Abstract
Abstract In this paper, a non-dimensional unsteady adiabatic flow of a plane or cylindrical strong shock wave propagating in plasma is studied. The plasma is assumed to be an ideal gas with infinite electrical conductivity permeated by a transverse magnetic field. A self-similar solution of the problem is obtained in terms of density, velocity and pressure in the presence of magnetic field. We use the method of Lie group invariance to determine the class of self-similar solutions. The arbitrary constants, occurring in the expressions of the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. A particular case of the collapse of an imploding shock is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of flow variables. Our results are found in good agreement with the known results. All computational work is performed by using software package MATHEMATICA.
Highlights
The spread of shock waves under the control of strong magnetic field is a problem of great interest to researchers in a variety of fields such as nuclear science, geophysics, plasma physics and astrophysics
Singh et al [, ] used the method of Lie group of transformations to obtain an exact solution for unsteady equation of non-ideal gas and magnetogasdynamics
We use the method of Lie group invariance under infinitesimal point transformations [ – ] to study the problem of propagation of strong shock waves in a radiating and electrically conducting gas permeated by a transverse magnetic field
Summary
The spread of shock waves under the control of strong magnetic field is a problem of great interest to researchers in a variety of fields such as nuclear science, geophysics, plasma physics and astrophysics. Chisnell [ ] provided analytical solutions to the problem of converging shock waves by using the singular points method. Singh et al [ , ] used the method of Lie group of transformations to obtain an exact solution for unsteady equation of non-ideal gas and magnetogasdynamics. We consider the problem of propagation of a onedimensional unsteady flow of an inviscid ideal gas permeated by a transverse magnetic field with infinite electrical conductivity as it approaches the surface of a star. We use the method of Lie group invariance under infinitesimal point transformations [ – ] to study the problem of propagation of strong shock waves in a radiating and electrically conducting gas permeated by a transverse magnetic field. The Rankine-Hugoniot jump conditions for the strong shocks are as follows (Whitham [ ]): u= V, γ +
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