Similarity solution for the magnetogasdynamic shock wave in a self-gravitating and rotating ideal gas under the influence of radiation heat flux

Abstract

The Lie invariance method is used to analyze the one-dimensional, unsteady flow of a cylindrical shock wave in a rotating, self-gravitating, radiating ideal gas under the influence of an axial or azimuthal magnetic field, with an emphasis on adiabatic conditions. The analysis assumes a stationary environment just ahead of the shock wave and considers variations in fluid velocity, magnetic field, and density within the perturbed medium just behind the shock front. In the governing equations, the impact of thermal radiation under an optically thin limit is integrated into the energy equation. Utilizing the Lie invariance method, the set of partial differential equations governing the flow in this medium is transformed into a system of nonlinear ordinary differential equations (ODEs) using similarity variables. Two distinct cases of similarity solutions are obtained by selecting different values for the arbitrary constants associated with the generators. Among these cases, one yields similarity solutions assuming a power-law shock path and the other an exponential-law shock path. For both cases, the resulting set of nonlinear ODEs are numerically solved using the 4th-order Runge–Kutta method in MATLAB software. The article thoroughly explores the influence of various parameters, including γ (adiabatic index of the gas), Ma−2 (Alfvén–Mach number), σ (ambient density exponent), l1 (rotational parameter), and G0 (gravitational parameter) on the flow properties. The findings are visually presented to offer a comprehensive insight into the effects of these parameters.