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Similarity solutions for cylindrical shock wave in rotating ideal gas with or without magnetic field using Lie group theoretic method

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Abstract

The propagation of a cylindrical shock wave in rotating ideal gas under adiabatic flow condition is investigated using Lie group of transformation method. Both the cases of shock without magnetic field and under the influence of axial magnetic field are considered separately. The density and azimuthal fluid velocity in the case of shock without magnetic field and density, azimuthal fluid velocity and axial magnetic field in the case of shock under the influence of magnetic field are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the local Lie group of transformations bring about two different cases of solutions, i.e. with a power law and exponential law shock paths. Exact solutions are obtained in the case of power law shock path for both the cases of cylindrical shock with and without magnetic field. It is not possible to obtain the exact solution in the case of exponential law shock path. In this case, the numerical solutions can be obtained by using the respective boundary conditions. Distribution of gasdynamical and magnetogasdynamical flow quantities are illustrated through figures.

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References

  1. G. Nath, 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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

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

    ADS  MATH  Google Scholar 

  4. G.I. Taylor, The formation of a blast wave by a very intense explosion II. The atomic explosion of 1945. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 201(1065), 175–186 (1950)

    ADS  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Y.B. Zel’dovich, Y.P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, vol. II (Academic Press, New York, 1967)

    Google Scholar 

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

    Article  ADS  Google Scholar 

  8. D. Summers, 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)

    ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  11. J.P. Vishwakarma, A.K. Maurya, K.K. Singh, 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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  13. 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(9), 1463–1471 (2011)

    Article  ADS  Google Scholar 

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

    ADS  Google Scholar 

  15. J.P. Vishwakarma, N. Patel, 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 

  16. M. Nagasawa, Gravitational instability of the isothermal gas cylinder with an axial magnetic field. Prog. Theor. Phys. 77(3), 635–652 (1987)

    Article  ADS  Google Scholar 

  17. V.P. Korobeĭnikov, Problems in the Theory of Point Explosion in Gases, vol. 119 (AMS, Providence, 1976)

    Google Scholar 

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

    Article  Google Scholar 

  19. R.M. Lock, A.J. Mestel, Annular self-similar solutions in ideal magnetogasdynamics. J. Plasma Phys. 74(4), 531–554 (2008)

    Article  ADS  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  24. G. Nath, J.P. Vishwakarma, 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  26. W. Mostert, D.I. Pullin, R. Samtaney, V. Wheatley, Converging cylindrical magnetohydrodynamic shock collapse onto a power-law-varying line current. J. Fluid Mech. 793, 414–443 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. G. Nath, J.P. Vishwakarma, 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  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  29. G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations (Springer, Berlin, 1974)

    Book  MATH  Google Scholar 

  30. G.W. Bluman, S. Kumei, Symmetries and Differential Equations (Springer, New York, 1989)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  32. N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations (Wiley, Chichester, 1999)

    MATH  Google Scholar 

  33. P.J. Olver, Application of Lie Groups to Differential Equations, 2nd edn. (Springer, New York, 1993)

    Book  MATH  Google Scholar 

  34. P.E. Hydon, Symmetry Methods for Differential Equations: A Beginner’s Guide (Cambridge University Press, London, 2000)

    Book  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. H.A. Zedan, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  41. F. Oliveri, M.P. Speciale, 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. G.B. Whitham, On the propagation of shock waves through regions of non-uniform area or flow. J. Fluid Mech. 4(4), 337–360 (1958)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. P. Rosenau, S. Frankenthal, Equatorial propagation of axisymmetric magnetohydrodynamic shocks. Phys. Fluids 19(12), 1889–1899 (1976)

    Article  ADS  MATH  Google Scholar 

  45. G. Nath, Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas. Shock Waves 24(4), 415–428 (2014)

    Article  ADS  Google Scholar 

  46. J.P. Vishwakarma, A.K. Yadav, Self-similar analytical solutions for blast waves in inhomogeneous atmospheres with frozen-in-magnetic field. Eur. Phys. J. B Condens. Matter Complex Syst. 34(2), 247–253 (2003)

    Article  Google Scholar 

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Acknowledgements

Sumeeta Singh gracefully acknowledge DST, New Delhi, India, for providing INSPIRE fellowship, IF No.: 150736, to pursue research work.

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

    G. Nath & Sumeeta Singh

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  1. G. Nath
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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, 929 (2020). https://doi.org/10.1140/epjp/s13360-020-00946-z

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