Similarity Solutions of Cylindrical Strong Shock in Self-Gravitating Medium Under the Monocromatic Radiation

Abstract

A class of self-similar solutions to the model of a cylindrical shock wave in non-uniform atmosphere in the presence of monochromatic radiation and gravitation in magneto gas dynamics has been obtained by using a similarity method. The propagation of a cylindrical shock wave in an ideal gas with monochromatic radiation and gravitating effects has been discussed. Through applying similarity transformations to the system of equations, we obtained the symmetry generators of the system. By using the symmetry generators and the surface invariance condition, we obtained the group invariant solution and then, with the help of group invariant solution, we converted the given system of PDEs to the system of ODEs together with the boundary condition. The obtained system of ODEs together with boundary condition has been solved numerically by using Runge–Kutta method of order four. The flow variables are analyzed graphically behind the shock with respect to the variation of parameters.

1. Introduction

The analysis of shock propagation and related flows in astrophysical phenomena such as supernova explosion is of great interest. The study of the propagation of strong shock waves in a self-gravitating medium behind the shock front has always received much attention by researchers in various fields such as nuclear science, geophysics, plasma physics, and astrophysics. In recent years, considerable attention has been given to the study of one-dimensional gas motion occurring in an inhomogeneous non-gravitating and self-gravitating medium on a gaseous substance in stellar interiors. Using the dimensionality method, Sedov [1] obtained self-similar solutions to the number of problems with disturbed energy release. Nath [2] presents a theoretical model of cylindrical shock waves in magnetogasdynamics in the presence of monochromatic radiation following the work of Khudyakov [3]. Furthermore, cylindrical shock waves in the presence of monochromatic radiation without a magnetic field effect have been discussed by Nath and Takhar [4]. The propagation of a shock wave produced by the sudden point expansion in an inhomogeneous non-gravitating medium has been analyzed by many authors; Taylor [5] and Cavaliere and Messina [6] have studied the similarity flow behind the shock wave. A blast/shock wave interacting with three objects was examined by Dey et al. [7] for the diaphragm pressure ratio of 56.

A theoretical study of converging shock waves for a cylindrically or spherically symmetric flow in an ideal gas was first performed by Guderley [8]. In particular, we refer to the work by Sakurai [9], Zeldovich and Raizer [10], and Lazarus [11]. Sharma and Radha [12] and Sharma and Arora [13] investigated the implosion problem to obtain more accurate results. Logan and Perez [14] derived the complete class of similarity solutions for the one-dimensional, time-dependent shock hydrodynamics problem using the Lie group method. Tomar et al. [15] and Chauhan et al. [16] obtained the global solution to the imploding shock wave problem for non-ideal gas in the absence and presence of a magnetic field by using the perturbation series technique. The method of similarity transformations has been applied by Arora et al. [17] to the problem of strong shock waves in magnetogasdynamics without thermal radiation. A theoretical study of the converging strong shock waves problem for cylindrically and spherically symmetric flow in non-ideal gas using a Mie–Gruneisen-type equation of state has been done by Narsimhulu et al. [18]. Wanga and Fakhar [19] applied Lie symmetry method to an extended (2 + 1)-dimensional Zakharov-Kuznetsov-Burgers equation. By taking the general form of the equation of state, Pandey et al. [20] applied the symmetry method to the system of PDEs governing one-dimensional unsteady motion of a viscous compressible fluid. Khalique and Biswas [21] determined the stationary solutions of the nonlinear Klein–Gordon equations in $(1+1)$ dimensions by applying the Lie symmetry approach. Bajargaan and Patel [22] obtained the self-similar solution to the cylindrically symmetric shock wave in self-gravitating, rotating, axisymmetric dusty gas in which they also considered the presence of heat conduction and radiation heat flux.

The mathematical model investigated in this study describes the propagation of a cylindrically symmetric shock wave in a non-uniform medium under the combined effects of gravitation, magnetic field, and monochromatic radiation. Such models are of considerable importance in understanding several astrophysical and high-energy plasma phenomena where cylindrical symmetry is more appropriate than spherical symmetry. The role of monochromatic radiation in modifying shock dynamics was first systematically examined by Onkar Nath, who studied self-similar cylindrical MHD shock waves in the presence of radiation and demonstrated its significant influence on the flow field and shock strength [2]. Subsequently, Nath extended this analysis to rotating atmospheres and highlighted the combined effects of rotation and radiation on cylindrical shock propagation [23]. In many realistic physical situations, such as astrophysical jets, supernova remnants, accretion-driven outflows, and laboratory plasma experiments, shock waves propagate through magnetized, radiating, and gravitationally influenced media. The inclusion of a magnetic field alters the post-shock structure through magnetic pressure and tension effects, while monochromatic radiation accounts for radiative energy transfer in optically active media. Motivated by these earlier studies, the present work extends the existing literature by incorporating gravitation into the cylindrical magnetogasdynamics framework with monochromatic radiation and by employing Lie symmetry analysis to obtain self-similar solutions. This approach provides deeper insight into the coupled influence of magnetogasdynamics, radiation, and gravitation on the evolution of cylindrical shock waves. This study is the first to perform a Lie symmetry analysis of strong cylindrical shocks incorporating gravity, radiation, and MHD effects simultaneously, extending the works of Nath [2,4] and Bajargaan a Patel [22].

In this paper, we applied the Lie group transformation method, as discussed by Bluman et al. [24] and Logan [14], to obtain similarity solutions for the propagation of cylindrical shock waves in a non-rotating, non-uniform atmosphere under the influence of monochromatic radiation and gravitation. In addition, the effect of the magnetic field has been taken into account as a significant factor in the analysis.

2. Basic Equations and Shock Conditions

In magnetogasdynamics, the governing equations for one-dimensional cylindrically symmetric motion of an inviscid perfect gas in a non-uniform medium, under the influence of monochromatic radiation and gravitational forces, can be expressed as (cf. Khudyakov [3]; Whitham [25]; Nath et al. [26]; Zedan [27]):

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \rho }{\partial t}}+u{\displaystyle \frac{\partial \rho }{\partial r}}+\rho {\displaystyle \frac{\partial u}{\partial r}}+{\displaystyle \frac{\rho u}{r}}=0,\\ \hfill {\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial r}}+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial r}}+{\displaystyle \frac{h}{\rho }}{\displaystyle \frac{\partial h}{\partial r}}+{\displaystyle \frac{h^2}{\rho }}+{\displaystyle \frac{Gm}{r}}=0,\\ \hfill {\displaystyle \frac{\partial h}{\partial t}}+u{\displaystyle \frac{\partial h}{\partial r}}+h{\displaystyle \frac{\partial u}{\partial r}}+{\displaystyle \frac{uh}{r}}=0,\\ \hfill {\displaystyle \frac{\partial p}{\partial t}}+u{\displaystyle \frac{\partial p}{\partial r}}-{\displaystyle \frac{\gamma p}{\rho }}({\displaystyle \frac{\partial \rho }{\partial t}}+u{\displaystyle \frac{\partial \rho }{\partial r}})\\ \hfill +{\displaystyle \frac{(\gamma -1)}{\rho r}}(r{\displaystyle \frac{\partial j}{\partial r}}+j)=0,\\ \hfill {\displaystyle \frac{\partial m}{\partial r}}=2\pi \rho r,\\ \hfill {\displaystyle \frac{\partial j}{\partial r}}=Kj,\end{array}$$

(1)

here, $\rho$ denotes the density, p the pressure, u the velocity, h the magnetic field, and j the monochromatic radiation flux. The symbol e represents the energy per unit mass at a radial distance r. The parameter $\gamma$ is the ratio of specific heats, m is the mass per unit length, and r is the radial distance from the centre. Time is denoted by t, K is the absorption coefficient, and G represents the gravitational constant. The internal energy is given by

$$\begin{array}{c}\hfill e={\displaystyle \frac{p}{\rho (\gamma -1)}}(\mathrm{with}\gamma =\mathrm{constant}).\end{array}$$

(2)

For the ideal gas, the equation of state is

$$\begin{array}{c}\hfill p=\rho RT,\end{array}$$

(3)

where T and R respectively denote the absolute temperature and the gas constant which obeys the thermodynamic relations $R=C_p-C_v$ and $e=C_{v}T$. Here, $\gamma$ is the ratio $C_p:C_v$, where $C_v={\displaystyle \frac{R}{(\gamma -1)}}$ and $C_p$ represents the specific heat at constant volume and constant pressure, respectively. Let the initial condition at time $t=0$ be given by $u=0$, $p=p_0\left(r\right)$, $\rho ={\rho }_0\left(r\right)$, $h=h_0\left(r\right)$, $j=j_0\left(r\right)$ and $m=m_0\left(r\right)$, which are all the functions of r. The Rankine–Hugoniot (R-H) jump conditions at the shock are given by (Whitham [25] ):

$$\begin{array}{ccc}\hfill u_1& =& {\displaystyle \frac{2V}{\gamma +1}},\hfill \\ \hfill {\rho }_1& =& {\displaystyle \frac{\gamma +1}{\gamma -1}}{\rho }_0,\hfill \\ \hfill p_1& =& {\displaystyle \frac{2V^2{\rho }_0}{\gamma +1}},\hfill \\ \hfill h_1& =& {\displaystyle \frac{\gamma +1}{\gamma -1}}{h}_0,\hfill \\ \hfill m_1& =& m_0={\displaystyle \frac{2\pi {\rho }^{*}{r}^{2+\beta }}{2+\beta }},\hfill \\ \hfill j_1& =& j_0,\hfill \end{array}$$

(4)

where ${\rho }_0$, $p_0$, $h_0$, $m_0$ and $j=j_0$ respectively denote the density, pressure, magnetic field, mass and radiation flux in the undisturbed medium, and V denotes the velocity of shock wave. The parameters just behind the shock is represented by the suffix 1.

The absorption coefficient is given as

$$\begin{array}{ccc}\hfill K& =& k_0{\rho }^n{p}^m{j}^q{r}^s{t}^l,\hfill \end{array}$$

(5)

where the dimensions of this constant K are given by

$$\begin{array}{c}\hfill [K]=M^{-n-m-q}{L}^{3n+m-s}{T}^{2m+3q-l}.\end{array}$$

(6)

The constant values taken by Nath and Takhar [4] are $n=-{\displaystyle \frac{1}{2}},m={\displaystyle \frac{3}{2}},q=0,s+l=-1.$ Moreover, the dimensionless constants $j_0$, $p_0$, ${\rho }_0$ are related as follows:

$$\begin{array}{ccc}\hfill j_0& =& p_0^{3/2}{\rho }_0^{-1/2}.\hfill \end{array}$$

(7)

3. Lie Group of Transformation

Similarity methods for obtaining self-similar solutions of partial differential equations are typically based on the principle of reducing the number of independent variables. In multidimensional problems, a one-parameter Lie group of transformations reduces the one independent variable at each step, thereby generating a new equation with one fewer variable than in the previous step. The transformed equation at each stage must remain invariant under the corresponding Lie group of transformations. Once the Lie group of transformations that leave the partial differential equations invariant is determined, it becomes possible to construct a solution that is invariant under these transformations [28,29,30,31,32]. In this work, we investigate the motion of a converging shock wave in a self-gravitating medium under the influence of monochromatic radiation using the similarity method.

$$\begin{array}{c}\hfill r^{*}=r+\epsilon R(r,t,u,p,h,\rho ,m,j),\\ \hfill t^{*}=t+\epsilon T(r,t,u,p,h,\rho ,m,j),\\ \hfill u^{*}=u+\epsilon U(r,t,u,p,h,\rho ,m,j),\\ \hfill p^{*}=p+\epsilon P(r,t,u,p,h,\rho ,m,j),\\ \hfill {\rho }^{*}=\rho +\epsilon S(r,t,u,p,h,\rho ,m,j),\\ \hfill h^{*}=h+\epsilon H(r,t,u,p,h,\rho ,m,j),\\ \hfill j^{*}=j+\epsilon J(r,t,u,p,h,\rho ,m,j),\\ \hfill m^{*}=m+\epsilon W(r,t,u,p,h,\rho ,m,j),\end{array}$$

(8)

The symmetry generators $R,T,U,P,S,H,W,J$ are functions of the variables $r,t,u,p,h,\rho ,m,$ and j. The one-parameter infinitesimal transformation (8) is constructed in such a way that the system of partial differential equations given by Equation (1), together with the shock conditions specified in Equation (4), remains invariant. The parameter $\epsilon$ is chosen to be sufficiently small so that its square and higher-order terms can be neglected. This invariance group reduces the number of independent variables in the system of PDEs by one, thereby enabling the reduction of the PDE system (1) into a corresponding system of ODEs.

For simplicity, let us take $x_1=t,x_2=r,u_1=u,u_2=p,u_3=h,u_4=\rho ,u_5=m,u_6=j$ and $p_j^i={\displaystyle \frac{\partial u_i}{\partial x_j}}$, where $i=1,2,3,4,5,6$ and $j=1,2$.

The system of basic Equation (1) can be represented as

$$G_k(x_j,u_i,p_j^i)=0,k=1,2,3,4,5,6,$$

which remains constantly conformally invariant under the Lie group of transformation (8), if there exist constants ${\alpha }_{kr}(k,r=1,2,3,4,5,6)$ such that

$$\begin{array}{ccc}\hfill L{G}_k& =& {\alpha }_{kr}{G}_r,\hfill \end{array}$$

(9)

holds for all smooth surfaces, $u_i=u_i\left(x_j\right)$. The Lie derivative L in the direction of the extended vector field is given by

$$\begin{array}{ccc}\hfill L& =& {\xi }_x^j{\displaystyle \frac{\partial }{\partial x_j}}+{\xi }_u^i{\displaystyle \frac{\partial }{\partial u_i}}+{\xi }_{p_j}^i{\displaystyle \frac{\partial }{\partial p_j^i}},\hfill \end{array}$$

with ${\xi }_x^1=T$, ${\xi }_x^2=R$, ${\xi }_u^1=U$, ${\xi }_u^2=P$, ${\xi }_u^3=H$, ${\xi }_u^4=S$, ${\xi }_u^5=W$, ${\xi }_u^6=J$, and

$$\begin{array}{ccc}\hfill {\xi }_{p_j}^i& =& {\displaystyle \frac{\partial {\xi }_u^i}{\partial x_j}}+{\displaystyle \frac{\partial {\xi }_u^i}{\partial u_k}}{p}_j^k-{\displaystyle \frac{\partial {\xi }_x^l}{\partial x_j}}{p}_l^i-{\displaystyle \frac{\partial {\xi }_x^l}{\partial u_m}}{p}_l^i{p}_j^m,\hfill \end{array}$$

(10)

where l = 1, 2, j = 1, 2, i = 1, 2, 3, 4, 5, 6, m = 1, 2, 3, 4, 5, 6 and k = 1, 2, 3, 4, 5, 6. Here, summation convention is represented by repeated indices and ${\xi }_{p_j}^i$ represents the generalised derivative transformation.

Equation (9) implies

$${\xi }_x^j{\displaystyle \frac{\partial G_k}{\partial x_j}}+{\xi }_u^i{\displaystyle \frac{\partial G_k}{\partial u_i}}+{\xi }_{p_j}^i{\displaystyle \frac{\partial G_k}{\partial p_j^i}}={\alpha }_{kn}{G}_n,$$

(11)

$$wherek=1,2,3,4,5,6,n=1,2,3,4,5,6.$$

By putting the value of ${\xi }_{p_j}^i$ from Equation (10) into Equation (11), we found a polynomial equation in $p_j^i$. On setting all the coefficients of $p_j^i$ and $p_j^i{p}_l^m$ to zero, we obtain a system of first order, linear PDEs, which are considered a system for determining equations in terms of symmetry generators $T,R,U,H,P,S,W,J$. Further, we solved the system of determining equations, which yields the invariant group (8) completely.

We apply the above procedure to the system of PDEs (1). We derived the following determining equations from the invariance of the continuity Equation $(1)$:

$$\begin{array}{c}\hfill S_u-\rho T_r={\alpha }_{12},\\ \hfill S_p={\alpha }_{14},\\ \hfill S_{\rho }-T_t-u{T}_r={\alpha }_{11},\\ \hfill S_h={\alpha }_{13},S_m=0,S_j=0,\\ \hfill S+u{S}_u+\rho U_u-\rho R_r\\ \hfill ={\alpha }_{11}\rho +{\alpha }_{12}u+{\alpha }_{13}h+\gamma p{\alpha }_{14},\\ \hfill u{S}_p+\rho U_p={\displaystyle \frac{1}{\rho }}{\alpha }_{12}+{\alpha }_{14}u,\\ \hfill -R_t+U+u{S}_{\rho }-u{R}_r+\rho U_{\rho }={\alpha }_{11}u,\\ \hfill u{S}_h+\rho U_h={\displaystyle \frac{h}{\rho }}{\alpha }_{12}+u{\alpha }_{13},\\ \hfill u{S}_m+\rho U_m={\alpha }_{15},\\ \hfill u{S}_j+\rho U_j={\alpha }_{14}{\displaystyle \frac{(\gamma -1)}{\rho }}+{\alpha }_{16},\\ \hfill S_t+{\displaystyle \frac{Su}{r}}+{\displaystyle \frac{\rho U}{r}}-{\displaystyle \frac{\rho uR}{r^2}}+\rho U_r+u{S}_r\\ \hfill ={\alpha }_{11}{\displaystyle \frac{\rho u}{r}}+{\alpha }_{12}({\displaystyle \frac{Gm}{r}}+{\displaystyle \frac{h^2}{jr}})\\ \hfill +{\alpha }_{13}{\displaystyle \frac{uh}{r}}+{\alpha }_{14}{\displaystyle \frac{(\gamma -1)j}{\rho r}}-{\alpha }_{15}2\pi r\rho -{\alpha }_{16}Kj.\end{array}$$

Similarly, from the invariance of the momentum Equation $(1)$, we derived the following determining equations:

$$\begin{array}{c}\hfill U_u-T_t-u{T}_r={\alpha }_{22},\\ \hfill U_p-{\displaystyle \frac{1}{\rho }}{T}_r={\alpha }_{24},\\ \hfill U_{\rho }={\alpha }_{21},U_m=0,\\ \hfill U_h-{\displaystyle \frac{h}{\rho }}{T}_r={\alpha }_{23},U_j=0,\\ \hfill -R_t+U+u{U}_u-u{R}_r+{\displaystyle \frac{1}{\rho }}{P}_u+{\displaystyle \frac{h}{\rho }}{H}_u\\ \hfill ={\alpha }_{21}\rho +u{\alpha }_{22}+h{\alpha }_{23}+{\alpha }_{24}\gamma p,\\ \hfill -{\displaystyle \frac{S}{{\rho }^2}}+u{U}_p+{\displaystyle \frac{1}{\rho }}{P}_p-{\displaystyle \frac{1}{\rho }}{R}_r+{\displaystyle \frac{h}{\rho }}{\displaystyle \frac{\partial H}{\partial p}}\\ \hfill ={\displaystyle \frac{{\alpha }_{22}}{\rho }}+u{\alpha }_{24},\\ \hfill u{U}_{\rho }+{\displaystyle \frac{1}{\rho }}{P}_{\rho }+{\displaystyle \frac{h}{\rho }}{H}_p=u{\alpha }_{21},\\ \hfill u{U}_m+{\displaystyle \frac{1}{\rho }}{P}_m+{\displaystyle \frac{h}{\rho }}{H}_{\rho }={\alpha }_{25},\\ \hfill u{U}_j+{\displaystyle \frac{1}{\rho }}{P}_j+{\displaystyle \frac{h}{\rho }}{H}_m\\ \hfill ={\alpha }_{24}{\displaystyle \frac{(\gamma -1)}{\rho }}+{\alpha }_{26},\\ \hfill {\displaystyle \frac{H}{\rho }}+u{U}_h+{\displaystyle \frac{1}{\rho }}{H}_j+{\displaystyle \frac{h}{\rho }}{H}_h-{\displaystyle \frac{h}{\rho }}{R}_r-{\displaystyle \frac{hS}{{\rho }^2}}\\ \hfill ={\alpha }_{22}{\displaystyle \frac{h}{\rho }}+u{\alpha }_{23},\\ \hfill U_t+u{U}_r+{\displaystyle \frac{1}{\rho }}{P}_r+{\displaystyle \frac{h}{\rho }}{H}_r+{\displaystyle \frac{2hH}{\rho r}}-{\displaystyle \frac{h^{2}S}{{\rho }^{2}r}}-{\displaystyle \frac{h^{2}R}{\rho r^2}}+{\displaystyle \frac{GW}{r}}-{\displaystyle \frac{GmR}{r^2}}\\ \hfill ={\alpha }_{21}{\displaystyle \frac{\rho u}{r}}+{\alpha }_{22}{\displaystyle \frac{h^2}{\rho r}}+{\alpha }_{22}{\displaystyle \frac{Gm}{r}}\\ \hfill +{\alpha }_{23}{\displaystyle \frac{uh}{r}}+{\alpha }_{24}{\displaystyle \frac{\gamma pu}{r}}+{\alpha }_{24}{\displaystyle \frac{(\gamma -1)j}{\rho r}}-{\alpha }_{25}2\pi r\rho -{\alpha }_{26}Kj.\end{array}$$

Next, the invariance of the magnetic field equation, $(1)$, yields the following determining equations:

$$\begin{array}{c}\hfill H_u-h{T}_r={\alpha }_{32},\\ \hfill H_p={\alpha }_{34},H_{\rho }={\alpha }_{31},\\ \hfill H_h-T_t-h{T}_r={\alpha }_{33},\\ \hfill H_m=0,H_j=0,\\ \hfill H+u{H}_u+h{U}_u-h{R}_r\\ \hfill =\rho {\alpha }_{31}+u{\alpha }_{32}+h{\alpha }_{33}+\gamma p{\alpha }_{34},\\ \hfill u{H}_p+h{U}_p={\displaystyle \frac{{\alpha }_{32}}{\rho }}+u{\alpha }_{34},\\ \hfill U-R_t+u{H}_h-u{R}_r+h{U}_h\\ \hfill ={\displaystyle \frac{h{\alpha }_{32}}{\rho }}+{\alpha }_{33}u,\\ \hfill u{H}_{\rho }+h{U}_{\rho }=u{\alpha }_{31},\\ \hfill u{H}_m+h{U}_m={\alpha }_{35},\\ \hfill u{H}_j+h{U}_j\\ \hfill ={\displaystyle \frac{(\gamma -1)}{\rho }}{\alpha }_{34}+{\alpha }_{36},\\ \hfill H_t+u{H}_r+h{U}_r+{\displaystyle \frac{Uh}{r}}+{\displaystyle \frac{uH}{r}}-{\displaystyle \frac{uhR}{r^2}}\\ \hfill ={\alpha }_{31}{\displaystyle \frac{\rho u}{r}}+{\alpha }_{32}{\displaystyle \frac{Gm}{r}}+{\alpha }_{33}{\displaystyle \frac{uh}{r}}\\ \hfill +{\alpha }_{34}({\displaystyle \frac{\gamma pu}{r}}+{\displaystyle \frac{(\gamma -1)j)}{\rho r}})\\ \hfill -{\alpha }_{35}2\pi \rho r-Kj{\alpha }_{36}.\end{array}$$

Next, the invariance of the energy Equation $(1)$ yields the following determining equations:

$$\begin{array}{c}\hfill P_u-\gamma p{T}_r={\alpha }_{42},P_p-T_t-u{T}_r={\alpha }_{44},\\ \hfill P_{\rho }={\alpha }_{41},P_m=0,\\ \hfill P_h={\alpha }_{43},P_j-{\displaystyle \frac{(\gamma -1)}{\rho }}{J}_r=0,\\ \hfill u{P}_u+\gamma p{U}_u-\gamma p{R}_r+\gamma P+{\displaystyle \frac{(\gamma -1)}{\rho }}{J}_u\\ \hfill =\rho {\alpha }_{41}+u{\alpha }_{42}+{\alpha }_{43}h+\gamma p{\alpha }_{44},\\ \hfill U-R_t+u{P}_p-u{R}_r+\gamma p{U}_p+{\displaystyle \frac{(\gamma -1)}{\rho }}{J}_p\\ \hfill ={\displaystyle \frac{{\alpha }_{42}}{\rho }}+{\alpha }_{44}u,\\ \hfill u{P}_h+\gamma p{U}_h+{\displaystyle \frac{(\gamma -1)}{\rho }}{J}_h\\ \hfill ={\alpha }_{42}{\displaystyle \frac{h}{\rho }}+u{\alpha }_{43},\\ \hfill u{P}_{\rho }+\gamma p{U}_{\rho }+{\displaystyle \frac{(\gamma -1)}{\rho }}{J}_{\rho }={\alpha }_{41}u,\\ \hfill u{P}_m+\gamma p{U}_m+{\displaystyle \frac{(\gamma -1)}{\rho }}{J}_m={\alpha }_{45},\\ \hfill u{P}_j+\gamma p{U}_j+{\displaystyle \frac{(\gamma -1)}{\rho }}(J_j-R_r)-{\displaystyle \frac{(\gamma -1)S}{{\rho }^2}}\\ \hfill ={\displaystyle \frac{(\gamma -1)}{\rho }}{\alpha }_{44}+{\alpha }_{46},\\ \hfill P_t+u{P}_r+\gamma p{U}_r+{\displaystyle \frac{\gamma Pu}{r}}+{\displaystyle \frac{\gamma pU}{r}}\\ \hfill -{\displaystyle \frac{\gamma puR}{r^2}}+(\gamma -1)({\displaystyle \frac{J}{\rho r}}-{\displaystyle \frac{jS}{{\rho }^{2}r}}\\ \hfill -{\displaystyle \frac{jR}{\rho r^2}})+{\displaystyle \frac{(\gamma -1)}{\rho }}{J}_r\\ \hfill ={\alpha }_{41}{\displaystyle \frac{\rho u}{r}}+{\alpha }_{42}({\displaystyle \frac{h^2}{\rho r}}\\ \hfill +{\displaystyle \frac{Gm}{r}})+{\alpha }_{43}{\displaystyle \frac{uh}{r}}+{\alpha }_{44}{\displaystyle \frac{\gamma pu}{r}}\\ \hfill +{\alpha }_{44}{\displaystyle \frac{(\gamma -1)}{\rho }}{\displaystyle \frac{j}{r}}-{\alpha }_{45}2\pi r\rho -{\alpha }_{46}Kj.\end{array}$$

Next, the invariance of the Equation $(1)$ gives the following determining equations:

$$\begin{array}{c}\hfill {\alpha }_{52}=0,{\alpha }_{54}=0,{\alpha }_{51}=0,{\alpha }_{53}=0,\\ \hfill T_r=0,W_u={\alpha }_{51}\rho +{\alpha }_{52}u+{\alpha }_{53}h+{\alpha }_{54}\gamma p,\\ \hfill W_p={\displaystyle \frac{1}{\rho }}{\alpha }_{52}+u{\alpha }_{54},W_{\rho }={\alpha }_{51}u,\\ \hfill W_h={\displaystyle \frac{h}{\rho }}{\alpha }_{52}+u{\alpha }_{53},\\ \hfill W_m-R_r={\alpha }_{55},W_j={\displaystyle \frac{(\gamma -1)}{\rho }}{\alpha }_{54}+{\alpha }_{56},\\ \hfill W_r-2\pi Sr-2\pi \rho R=\\ \hfill {\displaystyle \frac{\rho u}{r}}{\alpha }_{51}+{\displaystyle \frac{Gm}{r}}{\alpha }_{52}+{\displaystyle \frac{uh}{r}}{\alpha }_{53}\\ \hfill +{\displaystyle \frac{h^2}{\rho r}}{\alpha }_{52}+{\alpha }_{54}({\displaystyle \frac{\gamma pu}{r}}+{\displaystyle \frac{(\gamma -1)j}{\rho r}})\\ \hfill -{\alpha }_{55}2\pi \rho r-{\alpha }_{56}Kj.\end{array}$$

Finally, the invariance of the Equation $(1)$ gives the following determining equations:

$$\begin{array}{c}\hfill {\alpha }_{61}=0,{\alpha }_{62}=0,{\alpha }_{63}=0,\\ \hfill {\alpha }_{64}=0,T_r=0,\\ \hfill J_u={\alpha }_{61}\rho +{\alpha }_{62}u+{\alpha }_{63}h+{\alpha }_{64}\gamma p,\\ \hfill J_p={\displaystyle \frac{1}{\rho }}{\alpha }_{62}+u{\alpha }_{64},J_{\rho }\\ \hfill ={\alpha }_{61}u,J_h={\displaystyle \frac{h}{\rho }}{\alpha }_{62}+u{\alpha }_{63},\\ \hfill J_j-R_r={\displaystyle \frac{(\gamma -1)}{\rho }}{\alpha }_{64}+{\alpha }_{66},\\ \hfill J_m={\alpha }_{65},\\ \hfill J_r-Kj({\displaystyle \frac{3}{2}}{\displaystyle \frac{P}{p}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{S}{\rho }}+{\displaystyle \frac{R}{r}}-2{\displaystyle \frac{T}{t}}+{\displaystyle \frac{J}{j}})\\ \hfill ={\alpha }_{61}{\displaystyle \frac{\rho u}{r}}+{\alpha }_{62}({\displaystyle \frac{h^2}{\rho r}}+{\displaystyle \frac{Gm}{r}})\\ \hfill +{\alpha }_{63}{\displaystyle \frac{uh}{r}}+{\alpha }_{64}({\displaystyle \frac{\gamma pu}{r}}\\ \hfill +{\displaystyle \frac{(\gamma -1)j}{\rho r}})-{\alpha }_{65}2\pi \rho r-{\alpha }_{66}Kj,\\ \hfill T=T(t),R=R(r,t).\end{array}$$

After solving the above systems of determining equations simultaneously, we get the following symmetry generators:

$$\begin{array}{c}\hfill R=({\alpha }_{22}+2a)r,T=at,\\ \hfill U=({\alpha }_{22}+a)u,P=(2{\alpha }_{22}+{\alpha }_{11}+3a)p,\\ \hfill H={\displaystyle \frac{1}{2}}(2{\alpha }_{22}+{\alpha }_{11}+3a)h,S=({\alpha }_{11}+a)\rho ,\\ \hfill W=(2{\alpha }_{22}+2a)m,J=(3{\alpha }_{22}+2{\alpha }_{11}+5a)j,\end{array}$$

(12)

where $a,b,{\alpha }_{11}$ and ${\alpha }_{22}$ are the arbitrary constants. Also, we have the relation ${\alpha }_{11}=-3a$ and $5{\alpha }_{22}={\alpha }_{11}$.

4. Construction of Similarity Solutions

Let us take the case of when $a\ne 0$ and ${\alpha }_{22}+2a\ne 0$.

The invariant surface conditions (see, Logan [14] and Bluman et al. [24] are as follows):

$$\begin{array}{c}\hfill R{u}_r+T{u}_t=U,R{p}_r+T{p}_t=P,\\ \hfill R{h}_r+T{h}_t=H,R{\rho }_r+T{\rho }_t=S,\\ \hfill R{m}_r+T{m}_t=W,R{j}_r+T{j}_t=J.\end{array}$$

(13)

On integrating the set of Equations (13) together with (12), we obtain the following forms of flow variable:

$$\begin{array}{c}\hfill u=t^{(\delta -1)}\widehat{U}(\xi ),p=t^{(2(\delta -1)+k)}\widehat{P}(\xi ),\\ \hfill h=t^{{\displaystyle \frac{1}{2}}(2(\delta -1)+k)}\widehat{H}(\xi ),\rho =t^k\widehat{S}(\xi ),\\ \hfill m=t^{(2\delta +k)}\widehat{W}(\xi ),j=t^{(3(\delta -1)+2k)}\widehat{J}(\xi ),\end{array}$$

(14)

where $\delta ={\displaystyle \frac{{\alpha }_{22}+2a}{a}}$, $k={\displaystyle \frac{{\alpha }_{11}+a}{a}}$. $\widehat{U},\widehat{P},\widehat{H},\widehat{S}$, $\widehat{W}$ and $\widehat{J}$ are the functions of similarity variable $\xi$, which are determined as follows:

$$\begin{array}{c}\hfill \xi ={\displaystyle \frac{r}{t^{\delta }}},\end{array}$$

(15)

and the similarity curve is defined as

$$\begin{array}{c}\hfill \overline{R}(t)=\xi t^{\delta }.\end{array}$$

(16)

Let the shock path $r=\overline{R}$ and the shock velocity V at the basic position of the shock $\xi =1$ be given as

$$\begin{array}{c}\hfill \overline{R}=t^{\delta },\end{array}$$

(17)

$$\begin{array}{c}\hfill V={\displaystyle \frac{\delta \overline{R}}{t}}=\delta t^{\delta -1}.\end{array}$$

(18)

The boundary conditions at the strong shock at which $\xi =1$ are as follows:

$$\begin{array}{c}\hfill {u|}_{\xi =1}=t^{(\delta -1)}\widehat{U}{(1),p|}_{\xi =1}=t^{(2(\delta -1)+k)}\widehat{P}(1),\\ \hfill {h|}_{\xi =1}=t^{{\displaystyle \frac{1}{2}}(2(\delta -1)+k)}\widehat{H}{(1),\rho |}_{\xi =1}=t^k\widehat{S}(1),\\ \hfill {m|}_{\xi =1}=t^{(2\delta +k)}\widehat{W}{(1),j|}_{\xi =1}=t^{(3(\delta -1)+2k)}\widehat{J}(1).\end{array}$$

(19)

Invariance of the jump condition in Equation (4) suggests that ${\rho }_0(r)$, $h_0(r)$, $m_0(r)$ and $j_0(r)$ must be of the following form:

$$\begin{array}{c}\hfill {\rho }_0(r)={\rho }_c{r}^{\theta },h_0=h_c{r}^{\eta },m_0(r)=m_c{r}^{\mu },j_0(r)=j_c{r}^{\varphi },\end{array}$$

(20)

where ${\rho }_c$, $h_c$, $m_c$ and $j_c$ are the arbitrary constants and

$$\begin{array}{c}\hfill \delta ={\displaystyle \frac{7}{5}},\theta =-{\displaystyle \frac{10}{7}},\eta =-{\displaystyle \frac{3}{7}},\mu ={\displaystyle \frac{4}{7}},\varphi =-2.\end{array}$$

(21)

On applying jump conditions (4) and using Equations (20) and (21), we obtained the following conditions on the functions $\widehat{U},\widehat{P},\widehat{H},\widehat{S}$, $\widehat{W}$ and $\widehat{J}$:

$$\begin{array}{c}\hfill \widehat{U}(1)={\displaystyle \frac{2\delta }{\gamma +1}},\widehat{P}(1)={\displaystyle \frac{2{\rho }_c{\delta }^2}{\gamma +1}},\\ \hfill \widehat{H}(1)={\displaystyle \frac{\gamma +1}{\gamma -1}}{h}_c,\widehat{S}(1)={\displaystyle \frac{\gamma +1}{\gamma -1}}{\rho }_c,\\ \hfill \widehat{W}(1)=m_c,\widehat{J}(1)=j_c,\end{array}$$

(22)

where $\delta$, ${\rho }_c$, $m_c$ and $j_c$ are the constants associated with the medium.

By using the Equations (17), (18) and (20), Equation (14) can be rewritten as:

$$\begin{array}{c}\hfill u=V{U}^{*}(\xi ),p={\rho }_0(\overline{R}(t))V^2{P}^{*}(\xi ),\\ \hfill h=h_0(\overline{R}(t))H^{*}(\xi ),\rho ={\rho }_0(\overline{R}(t))S^{*}(\xi ),\\ \hfill m=m_0(\overline{R}(t))W^{*}(\xi ),j=j_0{J}^{*}(\xi ),\end{array}$$

(23)

where $U^{*}(\xi )={\displaystyle \frac{\widehat{U}(\xi )}{\delta }}$, $P^{*}={\displaystyle \frac{\widehat{P}(\xi )}{{\rho }_c{\delta }^2}}$, $H^{*}={\displaystyle \frac{\widehat{H(\xi )}}{h_c}}$, $S^{*}={\displaystyle \frac{\widehat{S}(\xi )}{{\rho }_c}}$, $W^{*}(\xi )={\displaystyle \frac{\widehat{M}(\xi )}{m_c}}$, $J^{*}={\displaystyle \frac{\widehat{J}(\xi )}{j_c}}$ and $\delta \theta =-2$.

On substituting Equation (23) into the system (1) of governing equations and using Equations (15)–(18) and (20), we obtained the following system of ODEs in $U^{*},P^{*},H^{*},S^{*}$, $W^{*}$ and $J^{*}$ in the following form after dropping the asterisk sign:

$$\begin{array}{c}\hfill S\theta +(U-\xi )S\prime +SU\prime +{\displaystyle \frac{SU}{\xi }}=0,\\ \hfill ({\displaystyle \frac{\delta -1}{\delta }})SU+(U-\xi )U\prime S+P\prime +{\displaystyle \frac{h_c^{2}HH\prime }{{\rho }_c{\delta }^2}}+{\displaystyle \frac{h_c^2{H}^2}{{\rho }_c\xi {\delta }^2}}\\ \hfill +{\displaystyle \frac{WGS{m}_c}{\xi {\delta }^2}}=0,\\ \hfill \eta H+(U-\xi )H\prime +HU\prime +{\displaystyle \frac{HU}{\xi }}=0,\\ \hfill (2+\theta -{\displaystyle \frac{2}{\delta }})P+(U-\xi )P\prime \\ \hfill +\gamma P[U\prime +{\displaystyle \frac{U}{\xi }}]+{\displaystyle \frac{J_c(\gamma -1)}{S{\delta }^3}}[J\prime +{\displaystyle \frac{J}{\xi }}]=0,\\ \hfill W\prime -{\displaystyle \frac{2\pi S{\rho }_c\xi }{m_c}}=0,\\ \hfill J\prime -k_0{\rho }_c{\delta }^{3}J{P}^{3/2}{S}^{-1/2}\xi =0,\end{array}$$

(24)

where the prime have been used for the differentiation with respect to the similarity variable $\xi$. Also, we obtained the following conditions for the strong shock from the jump conditions (4),

$$\begin{array}{c}\hfill U(1)={\displaystyle \frac{2}{\gamma +1}},P(1)={\displaystyle \frac{2}{\gamma +1}},\\ \hfill H(1)={\displaystyle \frac{\gamma +1}{\gamma -1}},S(1)={\displaystyle \frac{\gamma +1}{\gamma -1}},\\ \hfill W(1)=1,J(1)=1.\end{array}$$

(25)

5. Results and Discussion

The values of the ambient density exponent $\theta$ and similarity exponent $\delta$ have been calculated from the relations given in Equation (21) as $\theta =-10/7$ and $\delta =7/5$. The set of differential equations (24) together with the boundary conditions (25) has been solved for $\gamma =4/3,7/5,3/2$ by using the well-known Runge–Kutta method of order four. The results are depicted in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6. We have depicted the variation of velocity in Figure 1, variation of pressure in Figure 2, variation of magnetic field in Figure 3, variation of density in Figure 4, variation of mass in Figure 5 and variation of radiation flux in Figure 6, with respect to the change in parameter $\gamma$. This helps us to understand the importance of gravitation on flow parameters.

From Figure 1, it is observed that the velocity behind the shock wave decreases monotonically for $\gamma$ = 4/3, 7/5, and 3/2. The results for $\gamma$ = 4/3 and 7/5 are in excellent agreement with those reported by Nath [2]. From Figure 2, Figure 3 and Figure 4, we see that pressure, magnetic field, and density decrease monotonically for $\gamma =4/3,7/5$, while for $\gamma =3/2$, all three, i.e., pressure, magnetic field, and density, first decrease monotonically and then begin to increase monotonically. From Figure 5 and Figure 6, we see that the mass and the radiation flux increase monotonically for $\gamma =7/5,3/2$, and decrease monotonically for $\gamma =4/3$ as we move towards the center of the implosion. The variation of flow parameters velocity, pressure, magnetic field, density, mass, and radiation flux have been shown in the figures given below.

6. Conclusions

In this study, we applied the similarity transformation method to investigate the propagation of cylindrical MHD shock waves in a non-uniform atmosphere under the combined influence of gravitation and monochromatic radiation. Employing the symmetry transformation method to the system of PDEs (1), we derive the corresponding symmetry generators, expressed in terms of arbitrary constants. Based on these constants, we categorized the possible self-similar solutions of the problem, which follow power law behavior. Using surface invariance conditions, the flow variables were obtained in terms of the symmetry generators as functions of the similarity variable. These flow variables allowed us to reduce the PDEs system into an ODEs system subject to appropriate boundary conditions. Furthermore, we established a relation between the similarity exponent $\delta$ and the ambient density exponent $\theta$. Finally, the system of ODEs is integrated using a fourth-order Runge–Kutta method with a step size $\Delta \xi ={10}^{-3}$; convergence and accuracy were verified by step-size refinement and by ensuring consistency with the shock boundary conditions $\gamma =4/3,7/5,3/2$, and the results have been presented graphically.