Differential Equations and Dynamical Systems

, Volume 27, Issue 1–3, pp 169–180 | Cite as

Similarity Solutions of Cylindrical Shock Waves in Non-Ideal Magnetogasdynamics with Thermal Radiation

  • Hariom SharmaEmail author
  • Rajan Arora
Original Research


In the present work we have taken one-dimensional unsteady flow of non-ideal gas with magnetic effect under the presence of thermal radiation. The system is hyperbolic in nature and solved by similarity method using Lie Group of Transformations under the assumption that the system is constantly conformally invariant under the transformations. The similarity solutions are investigated behind a cylindrical shock which is a consequence of a sudden explosion or produced by an expanding piston. The shock is assumed to be strong and propagating into the medium which is at rest, with uniform density. The total energy of the shock is assumed to be time dependent and obeying the power law. By means of similarity method our system of PDEs transformed into the system of ordinary differential equations (ODEs), which in general are nonlinear. The effects of thermal radiation on the the flow variables velocity, density, pressure and magnetic field are investigated behind the shock.


Shock waves Lie group of transformations Similarity solutions Non-ideal gas Magnetogasdynamics 

Mathematics Subject Classification

35Lxx 76Lxx 76Nxx 


  1. 1.
    Elliott, L.A.: Similarity methods in radiation and hydrodynamics. Proc. R. Soc. Lond. A 258(1294), 287–301 (1960)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Wang, K.C.: The Piston problem with thermal radiation. J. Fluid Mech. 20(3), 447–455 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Helliwell, J.B.: Self-similar piston problem with radiative heat transfer. J. Fluid Mech. 37(3), 497–512 (1969)CrossRefzbMATHGoogle Scholar
  4. 4.
    Singh, J.B., Mishra, S.K.: Modelling of self-similar cylindrical shock wave in radiation-magnetogasdynamics. Astrophys. Space Sci. 127, 33–43 (1986)CrossRefzbMATHGoogle Scholar
  5. 5.
    Vishwakarma, J.P., Srivastava, R.C., Kumar, A.: An exact similarity solution radiation-magnetogasdynamics for the flows behind a spherical shock. Astrophys. Space Sci. 129, 45–52 (1987)CrossRefzbMATHGoogle Scholar
  6. 6.
    Singh, L.P., Sharma, V.D., Ram, R.: Flow pattern induced by a piston impulsively moving in a perfectly conducting inviscid radiating gas. Phys. Fluids 3, 692–699 (1989)CrossRefGoogle Scholar
  7. 7.
    Ganguly, A., Jana, M.: Propagation of a shock wave in self-gravitating, radiative magnetohydrodynamic non-uniform rotating atmosphere. Bull. Calc. Math. Soc. 90, 77–82 (1998)zbMATHGoogle Scholar
  8. 8.
    Anisimov, S.I., Spiner, O.M.: Motion of an almost ideal gas in the presence of a strong point explosion. J. Appl. Math. Mech. 36, 883–887 (1972)CrossRefGoogle Scholar
  9. 9.
    Rangarao, M.P., Purohit, N.K.: Self-similar piston problem in non-ideal gas. Int. J. Eng. Sci. 14(1), 91–97 (1976)CrossRefzbMATHGoogle Scholar
  10. 10.
    Madhumita, G., Sharma, V.D.: Imploding cylindrical and spherical shock waves in a non-ideal medium. J. Hyperbol. Differ. Equ. 1, 521–530 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Arora, R., Sharma, V.D.: Convergence of strong shock in a van der Waals gas. SIAM J. Appl Math. 66, 1825–1837 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Vishwakarma, J.P., Nath, G.: Similarity solutions for unsteady flow behind an exponential shock in a dusty gas. Phys. Scr. 74, 493–498 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Roberts, P.H., Wu, C.C.: Structure and stability of a spherical implosion. Phys. Lett. 213, 59–64 (1996)CrossRefGoogle Scholar
  14. 14.
    Roberts, P.H., Wu, C.C.: The shock-wave theory of sonoluminescence. In: Srivastava, R.C., Leutloff, D., Takayama, K., Groning, H. (eds.) in Shock Focussing Effect in Medical Science and Sonoluminescence. Springer, New York (2003)Google Scholar
  15. 15.
    Vishwakarma, J.P., Maurya, A.K., Singh, K.K.: Self-similar adiabatic flow headed by magnetogasdynamics cylindrical shock wave in a rotating non-ideal gas. Geophys. Astrophys. Fluid Dyn. 101, 115–167 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Summers, D.: An idealised model of a magnetohydrodynamic spherical blast wave applied to a flare produced shock in the solar wind. Astron. Astrophys. 45(1), 151–158 (1975)Google Scholar
  17. 17.
    Rosenau, P., Frankenthal, S.: Equatorial propagation of axisymmetric magnetohydrodynamic shocks. Phys. Fluids 19(12), 1889–1899 (1976)CrossRefzbMATHGoogle Scholar
  18. 18.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, pp. 246–291. Springer, New York (1986)Google Scholar
  19. 19.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations, pp. 31–201. Springer, New York (1989)zbMATHGoogle Scholar
  20. 20.
    Arora, R., Sharma, A.: Similarity solutions of cylindrical shock waves in magnetogasdynamics with thermal radiation. J. Comput. Nonlinear Dyn. 11, 031001–031005 (2015)CrossRefGoogle Scholar
  21. 21.
    Singh, J.B.: Equatorial propagation of axisymmetric magnetogasdynamic shocks with thermal radiation, I. Astrophys. Space Sci. 96(1), 153–158 (1983)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Singh, J.B., Vishwakarma, P.R.: A self-similar flow behind a spherical shock wave with thermal radiation, I. Astrophys. Space Sci. 93(2), 261–265 (1983)CrossRefGoogle Scholar
  23. 23.
    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)CrossRefGoogle Scholar
  24. 24.
    Jena, J.: Self-similar solutions in a plasma with axial magnetic field (\(\theta \)-Pinch). Meccanica 47(5), 1209–1215 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2017

Authors and Affiliations

  1. 1.Department of Applied Science and EngineeringIIT RoorkeeSaharanpurIndia

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