Optimal system of Lie sub-algebras and numerical solution for shock wave in rotating non-ideal dusty gas with monochromatic radiation

In the present article, we have obtained the similarity solutions with the help of optimal system of Lie sub-algebras and numerical method for the cylindrical shock wave formed by a moving piston in a non-ideal dusty gas in a rotating medium with the impact of monochromatic radiation. The Lie group theoretic technique is used to obtain the optimal system of Lie sub-algebras for the fundamental equations in the case of 1-D (one-dimensional) flow. By using the Lie group theoretic technique, we are able to drive the similarity and numerical solution for both the power and exponential law shock paths. The similarity solutions with power law shock path in two different cases (i.e., Cases Ia, Ib) are obtained in Case I. Also, in Case II, the similarity solutions exist in two different cases with power law shock path (i.e., Cases IIa, IIc) and in two different cases with exponential law shock path (i.e., Cases IIb, IId) which are discussed in detail. The results presented in Cases Ib, IIa, IIb, IIc, and IId are newly obtained results. The numerical solutions in the power law shock path for Case Ia and the exponential law shock path in Case IIb are performed, and the distribution of the physical flow variables in the flow field region behind the shock front is obtained. The impacts of the several problematic physical parameters on the shock strength and on the physical flow variable are studied. In the present study, it is found that with an increment in the value of the initial ratio of the density of solid particles to the species density of the gas (G1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{1}$$\end{document}), the shock strength increases, whereas the shock strength decreases with an increment in the gas non-ideality parameter (omega<overline>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{\omega }$$\end{document}). For the lower value of G1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{1}$$\end{document} (i.e., for G1=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{1} = 6$$\end{document}), the shock strength decreases, whereas for a higher value of G1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{1}$$\end{document} (i.e. , for G1=20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{1} = 20$$\end{document} or 100), the shock strength increases with an increment in the mass concentration of the solid particles in the mixture (mu p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{p}$$\end{document}).