Similarity solution for magnetogasdynamic spherical shock wave in a self-gravitating non-ideal radiating gas using lie invariance method

In this article, a mathematical model describing the unsteady adiabatic flow of spherical shock waves in a self-gravitating, non-ideal radiating gas under the influence of an azimuthal magnetic field is formulated and similarity solutions are obtained. The ambient medium is assumed to be at rest with uniform density. The effect of thermal radiation under an optically thin limit is included in the energy equation of the governing system. By applying the Lie invariance method, the system of PDEs governing the flow in the said medium is transformed into a system of non-linear ODEs via similarity variables. All the four possible cases of similarity solution are obtained by selecting different values for the arbitrary constants involved in the generators. Among these four cases, only two possess similarity solutions, one by assuming the power-law shock path and other by exponential-law shock path. The set of non-linear ODEs obtained in the case of the power-law shock path is solved numerically using the Runge–Kutta method of 4th order in the MATLAB software. The effects of variation of various parameters such as non-ideal parameter (b¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\overline{b })$$\end{document}, adiabatic index of the gas (γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\gamma )$$\end{document}, Alfven-Mach number (Ma-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M}_{a}^{-2}$$\end{document}), ambient magnetic field variation index (ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\phi )$$\end{document}, and gravitational parameter (G0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({G}_{0})$$\end{document} on the flow quantities are discussed in detail and various results are portrayed in the figures. Furthermore, the article includes a detailed comparison made between the solutions obtained for cases with and without gravitational effects in the presence of magnetic field.