This article aims to obtain similarity solutions for self-gravitating perfect gas with variable density under the effect of axial or azimuthal magnetic field in rotating medium using Lie group theoretic method and optimal classification of sub-algebras.
One parameter infinitesimal group of transformation is introduced to obtain infinitesimal generators. Based on arbitrary constants present in the infinitesimal generators, a five-dimensional Lie algebra has been obtained. Governing set of PDEs (partial differential equations) has been transformed into a set of ODEs (ordinary differential equations) using a similarity transformation obtained by optimal classification of sub-algebras. The Runge-Kutta 4th-order method has been used to solve the system of ODEs.
Similarity solutions have been obtained for power law shock path and two cases of exponential law shock path. Effects of shock Cowling number, gravitation and rotation parameters, adiabatic index, ambient density variation index have been explored. Rotational parameter, shock Cowling number and adiabatic index have decaying effect on shock wave. Shock strength increases with an increment in gravitational parameter and change from azimuthal magnetic field to axial magnetic field.
The applicability of Lie group theoretic method depends on the invariance of system of PDE's and shock jump conditions. If these are invariant, then the similarity solution may exist.
The present study have its applications in formation of shock envelopes behind high-speed objects like meteors or missiles, gamma-ray bursts, complex filamentary structures in molecular clouds, galaxies and galaxy clusters, nova outburst, stellar winds, analysis of data from exploding wire experiments and pinch effect etc. (see Nath, 2021; Nath and Singh, 2017; Nath and Devi, 2020). Also, our results may be useful in detonation physics, star formation jets, fusion research and laser-induced shock.
Our result may be used to reduce the effect of shock wave on supersonic flight, space vehicle and human body.
All possible cases of similarity solutions have been obtained for self-gravitating perfect gas with variable density under the effect of axial or azimuthal magnetic field in rotating medium using Lie group theoretic method followed by optimal classification of Lie sub-algebras, which has not been discussed yet.
