Lie group theoretic method to cylindrical and spherical shock waves in self-gravitating dusty gas with magnetic field

Abstract

This study explores self-similar solutions to shock-wave propagation in a self-gravitating, perfectly conducting dusty gas influenced by a magnetic field, using the group-invariance technique. The system of partial differential equations is transformed to a system of ordinary differential equations, which are subsequently solved numerically by using the fourth-order Runge–Kutta scheme implemented in Mathematica. The analysis of physical flow parameters provides insights into the distribution of physical flow variables and the strength of shock waves. A notable advantage of employing the Lie group-theoretic method is that it yields all possible solution cases, unlike other methods, such as similarity solutions using dimensional analysis, which typically address only one case at a time. The findings indicate that increases in the adiabatic exponent, magnetic field strength, and non-idealness parameter of the gas reduce the shock wave strength. Conversely, changing the geometry from cylindrical to spherical, as well as increasing the mass concentration of solid particles or the ratio of solid-particle density to the initial gas density, enhances the shock strength.