google Scholar 

  29. Lock, R.M., Mestel, A.J.: Annular self-similar solutions in ideal magnetogasdynamics. J. Plasma Phys. 74(4), 531 (2008)

    MATH  Article  Google Scholar 

  30. Hartmann, L.: Accretion Processes in Star Formation, vol. 32. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  31. Balick, B., Frank, A.: Shapes and shaping of planetary nebulae. Ann. Rev. Astron. Astrophys. 40(1), 439–486 (2002)

    Article  Google Scholar 

  32. Lerche, I.: Mathematical theory of one-dimensional isothermal blast waves in a magnetic field. Aust. J. Phys. 32(5), 491–502 (1979)

    MathSciNet  Article  Google Scholar 

  33. Lerche, I.: Mathematical theory of cylindrical isothermal blast waves in a magnetic field. Aust. J. Phys. 34(3), 279–302 (1981)

    MathSciNet  Article  Google Scholar 

  34. Pullin, D.I., Mostert, W., Wheatley, V., Samtaney, R.: Converging cylindrical shocks in ideal magnetohydrodynamics. Phys. Fluids 26(9), 097103 (2014)

    Article  Google Scholar 

  35. Mostert, W., Pullin, D.I., Samtaney, R., Wheatley, V.: Converging cylindrical magnetohydrodynamic shock collapse onto a power-law-varying line current. J. Fluid Mech. 793, 414 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  36. Nath, G., Vishwakarma, J.P.: Magnetogasdynamic spherical shock wave in a non-ideal gas under gravitational field with conductive and radiative heat fluxes. Acta Astronaut. 128, 377–384 (2016)

    Article  Google Scholar 

  37. Nath, G., Singh, S.: Flow behind magnetogasdynamic exponential shock wave in self-gravitating gas. Int. J. Non-Linear Mech. 88, 102–108 (2017)

    Article  Google Scholar 

  38. Bluman, G.W., Cole, J.D.: Similarity Methods for Differential Equations, vol. 13. Springer, Berlin (2012)

    MATH  Google Scholar 

  39. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations, vol. 81. Springer, Berlin (2013)

    MATH  Google Scholar 

  40. Stephani, H.: Differential Equations: Their Solution Using Symmetries. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  41. Ibragimov, N.K., Ibragimov, N.K.: Elementary Lie Group Analysis and Ordinary Differential Equations, vol. 197. Wiley, New York (1999)

    MATH  Google Scholar 

  42. Olver, P.J.: Applications of Lie Groups to Differential Equations, vol. 107. Springer, Berlin (2000)

    MATH  Google Scholar 

  43. Hydon, P.E., Hydon, P.E.: Symmetry Methods for Differential Equations: A Beginner’s Guide (No. 22). Cambridge University Press, Cambridge (2000)

    MATH  Book  Google Scholar 

  44. Logan, J.D., Perez, J.D.J.: Similarity solutions for reactive shock hydrodynamics. SIAM J. Appl. Math. 39(3), 512–527 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  45. Donato, A.: Similarity analysis and non-linear wave propagation. Int. J. Non-Linear Mech. 22(4), 307–314 (1987)

    MATH  Article  Google Scholar 

  46. Torrisi, M.: Similarity solution and wave propagation in a reactive polytropic gas. J. Eng. Math. 22(3), 239–251 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  47. Zedan, H.A.: Applications of the group of equations of the one-dimensional motion of a gas under the influence of monochromatic radiation. Appl. Math. Comput. 132(1), 63–71 (2002)

    MathSciNet  MATH  Google Scholar 

  48. Donato, A., Oliveri, F.: Reduction to autonomous form by group analysis and exact solutions of axisymmetric MHD equations. Math. Comput. Model. 18(10), 83–90 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  49. Zayed, E.M.E., Zedan, H.A.: Autonomous forms and exact solutions of equations of motion of polytropic gas. Int. J. Theor. Phys. 40(6), 1183–1196 (2001)

    MATH  Article  Google Scholar 

  50. Oliveri, F., Speciale, M.P.: Exact solutions to the ideal magneto-gas-dynamics equations through Lie group analysis and substitution principles. J. Phys. A: Math. Gen. 38(40), 8803 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  51. Nath, G., Singh, S.: Similarity solutions for magnetogasdynamic shock waves in a rotating ideal gas using the Lie group-theoretic method. J. Eng. Math. 126(1), 1–22 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  52. Nath, G., Singh, S.: Similarity solutions for magnetogasdynamic cylindrical shock wave in rotating ideal gas using Lie Group theoretic method: Isothermal flow. Int. J. Geometr. Methods Mod. Phys. 17(08), 2050123 (2020)

    Article  Google Scholar 

  53. Nath, G., Singh, S.: Similarity solutions for cylindrical shock wave in rotating ideal gas with or without magnetic field using Lie group theoretic method. Eur. Phys. J. Plus 135(11), 1–18 (2020)

    Article  Google Scholar 

  54. Nath, G.: Similarity solutions for unsteady flow behind an exponential shock in an axisymmetric rotating non-ideal gas. Meccanica 50(7), 1701–1715 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  55. Levin, V.A., Skopina, G.A.: Detonation wave propagation in rotational gas flows. J. Appl. Mech. Tech. Phys. 45(4), 457–460 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  56. Nath, G.: Propagation of a strong cylindrical shock wave in a rotational axisymmetric dusty gas with exponentially varying density. Res. Astron. Astrophys. 10(5), 445 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sumeeta Singh.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Verify currency and authenticity via CrossMark

Cite this article

Singh, S. Similarity solutions for strong magnetogasdynamic cylindrical shock wave in rotating axisymmetric ideal gas with radiation heat flux using Lie group theoretic method. Ricerche mat (2022). https://doi.org/10.1007/s11587-022-00697-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11587-022-00697-2

Keywords

  • Shock waves
  • Lie group of Transformations
  • Magnetogasdynamics
  • Rotating medium
  • Radiation Heat Flux