Exact and Numerical Solution Using Lie Group Analysis for the Cylindrical Shock Waves in a Self-Gravitating Ideal Gas with Axial Magnetic Field

In this paper, we seek the exact and numerical solutions using Lie group analysis for one dimensional unsteady adiabatic flow in a self-gravitating ideal gas behind a cylindrical shock wave with axial magnetic field. With the help of Lie group analysis, the one-dimensional optimal system of sub-algebra is obtained for the system of equations of motion. With the help of optimal classes of infinitesimal generators, we constructed the similarity variable and transformation of the flow variables, which convert the system of partial differential equations into system of ordinary differential equations. In three particular cases, we have derived a general framework to solve the fundamental equations and exact feasible solutions are obtained. In two cases, the similarity solutions with exponential law and power law shock path are discussed. The similarity solution is obtained using numerical method in the case of power law shock path. It is obtained that the increase in the values of magnetic field strength and adiabatic index have the decaying effect on the shock wave. Also, increase in the strength of shock wave is witnessed with the rise in the gravitation parameter value. The effects of variation of magnetic field strength, adiabatic exponent and gravitational parameter on the flow variables are analyzed graphically.