Abstract
Propagation of cylindrical shock wave in a self-gravitating perfect gas under the influence of axial magnetic field using Lie group of transformation method is investigated. The flow is considered to be isothermal. Density and magnetic field are assumed to be varying in the undisturbed medium. Two different cases of solutions are brought out by the arbitrary constants appearing in the expressions of infinitesimals of local Lie group of transformations. One is with a power law shock path and the other one is with an exponential law shock path. Numerical solutions are obtained for both the cases of power law and exponential law shock paths. The effects of variation in Alfven-Mach number, gravitational parameter and ambient density variation index for power law shock path and effects of variation in Alfven-Mach number, gravitational parameter and ambient magnetic field variation index on the flow variables in the case of exponential law shock path are studied. Also the effects of increase in value of gravitational parameter and in the strength of ambient magnetic field on the shock strength are investigated. The increase in value of Alfven-Mach number leads to the increase in the density ratio which infers to the decrease in shock strength.
Similar content being viewed by others
References
S.C. Lin, J. Appl. Phys. 25, 54 (1954)
G. Nath, Astrophys. Space Sci. 361, 31 (2016)
G. Nath, J.P. Vishwakarma, Acta Astronaut. 123, 200 (2016)
G. Nath, S. Singh, Int. J. Non-linear Mech. 88, 102 (2017)
P.E. Hydon, Symmetry Methods for Differential Equations: A Beginner’s Guide (Cambridge University Press, London, 2000)
G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations (Springer, Berlin, 1974)
G.W. Bluman, S. Kumei, Symmetries and Differential Equations (Springer, New York, 1989)
H. Stephani, Differential Equations: Their Solution Using Symmetries (Cambridge University Press, New York, 1989)
N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations (Wiley, Chichester, 1999)
P.J. Olver, Application of Lie Groups to Differential Equations, 2nd edn. (Springer, New York, 1993)
L.V. Ovsiannikov, Group Analysis of Differential Equations (Academic Press, New York, 1982)
J.D. Logan, J.D.J. Pérez, SIAM J. Appl. Math. 39, 512 (1980)
H.A. Zedan, Appl. Math. Comput. 132, 63 (2002)
P. Basarab-Horwath, R.Z. Zhdanov, J. Math. Phys. 42, 376 (2001)
M.J. Englefield, Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics (Kluwer Academic, Dordrecht, 1992), p. 203
D.D. Laumbach, R.F. Probstein, Phys. Fluids 13, 1178 (1970)
P.L. Sachdev, S. Ashraf, Zeitschrift für angewandte Mathematik und Physik ZAMP 22, 1095 (1971)
V.P. Korobenikov, Problems in the theory of point explosion in gases (American Mathematical Soc., 1976)
T.A. Zhuravskaya, V.A. Levin, J. Appl. Math. Mech. 60, 745 (1996)
G. Nath, Adv. Space Res. 47, 1463 (2011)
I. Lerche, Aust. J. Phys. 32, 491 (1979)
I. Lerche, Aust. J. Phys. 34, 279 (1981)
S.C. Purohit, J. Phys. Soc. Jpn. 36, 288 (1974)
J.B. Singh, P.R. Vishwakarma, Astrophys. Space Sci. 95, 99 (1983)
J.K. Truelove, R.I. Klein, C.F. McKee, J.H. Holliman II, L.H. Howell, J.A. Greenough, D.T. Woods, Astrophys. J. 495, 821 (1998)
R.I. Klein, C.F. McKee, P. Colella, Astrophys. J. 420, 213 (1994)
B.P. Rybakin, V.B. Betelin, V.R. Dushin, E.V. Mikhalchenko, S.G. Moiseenko, L.I. Stamov, V.V. Tyurenkova, Acta Astronaut. 119, 131 (2016)
G.I. Taylor, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 201, 159 (1950)
G.I. Taylor, Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 201, 175 (1950)
L.I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic Press, New York, 1959)
Y.B. Zel’dovich, P. Yu Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, vol. II (Academic Press, New York, 1967)
G. Nath, A.K. Sinha, Physics Research International (2011). https://doi.org/10.1155/2011/782172
S. Galtier, Introduction to Modern Magnetohydrodynamics (Cambridge University Press, Cambridge, 2016)
P. Rosenau, S. Frankenthal, Phys. Fluids 19, 1889 (1976)
G.J. Hutchens, J. Appl. Phys. 77, 2912 (1995)
G. Nath, Shock Waves 24, 415 (2014)
Acknowledgements
Sumeeta Singh gracefully acknowledges DST, New Delhi, India, for providing INSPIRE fellowship, IF No.: 150736, to pursue research work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nath, G., Singh, S. Similarity solutions using Lie group theoretic method for cylindrical shock wave in self-gravitating perfect gas with axial magnetic field: isothermal flow. Eur. Phys. J. Plus 135, 316 (2020). https://doi.org/10.1140/epjp/s13360-020-00292-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-020-00292-0