Similarity solution via one-dimensional Lie subalgebras for shock waves in a self-gravitating and rotating gas with magnetic field and radiation heat flux

Nath, G.; Upadhyay, P.

doi:10.1063/5.0232628

The dynamical effects of a cylindrically symmetric moving shock wave in a self-gravitating and rotating ideal gas medium influenced by a magnetic field and radiation heat flux are investigated. The dynamics of the shock wave are governed by the one-dimensional motion of the gas, and the energy equation contains the effect of thermal radiation in the setting of an optically thick limit. A mathematical model is established using the Lie symmetry method, and all possible cases of similarity solutions are obtained by choosing different subalgebras from the optimal system. The numerical computations are performed using a fourth-order Runge–Kutta method. For a power law shock path, the contour plots of flow variables such as density and velocity are presented for convenient visualization. The variations of the shock strength and the flow variables in the physical flow field region behind the shock front with changes in the values of the gravitational parameter, rotational parameter, adiabatic index, ambient magnetic field strength, similarity exponent, and radiative heat transfer parameter are investigated. It is found that increases in the ambient magnetic field strength and the radiative heat transfer parameter lead to decay of the shock wave. For a power law shock path, it is found that increases in the gravitational parameter and adiabatic index cause decay of the shock wave, whereas for an exponential law shock path, the shock strength is increased by increases in the values of these parameters.

I. INTRODUCTION

Experimental investigations and observations in astrophysics reveal that the outer atmospheres of planets, stars, and galaxies undergo rotational motion due to the spin of these objects. This rotational motion induces macroscopic motion at supersonic velocities within the atmospheres, leading to the generation of shock waves. Consequently, the rotation of planets and stars exerts a significant influence on the processes occurring in their outer layers, and thus explosions within rotating gaseous atmospheres are of considerable astrophysical interest.^1–4 Alzahrani and Khan⁵ discussed entropy generation, Joule heating, and homogenous and heterogeneous chemical reactions in Ree–Eyring fluid flow between double rotating disks. Chaturani⁶ examined the propagation of cylindrical shock waves through a gas experiencing solid body rotation using the similarity method of Sedov.⁷ Levin and Skopina⁸ obtained a condition for the existence of Chapman–Jouguet waves and studied shock and detonation waves in rotating gas flows. Nath⁹ discussed the similarity solution for a strong shock moving with an exponential-law shock path in rotating dusty gas flow.

During the last few decades, there has been considerable research interest in electrically conducting fluid flows, such as those involved in the motion of electrically conducting fluids, supersonic motion of interstellar gas, supernova explosions, the Earth’s magnetic field, and flow through magnetized nozzles. Problems related to the existence and properties of magnetic fields are extremely important in the study of geological and astrophysical events.^10–18 Raza et al.¹⁹ studied the impact of magnetic fields on entropy generation in magnetohydrodynamic (MHD) nanofluids using a neural network approach. Mishra et al.²⁰ studied the effects of magnetic field in dusty nanofluids containing nanoparticles such as copper and silver as the conducting medium. In studies of high-temperature and low-pressure phenomena, the gaseous medium is often assumed to follow the ideal gas law, which reduces the complexity of the physical model. We shall adopt this assumption in the present study. There have been many investigations of implosion shock propagation in a perfect gas with or without a magnetic field.^21–25 The acceleration of plane shock waves described by solid body motion in a perfect gas in the presence of a magnetic field were discussed by Golubyatnikov and Kovalevskaya.²⁶ Nath²⁷ studied the behavior of a shock wave in rotating perfect gas flow in the presence of an azimuthal magnetic field and obtained a similarity solution.

Thermal radiation is a mode of transfer of heat energy as electromagnetic waves. The effect of radiation becomes significant for high-temperature phenomena in medium at low density, such those encountered in blast waves due to powerful explosions, large electrical discharges in rotating medium, the solar corona and solar wind, and supernova explosions.²⁸ The interaction of radiation heat flux and gasdynamics in the absence or presence of magnetic fields was analyzed by Pai,²⁹ and the behavior of shock waves in the presence of radiation was studied by Marshak.³⁰ Raza et al.³¹ discussed the thermal properties of graphene oxide (GO) and molybdenum disulphide (MoS₂) nanoparticles with the help of fractional derivatives. There have been a number of studies of thermal radiative phenomenon with the aim of improving the heat transfer behavior. Sulochana et al.³² investigated the impact of multiwalled carbon nanotubes (MWCNTs) and hybrid biodiesel on a compression ignition engine with regard to emission, combustion, and performance. Khan et al.³³ analyzed the thermal mechanisms involved in mixed convective flow of a Reiner–Philippoff nanofluid, with an assessment of entropy generation. Thermal performance is affected by thermal radiation, Ohmic dissipation, and nonuniform heat sources and sinks. Nazir et al.³⁴ examined the convective flow of a nanofluid in a square container under the influence of thermal boundary conditions. Nicastro³⁵ studied the gasdynamic equations in the presence of radiation under the assumption that the gas was optically thick and obtained a similarity solution. When the mean free path of radiation in a gas is small, the gas is said to be optically thick, and conversely, when the mean free path is long, the gas is optically thin. For an optically thick gas, the expression for the radiation heat flux takes a very simple form. The mean free path of radiation represents the mean distance over which a photon travels before it is absorbed by the molecules of the gas. Thus, it is similar to the mean free path considered in the kinetic theory of gases, which represents the mean distance between collisions. The mean free path of radiation is defined as N_R = 1/κρ, where κ is the Rosseland mean absorption coefficient and ρ is the density of the medium. Therefore, if a gas is optically thick, then the value of the absorption coefficient is high, and the effects of radiation in the medium are significant.²⁹ Summers³⁶ and Hutchens³⁷ discussed blast waves generated owing to the release of huge amounts of energy in conducting and nonconducting media, respectively. Chu et al.³⁸ discussed the effects of magnetic fields and radiation on steady MHD flow of Maxwell fluid generated by the stretching of two infinite disks. Vishwakarma et al.³⁹ and Vishwakarma and Patel⁴⁰ discussed shock wave propagation in a nonrotating and a rotating nonideal gas in the presence of a magnetic field and radiation heat flux. Vishwakarma and Nath⁴¹ and Nath⁴² studied the behavior of shock waves in the presence of radiation and conduction heat fluxes in rotating dusty gas. The propagation of MHD waves in the presence of a magnetic field was discussed by Rosenau and Frankenthal,^43,44 with and without consideration of the thermal conductivity of the medium. Cioffi et al.⁴⁵ studied supernova remnants in the interstellar medium produced by blast waves due to supernova explosion. Nath et al.⁴⁶ investigated the effects of magnetic field, radiation, and gravitation on shock wave propagation, using the similarity method of Sedov.⁷ Nath and Vishwakarma⁴⁷ discussed shock wave propagation in a nonideal gas in the presence of an azimuthal magnetic field, radiation, and heat conduction, without taking into account the effects of a rotating medium. Nath⁴⁸ extended the problem studied by Nath and Vishwakarma⁴⁷ by considering these rotational effects.

The effects of a self-gravitating gaseous medium are important for the formation of stars and galaxies and for sudden expansion of instantaneous line explosions. Gravitational effects on shock wave propagation have been the subjects of a number of studies.^49–52 Larson⁵³ numerically solved the hydrodynamic equations governing the evolution of a protostar during its collapse under self-gravitation. Rogers⁵⁴ obtained analytical solutions for blast wave propagation in an atmosphere with varying density ahead of the wavefront. Sakurai⁵⁵ extended the work of Kopal⁵⁰ by considering variable shock strength. Shock wave propagation in a self-gravitating gas with or without magnetic field effects has been studied using a variety of techniques.^56–58 Similarity solutions for the propagation of cylindrical shock waves in gravitating or nongravitating perfect gas with radiation heat flux were obtained by Singh⁵⁹ and Singh and Vishwakarma,⁶⁰ and Vishwakarma and Singh⁶¹ extended this work by considerating radiation as well as heat conduction fluxes. An exact solution was obtained by Nath et al.⁶² for a shock wave in a self-gravitating perfect gas with radiation flux and magnetic field, but without considering the effects of a rotating medium.

In most of the above studies, the method of dimensional analysis developed by Sedov⁷ was used to discuss the propagation of shock waves. An alternative approach is represented by the use of Lie symmetries. The Lie symmetry method was originally developed by Sophus Lie in the 19th century.⁶³ It is a powerful method for solving nonlinear partial differential equations (PDEs) and for studying nonlinear mathematical models. This method is based on the invariance properties of differential equations under an infinitesimal group of transformations. Using invariance conditions on differential equations, the infinitesimal generators of the transformation group containing arbitrary constants are obtained. Different choices of arbitrary constants in the infinitesimal generators provide all possible solutions of the system of basic equations of motion.^63,64 A brief discussion of the Lie theoretic method and its applications in different areas can be found in Refs. 63 and 65–68. Similarity solutions for the propagation of shock wave in gases were obtained by Nath and Devi⁶⁹ and Singh⁷⁰ using a Lie group transformation method. Using the Lie symmetry method, an optimal system was discussed by Ovsiannikov,⁷¹ who constructed a global matrix for the adjoint transformation. The concept of an optimal system is very useful for minimizing the search required to obtain all different solutions of a system of PDEs that leave the system invariant. Recently, Nath and Maurya⁷² and Nath and Kadam⁷³ used the Lie symmetry method to obtain the optimal system for the propagation of shock waves in a self-gravitating nonideal or ideal gas in the absence of presence of monochromatic radiation in a rotating medium. Hu et al.⁷⁴ proposed a promising new approach for finding the optimal system of Lie subalgebras for a one-dimensional general system of ordinary differential equations (ODEs). An optimal system of one-, two-, and three-dimensional Lie subalgebras for two-phase mass flow model was discussed by GhoshHajra et al.,⁷⁵ and a similar approach for three-dimensional gasdynamical equations was presented by Mandal et al.⁷⁶ Using a Lie symmetry method, Nadjafikhah and Zaeim⁷⁷ studied the classification of subalgebras in a hyperbolic space for wave equations.

In the problem considered in the present paper, the Lie symmetry method is used to derive all possible similarity solutions of a mathematical model that describes shock wave propagation in a self-gravitating perfect gas under the influence of a magnetic field, the rotation of the medium, and a radiation heat flux. We determine the inequivalent classes of subalgebras (optimal classes), of which there are 14 in the problem under consideration. Then, from these 14 inequivalent optimal classes of subalgebras, we obtain the optimal system. For each optimal class, group-invariant solutions of the problem exist, but our aim is to obtain the similarity solution, and it turns out that of the 14 cases, only three allow the existence of similarity solutions.

The present study generalizes the work of Singh⁵⁹ by taking into consideration the rotational effect of the medium and the presence of a magnetic field as well as a gravitational force. It also represents a generalization of the work of Singh and Vishwakarma,⁶⁰ again by taking into account the effect of a magnetic field and the rotational effect of the medium. In Refs. 59 and 60, the similarity solution for the power-law shock path was obtained by using the dimensional analysis method develop by Sedov,⁷ but we obtain similarity solutions in three cases, two with a power-law shock path and one with an exponential-law shock path, using a Lie symmetry method.^63,71 Our mathematical model may provide an improved basis for understanding a variety of astrophysical and aerodynamic phenomena, such as supernova explosions, the central regions of starburst galaxies, nuclear explosions, and exploding wire experiments on the pinch effect in rotating media.^1,2,29

To obtain similarity solutions of the problem under consideration, we assume that the azimuthal and axial fluid velocities and the density and magnetic field in the ambient medium all vary with the spatial coordinate. We also assume that the fluid is inviscid and perfectly conducting. In addition, we take the radial fluid velocity in the ambient medium to be zero.

Under these assumptions, numerical solution of the problem indicates that when the value of the gravitational parameter is increased, the shock wave strength decreases in the case of a power-law shock path and increases in the case of an exponential-law shock path. It is also found that the impact of the rotational parameter on the shock strength is the reverse of that of the gravitational parameter. However, the impacts of the magnetic field strength and radiation parameter on shock strength are similar for both power-law and exponential-law shock paths. In addition, the shock strength varies with a change in the magnetic field from axial to azimuthal.

II. GOVERNING EQUATIONS AND SHOCK JUMP CONDITIONS

The flow variables are denoted as follows: density by ρ, pressure by p, magnetic field by h, radial, azimuthal, and axial components of fluid velocity

\vec{Z}

by u, v, and w, respectively, mass of fluid within a cylinder of unit length by m, and radiation heat flux by J. We assume that the gas is a perfect gas, and therefore the equation of state and internal energy per unit mass are expressed as^6,10,56

p = ρ R^{*} T, I = p / (γ - 1) ρ,

(1)

where I, γ, R*, and T denote the internal energy per unit mass, adiabatic index, gas constant, and absolute temperature, respectively. We also assume that the gas is inviscid and with infinite electrical conductivity. We assume that the gaseous motion is one-dimensional unsteady adiabatic motion of a self-gravitating perfect gas and that a diverging cylindrical shock wave propagates symmetrically about the line of symmetry in the rotating medium under the influence of the radiation heat flux and the magnetic field. The mathematical model for such a type of problem is governed by the conservation laws of mass, momentum, energy, charge, etc. In an Eulerian formulation, the governing equations are expressed as^{30,40,42,47,60–62,70}

\frac{d ρ}{d t} + ρ \frac{\partial u}{\partial r} + \frac{ρ u}{r} = 0,

(2)

\frac{d u}{d t} + \frac{1}{ρ} (\frac{\partial p}{\partial r} + μ h \frac{\partial h}{\partial r} + \frac{χ μ h^{2}}{r}) - \frac{v^{2}}{r} + \frac{G m}{r} = 0,

(3)

\frac{d v}{d t} + \frac{v u}{r} = 0,

(4)

\frac{d w}{d t} = 0,

(5)

\frac{d I}{d t} - \frac{p}{ρ^{2}} (\frac{d ρ}{d t}) + \frac{1}{ρ r} \frac{\partial}{\partial r} (r J) = 0,

(6)

\frac{d h}{d t} + h \frac{\partial u}{\partial r} + (1 - χ) \frac{h u}{r} = 0,

(7)

\frac{\partial m}{\partial r} = 2 π ρ r,

(8)

where d/dt = ∂/∂t + u∂/∂r, r and t are space and time coordinates, G and μ denote the constant of gravitation and magnetic permeability, respectively, and χ = 0 and 1 indicate an axial and azimuthal magnetic field, respectively.

Assuming local thermodynamic equilibrium and using the radiative diffusion model for an optically thick gray gas, the radiative heat flux J may be obtained from the differential approximation of the radiation transport equation in the diffusion limit as^{28,47,48,59,60}

J = - \frac{16}{3} (\frac{ϰ}{κ}) T^{3} \frac{\partial T}{\partial r},

(9)

where ϰ is the Stefan–Boltzmann constant, and κ is the Rosseland mean absorption coefficient, the value of which is high for an optically thick gray gas and therefore the effect of radiation on the medium is significant.²⁹

The absorption coefficient κ is assumed to vary with temperature and density:^48,60

κ = κ_{o} ρ^{τ_{1}} T^{τ_{2}},

(10)

where κ_o is a dimensional constant, and τ₁ and τ₂ are nondimensional constants. In Eq. , the exponents τ₁, τ₂ and the constant κ_o are to be determined from the gas-property data within the appropriate temperature range; if a self-similar solution is sought, they must also satisfy the similarity requirements.

The relation between the azimuthal fluid velocity v and the angular velocity of the medium Ω is expressed as

v = r Ω .

(11)

The vorticity vector is defined as^8,42

\vec{L} = \frac{1}{2} curl \vec{Z},

(12)

and its components are given by

L_{r} = 0, L_{θ} = - \frac{1}{2} \frac{\partial w}{\partial r}, L_{z} = \frac{1}{2 r} \frac{\partial (r v)}{\partial r} .

(13)

We assume that r = R(t) is the locus of the outwardly moving shock front and U_s = dR/dt is its velocity. The pre-assumptions for the flow variables immediately ahead of the shock front are taken as

\begin{gathered} u_{a} = 0, v_{a} = v_{a} (r), w_{a} = w_{a} (r), ρ_{a} = ρ_{a} (r), \\ h_{a} = h_{a} (r), m_{a} = m_{a} (r), p_{a} = p_{a} (r), \\ r = R (t), J_{a} = 0 (see Ref. 78), \end{gathered}

(14)

where a subscript “a” indicates the value of a flow variable ahead of the shock front.

We consider that the shock is isothermal (according to the approximation adopted here, a proportionality between the heat flux and the temperature gradient leads to the creation of an isothermal shock, i.e., the jump in temperature is negligible²⁸). Thus, for a magnetogasdynamic isothermal shock, the Rankine–Hugoniot jump conditions are expressed as^27,29,46,61

\begin{gathered} ρ_{a} U_{s} = (U_{s} - u_{b}) ρ_{b}, h_{a} U_{s} = (U_{s} - u_{b}) h_{b}, \\ p_{a} + \frac{μ h_{a}^{2}}{2} + ρ_{a} U_{s}^{2} = p_{b} + \frac{μ h_{b}^{2}}{2} + ρ_{b} {(U_{s} - u_{b})}^{2}, \\ I_{a} + \frac{p_{a}}{ρ_{a}} + \frac{μ h_{a}^{2}}{ρ_{a}} + \frac{1}{2} U_{s}^{2} = I_{b} + \frac{p_{b}}{ρ_{b}} + \frac{μ h_{b}^{2}}{ρ_{b}} + \frac{1}{2} {(U_{s} - u_{b})}^{2} - \frac{J_{b}}{U_{s} ρ_{a}}, \\ v_{a} = v_{b}, w_{a} = w_{b}, m_{a} = m_{b}, T_{a} = T_{b}, \end{gathered}

(15)

where a subscript “b” indicates the value of a flow variable immediately behind the shock wave front.

Using Eqs. and , we obtain the shock jump conditions across the shock front as⁷²

\begin{gathered} u_{b} = (1 - β) U_{s}, h_{b} = \frac{h_{a}}{β}, \\ T_{b} = T_{a}, m_{b} = m_{a}, v_{b} = v_{a}, w_{b} = w_{a}, \\ p_{b} = L_{*} ρ_{a} U_{s}^{2}, J_{b} = F_{*} ρ_{a} U_{s}^{3}, \end{gathered}

(16)

where

\begin{gathered} L_{*} = \frac{1}{γ {(S_{M})}^{2}} + (1 - β) + \frac{A_{M}^{- 2}}{2} (1 - \frac{1}{β^{2}}), \\ F_{*} = [A_{M}^{- 2} (\frac{1}{β} - 1) + \frac{1}{2} (β^{2} - 1)], \end{gathered}

A_{M} = {[ρ_{a} U_{s}^{2} / μ {(h_{a})}^{2}]}^{1 / 2}

is the Alfvén Mach number,

S_{M} = {(ρ_{a} U_{s}^{2} / γ p_{a})}^{1 / 2}

is the shock Mach number referred to the frozen speed of sound

{(γ p_{a} / ρ_{a})}^{1 / 2}

⁠, and ρ_a/ρ_b = β is the density ratio across the shock front, the value of which lies within the range from 0 to 1 for the existence of a shock wave.

The value of β is obtained from the quadratic equation

β^{2} - β [\frac{1}{γ {(S_{M})}^{2}} + \frac{A_{M}^{- 2}}{2}] - \frac{A_{M}^{- 2}}{2} = 0 .

(17)

III. LIE SYMMETRY ANALYSIS

To discuss the similarity solutions of the present problem, the Lie symmetry method is used. To apply this method, the one-parameter group of infinitesimal transformations corresponding to the flow variables involved in the problem is taken into account. These are obtained through a Taylor’s series expansion of the one-parameter group of global transformations about the identity transformation in terms of a small quantity ϵ. We assume that the basic governing Eqs. – and the shock jump conditions are invariant under the infinitesimal transformations. Thus, the one-parameter group of infinitesimal transformations is taken as consisting of ^{65–67,72,73}

\begin{gathered} \hat{t} = t + ϵ η^{1} + o (ϵ^{2}), \hat{r} = r + ϵ η^{2} + o (ϵ^{2}), \\ \hat{ρ} = ρ + ϵ ξ^{1} + o (ϵ^{2}), \hat{u} = u + ϵ ξ^{2} + o (ϵ^{2}), \\ \hat{v} = v + ϵ ξ^{3} + o (ϵ^{2}), \hat{w} = w + ϵ ξ^{4} + o (ϵ^{2}), \\ \hat{p} = p + ϵ ξ^{5} + o (ϵ^{2}), \hat{h} = h + ϵ ξ^{6} + o (ϵ^{2}), \\ \hat{m} = m + ϵ ξ^{7} + o (ϵ^{2}), \hat{J} = J + ϵ ξ^{8} + o (ϵ^{2}), \end{gathered}

(18)

where ξⁱ and η^j, which are functions of (r, t, ρ, u, v, w, p, h, m, J), are known as the infinitesimal generators of the transformations , where i = 1, 2, 3, 4, 5, 6, 7, 8; j = 1, 2. In Eq. , ϵ is a small quantity, and so we can neglect second- and higher-order terms in ϵ.

We now introduce new symbols for the variables and their first-order partial derivatives as follows: t = x₁, r = x₂, ρ = X₁, u = X₂, v = X₃, w = X₄, p = X₅, h = X₆, m = X₇, J = X₈, and $Y_{j}^{i} = \partial X_{i} / \partial x_{j}$ ⁠; i = 1, …, 8 and j = 1, 2.

The fundamental Eqs. – can be written as

F_{l} (x_{j}, X_{i}, Y_{j}^{i}) = 0 .

(19)

This system of equations is constantly and conformally invariant with respect to the transformations if

L F_{l} = σ_{l k} F_{k},

(20)

where k, l = 1, …, 8, and the σ_kl are real numbers.

L

is called the Lie derivative and it defined as

L = η^{j} \frac{\partial}{\partial x_{j}} + ξ^{i} \frac{\partial}{\partial X_{i}} + η_{Y_{j}}^{i} \frac{\partial}{\partial Y_{j}^{i}},

(21)

with

η_{Y_{j}}^{i} = \frac{\partial ξ^{i}}{\partial x_{j}} + \frac{\partial ξ^{i}}{\partial X_{s}} Y_{j}^{s} - \frac{\partial η^{q}}{\partial x_{j}} Y_{q}^{i} - \frac{\partial η^{q}}{\partial X_{e}} Y_{q}^{i} Y_{j}^{e},

(22)

where q = 1, 2 and e, s = 1, 2, 3, …, 8.

From Eqs. –, we obtain

\begin{align} η^{j} \frac{\partial F_{l}}{\partial x_{j}} + ξ^{i} \frac{\partial F_{l}}{\partial X_{i}} + (\frac{\partial ξ^{i}}{\partial x_{j}} + \frac{\partial ξ^{i}}{\partial X_{s}} Y_{j}^{s} - \frac{\partial η^{q}}{\partial x_{j}} Y_{q}^{i} \\ - \frac{\partial η^{q}}{\partial X_{e}} Y_{q}^{i} Y_{j}^{e}) \frac{\partial F_{l}}{\partial Y_{j}^{i}} = σ_{l k} F_{k} . \end{align}

(23)

Substituting the governing Eqs. – one by one into Eq. and setting the coefficients of

Y_{q}^{i}

⁠,

Y_{q}^{i} Y_{j}^{e}

⁠, and the terms free from derivatives of dependent variables equal to zero gives a overdetermined system of equations in the infinitesimal generators called the system of determining equations. From these determining equations, the infinitesimal generators are obtained.

On solving the set of determining equations thereby obtained, the infinitesimal generators are obtained in the form

\begin{gathered} η^{1} = a t + c, η^{2} = (σ_{22} + 2 a) r, ξ^{1} = - 2 a ρ, \\ ξ^{2} = (σ_{22} + a) u, ξ^{3} = (σ_{22} + a) v, \\ ξ^{4} = (σ_{44} + a) w + k_{1}, ξ^{5} = 2 σ_{22} p, ξ^{6} = σ_{22} h, \\ ξ^{7} = 2 (σ_{22} + a) m, ξ^{8} = (3 σ_{22} + a) J + g (r, t), \end{gathered}

(24)

where c, σ₂₂, σ₄₄, k₁, and a are real constants, and g(r, t) is taken as an unknown function of r and t as an integration constant for ξ⁸.

IV. LIE BRACKETS AND OPTIMAL SYSTEM

The infinitesimal generators contain five constants, and therefore they generate a five-dimensional vector space (Lie algebra), which we denote by Π⁵. This is generated by five distinct Lie vectors, given by

\begin{gathered} V_{1} = \frac{\partial}{\partial t}, \\ V_{2} = r \frac{\partial}{\partial r} + u \frac{\partial}{\partial u} + v \frac{\partial}{\partial v} + 2 p \frac{\partial}{\partial p} + h \frac{\partial}{\partial h} + 2 m \frac{\partial}{\partial m} + 3 J \frac{\partial}{\partial J}, \\ V_{3} = w \frac{\partial}{\partial w}, \\ V_{4} = \frac{\partial}{\partial w}, \\ V_{5} = t \frac{\partial}{\partial t} + 2 r \frac{\partial}{\partial r} - 2 ρ \frac{\partial}{\partial ρ} + u \frac{\partial}{\partial u} + v \frac{\partial}{\partial v} + w \frac{\partial}{\partial w} + 2 m \frac{\partial}{\partial m} + J \frac{\partial}{\partial J} . \end{gathered}

(25)

To obtain the optimal system for the solution of the fundamental Eqs. –, the method described by Wambura et al.⁶⁸ and Nadjafikhah and Zaeim⁷⁷ is used. We express the commutation relations of the Lie vectors

V_{α}

⁠, where α = 1, …, 5 using Lie brackets as

[V_{α}, V_{ξ}] = V_{α} V_{ξ} - V_{ξ} V_{α}, ξ = 1, \dots, 5 .

(26)

Further, the Lie series is defined by

A d \exp (ϵ_{α} V_{α}) V_{ξ} = R = V_{ξ} - ϵ_{α} [V_{α}, V_{ξ}] + \frac{ϵ_{α}^{2}}{2} [V_{α}, [V_{α}, V_{ξ}]] \dots,

(27)

where ϵ_α (α = 1, …, 5) are small quantities.

With the help of Eqs. (26) and (27), we obtain the commutator table and adjoint representation of the vector field (25) as shown in Tables I and II, respectively.

TABLE I.

Commutator table.

$[V_{α}, V_{ξ}]$	$V_{1}$	$V_{3}$	$V_{4}$	$V_{5}$
$V_{1}$	0	0	0	$V_{1}$
$V_{2}$	0	0	0	0
$V_{3}$	0	0	$- V_{4}$	0
$V_{4}$	0	$V_{4}$	0	$V_{4}$
$V_{5}$	$- V_{1}$	0	$- V_{4}$	0

$[V_{α}, V_{ξ}]$	$V_{1}$	$V_{3}$	$V_{4}$	$V_{5}$
$V_{1}$	0	0	0	$V_{1}$
$V_{2}$	0	0	0	0
$V_{3}$	0	0	$- V_{4}$	0
$V_{4}$	0	$V_{4}$	0	$V_{4}$
$V_{5}$	$- V_{1}$	0	$- V_{4}$	0

TABLE II.

Adjoint table.

$R$	$V_{1}$	$V_{2}$	$V_{3}$	$V_{4}$	$V_{5}$
$V_{1}$	$V_{1}$	$V_{2}$	$V_{3}$	$V_{4}$	$V_{5} - ϵ_{1} V_{1}$
$V_{2}$	$V_{1}$	$V_{2}$	$V_{3}$	$V_{4}$	$V_{5}$
$V_{3}$	$V_{1}$	$V_{2}$	$V_{3}$	$V_{4} e^{ϵ_{3}}$	$V_{5}$
$V_{4}$	$V_{1}$	$V_{2}$	$V_{3} - ϵ_{4} V_{4}$	$V_{4}$	$V_{5} - ϵ_{4} V_{4}$
$V_{5}$	$V_{1} e^{ϵ_{1}}$	$V_{2}$	$V_{3}$	$V_{4} e^{ϵ_{5}}$	$V_{5}$

A. Formation of one-dimensional subalgebra for optimal solution

The general adjoint matrix D is derived in the Appendix. With the help of this general adjoint matrix, the optimal system is obtained as^68,72

\frac{1}{ω} D {[\begin{matrix} z_{1} & z_{2} & z_{3} & z_{4} & z_{5} \end{matrix}]}^{Tr} = {[\begin{matrix} y_{1} & y_{2} & y_{3} & y_{4} & y_{5} \end{matrix}]}^{Tr},

or

\frac{1}{ω} (\begin{matrix} z_{1} e^{ϵ_{1}} \\ z_{2} \\ z_{3} - z_{4} ϵ_{4} e^{ϵ_{5}} \\ z_{4} e^{ϵ_{3} + ϵ_{5}} \\ - z_{1} ϵ_{1} e^{ϵ_{1}} - z_{4} ϵ_{4} e^{ϵ_{5}} + z_{5} \end{matrix}) = (\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \\ y_{5} \end{matrix}),

(28)

where ω ≠ 0,

y = {[\begin{matrix} y_{1} & y_{2} & y_{3} & y_{4} & y_{5} \end{matrix}]}^{Tr} \to {[\begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \end{matrix}]}^{Tr}

⁠, and superscript “Tr” indicates the transpose of a matrix. By solving Eq. for different choices of z₁, z₂, z₃, z₄ and z₅, we obtain all the possible subalgebras of Π⁵, as shown in Table III. It is important to note that some subalgebras are equal under translational transformation: for example, Φ₃ = Φ₁₃; Φ₅ = Φ₉ = Φ₁₄ = Φ₁₅ = Φ₂₀ = Φ₂₁ = Φ₂₅ = Φ₂₇; Φ₇ = Φ₁₉; Φ₁₀ = Φ₂₂; Φ₁₂ = Φ₁₈ = Φ₂₄ = Φ₂₈; Φ₁₆ = Φ₃₀; Φ₂₃ = Φ₂₆ = Φ₂₉ = Φ₃₁. This can easily be proved by applying a translational transformation to η₁ or ξ⁴.

TABLE III.

Construction of subalgebras of Lie algebra Π⁵.

S. no.	Case	Basis element (y)	Sub-algebra Φ_d
1	z₁ ≠ 0, z₂ = z₃ = z₄ = z₅ = 0. Let ϵ₁ = 0 and z₁/ω = c₁	(c₁, 0, 0, 0, 0)	$⟨ V_{1} ⟩$
2	z₁ = z₃ = z₄ = 0 = z₅, z₂ = ωc₂ ≠ 0	(0, c₂, 0, 0, 0)	$⟨ V_{2} ⟩$
3	z₁ = z₂ = z₄ = 0 = z₅, z₃ = ωc₃ ≠ 0	(0, 0, c₃, 0, 0)	$⟨ V_{3} ⟩$
4	z₁ = z₂ = z₃ = z₅ = 0, z₄ ≠ 0. Let ϵ₃ + ϵ₅ = log(ωc₄/z₄), ϵ₄ = 0	(0, 0, 0, c₄, 0)	$⟨ V_{4} ⟩$
5	z₁ = z₂ = z₃ = z₄ = 0, z₅ = ωc₅ ≠ 0	(0, 0, 0, 0, c₅)	$⟨ V_{5} ⟩$
6	z₃ = z₄ = z₅ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁, z₂/ω = c₂	(c₁, c₂, 0, 0, 0)	$⟨ c_{1} V_{1} + c_{2} V_{2} ⟩$
7	z₂ = z₄ = z₅ = 0, z₁ ≠ 0, z₃ = ωc₃ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁, z₃/ω = c₃	(c₁, 0, c₃, 0, 0)	$⟨ c_{1} V_{1} + c_{3} V_{3} ⟩$
8	z₂ = z₃ = z₅ = 0, z₁ = ωc₁ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(c₁, 0, 0, c₄, 0)	$⟨ c_{1} V_{1} + c_{4} V_{4} ⟩$
9	z₂ = z₃ = z₄ = 0, z₅ = ωc₅ ≠ 0, z₁ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁	(c₁, 0, 0, 0, c₅)	$⟨ c_{1} V_{1} + c_{5} V_{5} ⟩$
10	z₁ = z₄ = z₅ = 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0	(0, c₂, c₃, 0, 0)	$⟨ c_{2} V_{2} + c_{3} V_{3} ⟩$
11	z₁ = z₃ = z₅ = 0, z₂ = ωc₂ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, c₂, 0, c₄, 0)	$⟨ c_{2} V_{2} + c_{4} V_{4} ⟩$
12	z₁ = z₃ = z₄ = 0, z₂ = ωc₂ ≠ 0, z₅ = ωc₅ ≠ 0	(0, c₂, 0, 0, c₅)	$⟨ c_{2} V_{2} + c_{5} V_{5} ⟩$
13	z₁ = z₂ = z₅ = 0, z₃ = ωc₃ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, 0, c₃, c₄, 0)	$⟨ c_{3} V_{3} + c_{4} V_{4} ⟩$
14	z₁ = z₂ = z₄ = 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₃ ≠ 0	(0, 0, c₃, 0, c₅)	$⟨ c_{3} V_{3} + c_{5} V_{5} ⟩$
15	z₁ = 0, z₂ = 0, z₄ ≠ 0, z₃ = 0, z₅ = ωc₅ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, 0, 0, c₄, c₅)	$⟨ c_{4} V_{4} + c_{5} V_{5} ⟩$
16	z₄ = z₅ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁	(c₁, c₂, c₃, 0, 0)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} ⟩$
17	z₃ = z₅ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, c₂, 0, c₄, 0)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{4} V_{4} ⟩$
18	z₃ = z₄ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₅ = ωc₅ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁	(c₁, c₂, 0, 0, c₅)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{5} V_{5} ⟩$
19	z₂ = z₄ = 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0, z₁ ≠ 0. Let ϵ₁ = 0 = ϵ₄ = ϵ₅, ϵ₃ = log(ωc₄/z₄)	(c₁, 0, c₃, c₄, 0)	$⟨ c_{1} V_{1} + c_{3} V_{3} + c_{4} V_{4} ⟩$
20	z₂ = z₄ = 0, z₁ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0. Let ϵ₁ = 0	(c₁, 0, c₃, 0, c₅)	$⟨ c_{1} V_{1} + c_{3} V_{3} + c_{5} V_{5} ⟩$
21	z₂ = z₃ = 0, z₁ = ωc₁ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, 0, 0, c₄, c₅)	$⟨ c_{1} V_{1} + c_{4} V_{4} + c_{5} V_{5} ⟩$
22	z₁ = z₅ = 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, c₂, c₃, c₄, 0)	$⟨ c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} ⟩$
23	z₁ = z₄ = 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0.	(0, c₂, c₃, 0, c₅)	$⟨ c_{2} V_{2} + c_{3} V_{3} + c_{5} V_{5} ⟩$
24	z₁ = z₃ = 0, z₂ = ωc₂ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, c₂, 0, c₄, c₅)	$⟨ c_{2} V_{2} + c_{4} V_{4} + c_{5} V_{5} ⟩$
25	z₅ = ωc₅ ≠ 0, z₁ = 0 = z₂, z₃ = ωc₃ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, 0, c₃, c₄, c₅)	$⟨ c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩$
26	z₁ = 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, c₂, c₃, c₄, c₅)	$⟨ c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩$
27	z₂ = 0, z₁ = ωc₁ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, 0, c₃, c₄, c₅)	$⟨ c_{1} V_{1} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩$
28	z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₄ ≠ 0, z₅ = ωc₅ ≠ 0, z₃ = 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, c₂, 0, c₄, c₅)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{4} V_{4} + c_{5} V_{5} ⟩$
29	z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₄ = 0, z₅ = ωc₅ ≠ 0. Let ϵ₁ = 0	(c₁, c₂, c₃, 0, c₅)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{5} V_{5} ⟩$
30	z₅ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, c₂, c₃, c₄, 0)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} ⟩$
31	z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, c₂, c₃, c₄, c₅)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩$

S. no.	Case	Basis element (y)	Sub-algebra Φ_d
1	z₁ ≠ 0, z₂ = z₃ = z₄ = z₅ = 0. Let ϵ₁ = 0 and z₁/ω = c₁	(c₁, 0, 0, 0, 0)	$⟨ V_{1} ⟩$
2	z₁ = z₃ = z₄ = 0 = z₅, z₂ = ωc₂ ≠ 0	(0, c₂, 0, 0, 0)	$⟨ V_{2} ⟩$
3	z₁ = z₂ = z₄ = 0 = z₅, z₃ = ωc₃ ≠ 0	(0, 0, c₃, 0, 0)	$⟨ V_{3} ⟩$
4	z₁ = z₂ = z₃ = z₅ = 0, z₄ ≠ 0. Let ϵ₃ + ϵ₅ = log(ωc₄/z₄), ϵ₄ = 0	(0, 0, 0, c₄, 0)	$⟨ V_{4} ⟩$
5	z₁ = z₂ = z₃ = z₄ = 0, z₅ = ωc₅ ≠ 0	(0, 0, 0, 0, c₅)	$⟨ V_{5} ⟩$
6	z₃ = z₄ = z₅ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁, z₂/ω = c₂	(c₁, c₂, 0, 0, 0)	$⟨ c_{1} V_{1} + c_{2} V_{2} ⟩$
7	z₂ = z₄ = z₅ = 0, z₁ ≠ 0, z₃ = ωc₃ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁, z₃/ω = c₃	(c₁, 0, c₃, 0, 0)	$⟨ c_{1} V_{1} + c_{3} V_{3} ⟩$
8	z₂ = z₃ = z₅ = 0, z₁ = ωc₁ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(c₁, 0, 0, c₄, 0)	$⟨ c_{1} V_{1} + c_{4} V_{4} ⟩$
9	z₂ = z₃ = z₄ = 0, z₅ = ωc₅ ≠ 0, z₁ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁	(c₁, 0, 0, 0, c₅)	$⟨ c_{1} V_{1} + c_{5} V_{5} ⟩$
10	z₁ = z₄ = z₅ = 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0	(0, c₂, c₃, 0, 0)	$⟨ c_{2} V_{2} + c_{3} V_{3} ⟩$
11	z₁ = z₃ = z₅ = 0, z₂ = ωc₂ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, c₂, 0, c₄, 0)	$⟨ c_{2} V_{2} + c_{4} V_{4} ⟩$
12	z₁ = z₃ = z₄ = 0, z₂ = ωc₂ ≠ 0, z₅ = ωc₅ ≠ 0	(0, c₂, 0, 0, c₅)	$⟨ c_{2} V_{2} + c_{5} V_{5} ⟩$
13	z₁ = z₂ = z₅ = 0, z₃ = ωc₃ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, 0, c₃, c₄, 0)	$⟨ c_{3} V_{3} + c_{4} V_{4} ⟩$
14	z₁ = z₂ = z₄ = 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₃ ≠ 0	(0, 0, c₃, 0, c₅)	$⟨ c_{3} V_{3} + c_{5} V_{5} ⟩$
15	z₁ = 0, z₂ = 0, z₄ ≠ 0, z₃ = 0, z₅ = ωc₅ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, 0, 0, c₄, c₅)	$⟨ c_{4} V_{4} + c_{5} V_{5} ⟩$
16	z₄ = z₅ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁	(c₁, c₂, c₃, 0, 0)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} ⟩$
17	z₃ = z₅ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, c₂, 0, c₄, 0)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{4} V_{4} ⟩$
18	z₃ = z₄ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₅ = ωc₅ ≠ 0. Let ϵ₁ = 0, z₁/ω = c₁	(c₁, c₂, 0, 0, c₅)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{5} V_{5} ⟩$
19	z₂ = z₄ = 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0, z₁ ≠ 0. Let ϵ₁ = 0 = ϵ₄ = ϵ₅, ϵ₃ = log(ωc₄/z₄)	(c₁, 0, c₃, c₄, 0)	$⟨ c_{1} V_{1} + c_{3} V_{3} + c_{4} V_{4} ⟩$
20	z₂ = z₄ = 0, z₁ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0. Let ϵ₁ = 0	(c₁, 0, c₃, 0, c₅)	$⟨ c_{1} V_{1} + c_{3} V_{3} + c_{5} V_{5} ⟩$
21	z₂ = z₃ = 0, z₁ = ωc₁ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, 0, 0, c₄, c₅)	$⟨ c_{1} V_{1} + c_{4} V_{4} + c_{5} V_{5} ⟩$
22	z₁ = z₅ = 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, c₂, c₃, c₄, 0)	$⟨ c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} ⟩$
23	z₁ = z₄ = 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0.	(0, c₂, c₃, 0, c₅)	$⟨ c_{2} V_{2} + c_{3} V_{3} + c_{5} V_{5} ⟩$
24	z₁ = z₃ = 0, z₂ = ωc₂ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, c₂, 0, c₄, c₅)	$⟨ c_{2} V_{2} + c_{4} V_{4} + c_{5} V_{5} ⟩$
25	z₅ = ωc₅ ≠ 0, z₁ = 0 = z₂, z₃ = ωc₃ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, 0, c₃, c₄, c₅)	$⟨ c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩$
26	z₁ = 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₄ = 0, ϵ₃ + ϵ₅ = log(ωc₄/z₄)	(0, c₂, c₃, c₄, c₅)	$⟨ c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩$
27	z₂ = 0, z₁ = ωc₁ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, 0, c₃, c₄, c₅)	$⟨ c_{1} V_{1} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩$
28	z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₄ ≠ 0, z₅ = ωc₅ ≠ 0, z₃ = 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, c₂, 0, c₄, c₅)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{4} V_{4} + c_{5} V_{5} ⟩$
29	z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₄ = 0, z₅ = ωc₅ ≠ 0. Let ϵ₁ = 0	(c₁, c₂, c₃, 0, c₅)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{5} V_{5} ⟩$
30	z₅ = 0, z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, c₂, c₃, c₄, 0)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} ⟩$
31	z₁ ≠ 0, z₂ = ωc₂ ≠ 0, z₃ = ωc₃ ≠ 0, z₅ = ωc₅ ≠ 0, z₄ ≠ 0. Let ϵ₁ = 0 = ϵ₄, ϵ₅ + ϵ₃ = log(ωc₄/z₄)	(c₁, c₂, c₃, c₄, c₅)	$⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩$

Thus, we have 14 different optimal classes of subalgebras, namely,

\begin{gathered} Φ_{1}; Φ_{2}; Φ_{3} = Φ_{13}; Φ_{4}; \\ Φ_{5} = Φ_{9} = Φ_{14} = Φ_{15} = Φ_{20} = Φ_{21} = Φ_{25} = Φ_{27}; \\ Φ_{6}; Φ_{7} = Φ_{19}; Φ_{8}; Φ_{10} = Φ_{22}; Φ_{11}; \\ Φ_{12} = Φ_{18} = Φ_{24} = Φ_{28}; Φ_{16} = Φ_{30}; \\ Φ_{17}; Φ_{23} = Φ_{26} = Φ_{29} = Φ_{31} . \end{gathered}

(29)

After obtaining these 14 different subalgebras, we have to check for the existence of similarity solutions for each of them. The collection of all the obtained similarity solutions discussed in Sec. V constitute the optimal solutions to the considered problem.

V. SIMILARITY SOLUTION AND BOUNDARY CONDITION

The set of fundamental Eqs. (2)–(8) may admit a similarity solution by an appropriate choice of the subalgebra Φ_d, d = 1, 2, 3, …, 31, from Table III.

A. Case I

The subalgebras

Φ_{23} = ⟨ c_{2} V_{2} + c_{3} V_{3} + c_{5} V_{5} ⟩

⁠,

Φ_{29} = ⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{5} V_{5} ⟩

⁠, and

Φ_{26} = ⟨ c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩

or

Φ_{31} = ⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5} ⟩

generate the same invariant subalgebra under translational transformation. In this case, the associated Lagrange auxiliary equation is expressed as

\begin{align} \frac{d t}{a t} & = \frac{d r}{(σ_{22} + 2 a) r} = \frac{d ρ}{- 2 a ρ} = \frac{d u}{(σ_{22} + a) u} = \frac{d v}{(σ_{22} + a) v} \\ = \frac{d w}{(σ_{44} + a) w} = \frac{d p}{2 σ_{22} p} = \frac{d h}{σ_{22} h} = \frac{d m}{2 (σ_{22} + a) m} \\ = \frac{d J}{(3 σ_{22} + a) J} . \end{align}

(30)

From integration of this equation, we obtain

η = \frac{r}{A t^{θ}},

(31)

and the similarity transformations

\begin{gathered} ρ = t^{- 2} \hat{D} (η), u = t^{θ - 1} \hat{U} (η), v = t^{θ - 1} \hat{V} (η), \\ w = t^{(σ_{44} + a) / a} \hat{W} (η), p = t^{2 (θ - 2)} \hat{P} (η), \\ h = t^{θ - 2} \hat{H} (η), m = t^{2 (θ - 1)} \hat{M} (η), J = t^{3 θ - 5} \hat{F} (η), \end{gathered}

(32)

where θ = (σ₂₂ + 2a)/a is a real constant and A is a dimensional constant.

The shock wave moves in an outward direction from the axis of the cylinder, and if we denote the distance of the shock front from the axis of symmetry by R(t), which is the shock radius, and the shock velocity by U_s, then we have

R (t) = A t^{θ}, U_{s} = θ A t^{θ - 1} .

(33)

From Eqs. , , and , the flow variables just ahead of the shock front vary as

\begin{gathered} u_{a} = 0, ρ_{a} = ρ_{*} R^{σ_{1}}, h_{a} = h_{*} R^{- σ_{2}}, \\ v_{a} = v_{*} R^{σ_{3}}, w_{a} = w_{*} R^{σ_{4}}, m_{a} = m_{*} R^{σ_{1} + 2}, \\ p_{a} = p_{*} R^{- 2 σ_{2}}, J_{a} = 0, \end{gathered}

(34)

where

p_{*} = (1 / σ_{2}) [(χ - σ_{2}) μ h_{*}^{2} / 2 + G m_{*} ρ_{*} / 2 - v_{*}^{2} ρ_{*} / 2]

⁠, m_* = 2πρ_*/(σ₁ + 2), ρ_*, v_*, w_*, h_* are dimensional constants, σ₁, σ₂, σ₃, σ₄ are real constants, and σ₁, σ₂, σ₃ satisfy the relations σ₁ + σ₂ + 1 = 0 and 2σ₃ + σ₁ + 2σ₂ = 0 in the presence of a magnetic field. Also, in the absence of a magnetic field (i.e., h_* = 0),

p_{a} = p_{*} R^{2 (σ_{1} + 1)}

⁠, where

p_{*} = \frac{1}{2 (σ_{1} + 1)} / [(v_{*}^{2} ρ_{*} - G m_{*} ρ_{*})]

and σ₁, σ₃ satisfy the relation 2σ₃ − σ₁ − 2 = 0.

The boundary conditions at the shock front (η = 1) are obtained as

\begin{gathered} \hat{D} (1) = \frac{ρ_{*} A^{σ_{1}}}{β}, \hat{U} (1) = (1 - β) A θ, \\ \hat{V} (1) = v_{*} A^{σ_{3}}, \hat{W} (1) = w_{*} A^{σ_{4}}, \\ \hat{H} (1) = \frac{h_{*} A^{- σ_{2}}}{β}, \hat{P} (1) = L_{*} ρ_{*} A^{σ_{1} + 2} θ^{2}, \\ \hat{M} (1) = m_{*} A^{σ_{1} + 1}, \hat{F} (1) = F_{*} A^{σ_{3} + 3} ρ_{*} θ^{3}, \end{gathered}

(35)

provided that σ₁ = −2/θ, σ₃ = (θ − 1)/θ, σ₄ = (σ₄₄ + a)/aθ = (θ − 1)/θ, and σ₂ = −(θ − 2)/θ.

For the existence of a similarity solution, the Alfvén Mach number A_M and shock Mach number S_M must be constant, and therefore the expressions for A_M and S_M are obtained as

A_{M} = \frac{\sqrt{ρ_{*}} θ A^{σ_{1} / 2 + 1 + σ_{2}}}{μ {(h_{*})}^{2}}, S_{M} = \frac{\sqrt{ρ_{*}} θ A^{σ_{1} / 2 + 1 + σ_{2}}}{γ p_{*}} .

(36)

The relationship between S_M and A_M in the presence of a magnetic field (i.e., for

A_{M}^{- 2} \neq 0

⁠) is obtained as

S_{M} = {\{\frac{σ_{2}}{γ [(χ - σ_{2}) A_{M}^{- 2} + \frac{G_{p}}{θ^{2}} - R_{p}^{2}]}\}}^{1 / 2},

(37)

and in the absence of magnetic field (i.e., for

A_{M}^{- 2} = 0

⁠), this reduces to

S_{M} = {[\frac{2 σ_{1} + 2}{γ (R_{p}^{2} - G_{p} / θ^{2})}]}^{1 / 2},

(38)

where

R_{p} = v_{*} A^{σ_{3} - 1} / θ

and

G_{p} = G m_{*} A^{σ_{1}}

are taken as the nondimensional parameters for rotation and gravitation, respectively.

Using Eqs. and , the similarity transformations can be written as

\begin{gathered} ρ = ρ_{a} D (η), u = U_{s} U (η), v = U_{s} V (η), \\ w = w_{a} W (η), p = U_{s}^{2} ρ_{a} P (η), \\ \sqrt{μ} h = \sqrt{ρ_{a}} U_{s} H (η), m = m_{a} M (η), \\ J = U_{s}^{3} ρ_{a} Q (η), \end{gathered}

(39)

where

\begin{gathered} D (η) = \frac{\hat{D} (η)}{A^{σ_{1}} ρ_{*}}, U (η) = \frac{\hat{U} (η)}{A θ}, \\ Q (η) = \frac{\hat{Q} (η)}{A^{σ_{1} + 3} θ^{3} ρ_{*}}, V (η) = \frac{\hat{V} (η)}{A θ}, \\ W (η) = \frac{\hat{W} (η)}{w_{*} A^{σ_{4}}}, P (η) = \frac{\hat{P} (η)}{A^{σ_{1} + 2} θ^{2} ρ_{*}}, \\ H (η) = \frac{\sqrt{μ} \hat{H} (η)}{\sqrt{ρ_{*}} A^{\frac{σ_{1}}{2} + 1} θ}, M (η) = \frac{\hat{M} (η)}{m_{*} A^{σ_{1} + 2}} . \end{gathered}

(40)

Using Eq. and the transformations in the fundamental Eqs. –, we obtain a system of ordinary differential equations (ODEs) as

D^{'} (η) = - \frac{D}{U - η} (- \frac{2}{θ} + U^{'} + \frac{U}{η}),

(41)

V^{'} (η) = - \frac{V}{U - η} (\frac{θ - 1}{θ} + \frac{U}{η}),

(42)

W^{'} (η) = - \frac{θ - 1}{θ} \frac{W}{U - η},

(43)

\begin{align} P^{'} (η) & = \frac{H^{2} - D {(U - η)}^{2}}{U - η} U^{'} - \frac{θ - 1}{θ} U D + \frac{V^{2} D}{η} \\ + (\frac{θ - 2}{θ} + \frac{U + χ η - 2 χ U}{η}) \frac{H^{2}}{U - η} - \frac{G_{p} M D}{θ^{2} η}, \end{align}

(44)

U^{'} (η) = \frac{- \frac{D^{3 / 2} (U - η) Q}{Γ_{p} P (η)} + \frac{θ - 1}{θ} U (U - η) D - \frac{V^{2} D (U - η)}{η} - (\frac{θ - 2}{θ} + \frac{U + χ η - 2 χ U}{η}) H^{2} + \frac{G_{p} M (U - η) D}{θ^{2} η} - (\frac{- 2}{θ} + \frac{U}{η}) P}{H^{2} + P - D {(U - η)}^{2}},

(45)

H^{'} (η) = \frac{- H}{U - η} [\frac{θ - 1}{θ} + \frac{σ_{1}}{2} + U^{'} + \frac{(1 - χ) U}{η}],

(46)

\begin{align} Q^{'} (η) & = - \{\frac{γ P}{γ - 1} U^{'} + \frac{γ P U}{(γ - 1) η} \\ + \frac{1}{γ - 1} [(U - η) P^{'} + \frac{2 (θ - 1)}{θ} P - \frac{2}{θ} P] + \frac{Q (η)}{η}\}, \end{align}

(47)

M^{'} (η) = (- \frac{2}{θ} + 2) η D (η),

(48)

where

Γ_{p} = [16 ϰ θ / 3 κ_{o} {(R^{*})}^{2}] {(ρ_{*} A^{σ_{1}})}^{- 1 / 2}

is the radiative dimensionless heat transfer parameter. It is necessary to take

τ_{1} = - \frac{1}{2}

and τ₂ = 2 for the existence of a similarity solution.

Using Eqs. and , the jump conditions across the shock front are obtained as

\begin{gathered} D_{η = 1} = \frac{1}{β}, U_{η = 1} = 1 - β, V_{η = 1} = R_{p}, \\ W_{η = 1} = 1, P_{η = 1} = L_{*}, H_{η = 1} = \frac{1}{A_{M} β}, \\ M_{η = 1} = 1, Q_{η = 1} = F_{*} . \end{gathered}

(49)

The components of the vorticity vector in nondimensional form are expressed as l_r = L_rR/U_s, l_θ = L_θR/U_s, and l_z = L_zR/U_s. With the help of Eqs. , , , , and , the expressions to calculate l_r, l_θ, l_z are obtained as

\begin{gathered} l_{r} = 0, l_{θ} = \frac{(θ - 1) S_{o}}{2 θ} \frac{W (η)}{U - η}, \\ l_{z} = - \frac{2 θ - 1}{2 θ} \frac{V (η)}{U - η}, \end{gathered}

(50)

where

S_{o} = w_{*} A^{σ_{4} - 1} / θ

is a nondimensional constant.

B. Case II

The subalgebras

Φ_{16} = ⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} ⟩

and

Φ_{30} = ⟨ c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} ⟩

in Table III generate the same invariant subspace, which can be prove by using translation invariance. Thus, the associated characteristic equation corresponding to Φ₁₆ or Φ₃₀ is given by

\begin{align} \frac{d t}{c} & = \frac{d r}{σ_{22} r} = \frac{d ρ}{0} = \frac{d u}{σ_{22} u} = \frac{d v}{σ_{22} v} = \frac{d w}{σ_{44} w} \\ = \frac{d p}{2 σ_{22} p} = \frac{d h}{σ_{22} h} = \frac{d m}{2 σ_{22} m} = \frac{d J}{3 σ_{22} J} . \end{align}

(51)

From integration of this equation, we obtain the similarity variable as

η^{*} = \frac{r}{B e^{δ t}},

(52)

and the similarity transformations as

\begin{gathered} ρ = \bar{S} (η^{*}), u = e^{δ t} \bar{Ψ} (η^{*}), v = e^{δ t} \bar{Φ} (η^{*}), \\ w = e^{σ_{44} t / c} \bar{ϒ} (η^{*}), p = e^{2 δ t} \bar{P} (η^{*}), h = e^{δ t} \bar{Π} (η^{*}), \\ m = e^{2 δ t} \bar{Ω} (η^{*}), J = e^{3 δ t} \bar{Λ} (η^{*}), \end{gathered}

(53)

where δ = σ₂₂/c and B are dimensional constants. The shock radius and shock front velocity are expressed as

R (t) = B e^{δ t}, U_{s} = \frac{d R}{d t} = B δ e^{δ t} .

(54)

Using Eqs. and , the Alfvén–Mach number A_M and shock Mach number S_M are expressed as

A_{M} = \frac{\sqrt{ρ_{*}} δ}{\sqrt{μ} h_{*}}, S_{M} = \frac{\sqrt{ρ_{*}} δ}{\sqrt{γ p_{*}}} (with σ_{2} = - 1) .

(55)

In the presence of a magnetic field, the flow variables just ahead of the shock front vary according to Eq. (34), where $p_{*} = (1 / σ_{2}) [(χ - σ_{2}) μ h_{*}^{2} / 2 + G m_{*} ρ_{*} / 2 - v_{*}^{2} ρ_{*} / 2]$ ⁠, ρ_*, v_*, w_*, h_* are dimensional constants, σ₁, σ₂, σ₃, σ₄ are real constants, and σ₁, σ₂, σ₃ satisfy the relations σ₁ + σ₂ + 1 = 0 and 2σ₃ + σ₁ + 2σ₂ = 0.

The shock Mach number S_M and Alfvén Mach number A_M in the presence of a magnetic field (i.e.,

A_{M}^{- 2} \neq 0

⁠) are related by

S_{M} = \frac{\sqrt{2}}{\sqrt{γ} {[R_{e}^{2} - (1 + χ) A_{M}^{- 2} - G_{e}]}^{1 / 2}} .

(56)

In the absence of a magnetic field (i.e.,

A_{M}^{- 2} = 0

⁠), this expression reduces to

S_{M} = \frac{\sqrt{2}}{\sqrt{γ} {(R_{e}^{2} - G_{e})}^{1 / 2}},

(57)

and the initial values of the flow variables are given by Eq. with h_* = 0; in this case,

p_{a} = p_{*} R^{2 (σ_{1} + 1)}

⁠, where

p_{*} = v_{*}^{2} ρ_{*} - G m_{*} ρ_{*} / [2 (σ_{1} + 1)]

and σ₁, σ₃ satisfy the relation 2σ₃ − σ₁ − 2 = 0, whereas the shock Mach number S_M becomes nondimensional with the condition σ₁ = 0.

In Eqs. and , R_e = v_*/δ and G_e = Gπρ_*/δ² are taken as the nondimensional parameters for rotation and gravitation, respectively, in case of an exponential-law shock path. From Eq. , it is clear that R_e = 0 cannot hold, because, if it did, then the shock Mach number would be imaginary i.e., the shock wave would not exist. Thus, for the existence of a shock wave, a necessary condition is that

R_{e}^{2} > (1 + χ) A_{M}^{- 2} + G_{e}

in Eq. and

R_{e}^{2} > G_{e}

in Eq. . Also, our problem can be reduced to the nonmagnetic and nongravitating case (i.e.,

A_{M}^{- 2} = 0

and G_e = 0), or the nonmagnetic but gravitating case (i.e.,

A_{M}^{- 2} = 0

and G_e ≠ 0), or the magnetic but nongravitating case (i.e.,

A_{M}^{- 2} \neq 0

and G_e = 0). It cannot, however, be reduced to the nonrotating case (i.e., R_e = 0). The jump conditions across the shock front (i.e., at η* = 1) are obtained by using the transformations and Eqs. , , and , as

\begin{gathered} \bar{S} (1) = \frac{ρ_{*}}{β}, \bar{Ψ} (1) = (1 - β) B δ, \bar{Φ} (1) = v_{*} B, \\ \bar{ϒ} (1) = w_{*} B^{σ_{4}}, \bar{Π} (1) = \frac{h^{*} B}{β}, \bar{Ω} (1) = m_{*} B^{2}, \\ \bar{P} (1) = L_{*} ρ_{a} δ^{2} B^{2}, \bar{Λ} (1) = F_{*} ρ_{*} B^{3} δ^{3}, \end{gathered}

(58)

where m* = πρ_*, σ₁ = 0, σ₂ = −1, σ₃ = 1, and σ₄ = σ₄₄/cδ.

With the help of Eqs. and , the similarity transformations are expressed as

\begin{gathered} ρ = ρ_{a} S (η^{*}), u = U_{s} Ψ (η^{*}), v = U_{s} Φ (η^{*}), \\ w = w_{a} ϒ (η^{*}), p = U_{s}^{2} ρ_{a} P (η^{*}), m = m_{a} Ω (η^{*}), \\ \sqrt{μ} h = \sqrt{ρ_{a}} U_{s} Π (η^{*}), J = U_{s}^{3} ρ_{a} Λ (η^{*}), \end{gathered}

(59)

where

\begin{gathered} S (η^{*}) = \frac{\bar{S} (η^{*})}{ρ_{a}}, Ψ (η^{*}) = \frac{\bar{Ψ} (η^{*})}{B δ}, Φ (η^{*}) = \frac{\bar{Φ} (η^{*})}{B δ}, \\ ϒ (η^{*}) = \frac{\bar{ϒ} (η^{*})}{w_{*} B^{σ_{2}}}, P (η^{*}) = \frac{\bar{P} (η^{*})}{B^{2} δ^{2} ρ_{a}}, \\ Π (η^{*}) = \frac{\sqrt{μ} \bar{Π} (η^{*})}{\sqrt{ρ_{a}} B δ}, Ω (η^{*}) = \frac{\bar{Ω} (η^{*})}{m^{*} B^{2}}, Λ (η^{*}) = \frac{\bar{Λ} (η^{*})}{B^{3} δ^{3} ρ_{a}} . \end{gathered}

(60)

Using Eq. and the similarity transformations in the fundamental Eqs. – gives a system of ODEs that, after simplification, can be expressed as

S^{'} = - \frac{S}{Ψ - η^{*}} (Ψ^{'} + \frac{Ψ}{η^{*}}),

(61)

Φ^{'} = - \frac{Φ}{Ψ - η^{*}} (1 + \frac{Ψ}{η^{*}}),

(62)

ϒ^{'} = - \frac{σ_{4} ϒ}{Ψ - η^{*}},

(63)

Π^{'} = - \frac{Π}{Ψ - η^{*}} [1 + Ψ^{'} + (1 - χ) \frac{Ψ}{η^{*}}],

(64)

\begin{align} P^{'} & = \frac{Π^{2} - S {(Ψ - η^{*})}^{2}}{Ψ - η^{*}} Ψ^{'} - S Ψ + \frac{Φ^{2} S}{η^{*}} \\ + \frac{[(1 - 2 χ) Ψ + (χ + 1) η^{*}] Π^{2}}{η^{*} (Ψ - η^{*})} - \frac{G_{e} Ω S}{η^{*}}, \end{align}

(65)

\begin{align} Ψ^{'} (η) = \frac{\frac{S (Ψ - η^{*}) Λ}{Γ_{e} P (η)} - Ψ (Ψ - η^{*}) S - \frac{G_{e} Ω (Ψ - η^{*}) S}{η^{*}} + \frac{(1 - 2 χ) Ψ + (1 + χ) η^{*}}{η^{*}} Π^{2} + \frac{Φ^{2} S (Ψ - η^{*})}{η^{*}} + \frac{Ψ P}{η^{*}}}{S {(Ψ - η^{*})}^{2} - Π^{2} - P}, \end{align}

(66)

Λ^{'} = \frac{S {(Ψ - η^{*})}^{2} - Π^{2} - γ P}{γ - 1} Ψ^{'} + \frac{Z (η^{*})}{η^{*}},

(67)

Ω^{'} = 2 S η^{*},

(68)

where

\begin{array}{l} Z (η^{*}) & = η^{*} Ψ (Ψ - η^{*}) S - γ Ψ P - Π^{2} [(1 - 2 χ) Ψ \\ + (1 + χ) η^{*} η^{*}] - Φ^{2} S + G_{e} Ω (Ψ - η^{*}) S - (γ - 1) Λ \end{array}

and

Γ_{e} = \frac{16 ϰ δ}{3 κ_{0} {(R^{*})}^{2}}

is the nondimensional radiation parameter, and it has been necessary to take τ₁ = −1 and τ₂ = −2 for the existence of a similarity solution.

By using Eqs. and , we obtain the jump conditions at the shock front as

\begin{gathered} S (1) = \frac{1}{β}, Ψ (1) = 1 - β, Φ (1) = R_{e}, \\ P (1) = L_{*}, Π (1) = \frac{1}{A_{M} β}, ϒ (1) = 1, \\ Ω (1) = 1, Λ (1) = F_{*} . \end{gathered}

(69)

The vorticity components in nondimensional form are l_r = L_rR/U_s, l_θ = L_θR/U_s, and l_z = L_zR/U_s, and the expressions to calculate these are obtained with the use of Eqs. , , , , and as

l_{θ} = \frac{S_{o}^{*} ϒ (η^{*})}{Ψ - η^{*}}, l_{r} = 0, l_{z} = \frac{- Φ (η^{*})}{Ψ - η^{*}},

(70)

where

S_{o}^{*} = w_{*} / δ

is a nondimensional constant, and σ₄₄ = 1.

C. Case III

Here, the subalgebra is

Φ_{6} = ⟨ c_{1} V_{1} + c_{2} V_{2} ⟩

in Table III, and the associate characteristic equation is given by

\begin{align} \frac{d t}{c} & = \frac{d r}{σ_{22} r} = \frac{d ρ}{0} = \frac{d u}{σ_{22} u} = \frac{d v}{σ_{22} v} = \frac{d w}{0} = \frac{d p}{2 σ_{22} p} \\ = \frac{d h}{σ_{22} h} = \frac{d m}{2 σ_{22} m} = \frac{d J}{3 σ_{22} J} . \end{align}

(71)

Proceeding as in case II, we ontain the same similarity variable and similarity transformations as given in Eqs. (31) and (59), respectively, and the shock jump condition as given in Eq. (58) with σ₄ = 0 for the initial values of variables as given in Eq. (34). In this case, by using the similarity transformations (59) in the system of fundamental Eqs. (2)–(8) and jump conditions (58), we obtain the same system of ODEs (61)–(68) and shock jump conditions (69) with σ₄ = 0.

In the present case, from Eq. , for σ₄ = 0, the vorticity components in nondimensional form l_r = l_rR/U_s, l_θ = l_θR/U_s, and l_z = L_zR/U_s are obtained as

l_{r} = 0, l_{θ} = 0, l_{z} = - \frac{Φ (η^{*})}{U - η^{*}} .

(72)

D. Case IV

Here, the subalgebras are Φ₅, Φ₉, Φ₁₂, Φ₁₄, Φ₁₅, Φ₁₇, Φ₁₈, Φ₂₀, Φ₂₁, Φ₂₄, Φ₂₅, Φ₂₇, and Φ₂₈ in Table III. The similarity variable and the similarity transformations can be obtained, but the shock jump condition is not invariant for these subalgebras. Thus, a similarity solution does not exist in this case.

E. Case V

Here, the subalgebras are Φ₁, Φ₂, Φ₃, Φ₄, Φ₇, Φ₈, Φ₁₀, Φ₁₁, Φ₁₃, Φ₁₉, Φ₂₂ in Table III. For these, a similarity variable and similarity transformations cannot be obtained, and hence a similarity solution does not exist in this case.

VI. NUMERICAL SOLUTION AND DISCUSSION OF RESULTS

The systems of ODEs (41)–(48) and (61)–(68) for the cases of a power-law shock path (PLSP) and an exponential-law shock path (ELSP), respectively, are integrated numerically with respect to the similarity variables η and η* with boundary conditions (49) and (69), respectively. A fourth-order Runge–Kutta method is used for the integrations. For these numerical calculations, we have taken the values of the physical parameters as follows:^{39,56,60,70,72}

PLSP: $γ = \frac{4}{3}$ ⁠, $\frac{5}{3}$ ⁠; G_p = 0, 0.01, 0.025; $A_{M}^{- 2} = 0.08$ ⁠, 0.085; R_p = 0, 0.06, 0.08; S_o = 0, 1; Γ_p = 2, 4.
ELSP: $γ = \frac{4}{3}$ ⁠, $\frac{5}{3}$ ⁠; G_e = 0, 0.04, 0.06; $A_{M}^{- 2} = 0$ ⁠, 0.02, 0.03, 0.035, 0.04; R_e = 0.55, 0.58, 0.6; $S_{o}^{*} = 1$ ⁠; Γ_e = 0.5, 0.7.

For the PLSP, the value G_p = 0 represents the nongravitating case, for which our solution corresponds to that obtained by Singh.⁷⁰ The values R_p = 0, S_o = 0, and G_p = 0 for the PLSP represent the nongravitating and nonrotating case, for which our solution corresponds to that obtained by Vishwakarma et al.³⁹ for a perfect gas. The values $A_{M}^{- 2} = 0$ ⁠, R_p = 0, S_o = 0 and G_p = 0 represent the nonmagnetic, nonrotating, and nongravitating case, for which our solution corresponds to that obtained by Singh.⁵⁹ Thus, our present work for the PLSP represents a generalization of that in Refs. 39, 59, and 70. For the ELSP, our work represents a generalization of that by Singh⁷⁰ in that it additionally considers the effect of gravitation. Also, Singh⁷⁰ assumed an azimuthal magnetic field, but we have considered both axial and azimuthal magnetic fields. The value $γ = \frac{4}{3}$ corresponds to a relativistic gas and $γ = \frac{5}{3}$ to a fully ionized gas, both of which are appropriate in an the astrophysical context.⁶⁹ These values of γ characterize the most general ranges that are seen in real stars. Also, the value of γ (γ < 1 or γ > 1) is in the general range corresponding to different fluids present in the interstellar medium.^79,80 Low values of γ are expected to lead to the formation of dense clusters of low-mass stars, whereas γ > 1 probably results in the formation of isolated and massive stars.⁸¹ In our problem, the value of the parameter $A_{M}^{- 2}$ is taken in the range $0.01 < A_{M}^{- 2} < 1$ because Rosenau and Frankenthal^43,82 have shown that the magnetic field has a remarkable effect on flow variables and on shocks when $A_{M}^{- 2} \geq 0.01$ ⁠. Also, for $A_{M}^{- 2} = 1$ ⁠, the shock wave degenerates into a weak discontinuity that does not compress the gas. The choices of the nondimensional gravitational parameter $G_{p} = G m_{*} A^{σ_{1}}$ for the PLSP and G_e = Gπρ_*/δ² for the ELSP are typical values, because all quantities in this parameter are positive and no restrictions on the values of m_*, A, ρ_* and δ are obtained, whereas π is a fixed constant and the value of G depends on the mass of the object. Thus, the values G_p = 0, 0.01, 0.025 and G_e = 0, 0.04, 0.06 are taken such that these parameters have significant effects on the flow variables and on shocks. The values of the rotational parameters R_p and R_e are taken so that the conditions for the existence of shock waves as obtained from Eqs. (37) and (56), respectively, are satisfied: $R_{p}^{2} > (χ - σ_{2}) A_{M}^{- 2} + G_{p} / θ^{2} - σ_{2} / γ$ for the PLSP and $R_{e}^{2} > (χ + 1) A_{M}^{- 2} + G_{e} + 2 / γ$ for the ELSP. The choices of the values of the nondimensional radiative heat transfer parameters $Γ_{p} = [16 ϰ θ / 3 κ_{o} {(R^{*})}^{2}] {(ρ_{*} A^{σ_{1}})}^{- 1 / 2}$ for the PLSP and $Γ_{e} = 16 ϰ δ / 3 κ_{o} {(R^{*})}^{2}$ for the ELSP are typical. The values of the radiative heat transfer parameters Γ_p and Γ_e are inversely related to the absorption coefficient or opacity of the medium.

Tables IV and V show the values of the density ratio at the shock front and the position of the inner boundary surface (IBS) for the PLSP, and Tables VI and VII show the corresponding values for the ELSP. Physically, the IBS may result from an exploding wire in a swirling detonation engine, the surface of a stellar corona, condensed explosives, or a diaphragm retaining a very high-pressure driving gas. With the help of the data in these tables, we have plotted the flow variables between the IBS and shock front in Figs. 1 and 2 for a PLSP and in Figs. 3 and 4 for the ELSP.

TABLE IV.

For the PLSP, the values of β and η_p for different values of R_p, G_p, and $A_{M}^{- 2}$ ⁠, with χ = 1, θ = 1.4, Γ_p = 4, and $γ = \frac{5}{3}$ ⁠.

R_p	G_p	$A_{M}^{- 2}$	β	η_p
0	0	0.08	0.286 354	0.928 926
	0	0.085	0.298 305	0.921 712 6
	0.01	0.08	0.294 428	0.913 388
	0.01	0.085	0.306 431	0.905 639
	0.025	0.08	0.306 805	0.887 264
	0.025	0.085	0.318 876	0.878 726
0.06	0	0.08	0.280 745	0.939 016
	0	0.085	0.292 655	0.932 190
	0.01	0.08	0.288 716	0.924 503
	0.01	0.085	0.300 683	0.917 729
	0.025	0.08	0.300 944 8	0.899 981
	0.025	0.085	0.312 985	0.891 813
0.08	0	0.08	0.276 434	0.946 367
	0	0.085	0.288 311	0.939 853
	0.01	0.08	0.284 323	0.932 646
	0.01	0.085	0.296 26	0.925 569
	0.025	0.08	0.296 433	0.909 342
	0.025	0.085	0.384 48	0.901 462
2.305	9.9	0.5	0.661 646	0.993 885 3
	9.9	0.6	0.829 01	0.849 039 5
	10	0.5	0.740 471	0.889 249 1
	10	0.6	0.914 301	0.791 873

R_p	G_p	$A_{M}^{- 2}$	β	η_p
0	0	0.08	0.286 354	0.928 926
	0	0.085	0.298 305	0.921 712 6
	0.01	0.08	0.294 428	0.913 388
	0.01	0.085	0.306 431	0.905 639
	0.025	0.08	0.306 805	0.887 264
	0.025	0.085	0.318 876	0.878 726
0.06	0	0.08	0.280 745	0.939 016
	0	0.085	0.292 655	0.932 190
	0.01	0.08	0.288 716	0.924 503
	0.01	0.085	0.300 683	0.917 729
	0.025	0.08	0.300 944 8	0.899 981
	0.025	0.085	0.312 985	0.891 813
0.08	0	0.08	0.276 434	0.946 367
	0	0.085	0.288 311	0.939 853
	0.01	0.08	0.284 323	0.932 646
	0.01	0.085	0.296 26	0.925 569
	0.025	0.08	0.296 433	0.909 342
	0.025	0.085	0.384 48	0.901 462
2.305	9.9	0.5	0.661 646	0.993 885 3
	9.9	0.6	0.829 01	0.849 039 5
	10	0.5	0.740 471	0.889 249 1
	10	0.6	0.914 301	0.791 873

TABLE V.

For the PLSP, the values of β and η_p for different values of χ, γ, θ, and Γ_p, with R_p = 0.08, $A_{M}^{- 2} = 0.08$ ⁠, and G_p = 0.2.

χ	γ	θ	β	Γ_p	η_p
0	$\frac{4}{3}$	1.3	0.300 851	2	0.955 091
		1.3	0.300 851	4	0.903 736
		1.5	0.329 034	2	0.926 761
		1.5	0.329 034	4	0.812 226
	$\frac{5}{3}$	1.3	0.300 851	2	0.950 934
		1.3	0.300 851	4	0.895 308
		1.5	0.329 034	2	0.919 319
				4	0.804 934
				10	0.728 979 7
				50	0.691 769 4
	$\frac{4}{3}$	1.3	0.413 258	2	0.838 811
		1.3	0.413 258	4	0.828 904
		1.35	0.433 122	2	0.810 663
		1.35	0.433 122	4	0.798 072
1	$\frac{5}{3}$	1.3	0.413 258	2	0.814 824
		1.3	0.413 258	4	0.803 618
		1.35	0.433 122	2	0.782 794
				4	0.768 757
				10	0.118 781
				50	0.082 388 21

χ	γ	θ	β	Γ_p	η_p
0	$\frac{4}{3}$	1.3	0.300 851	2	0.955 091
		1.3	0.300 851	4	0.903 736
		1.5	0.329 034	2	0.926 761
		1.5	0.329 034	4	0.812 226
	$\frac{5}{3}$	1.3	0.300 851	2	0.950 934
		1.3	0.300 851	4	0.895 308
		1.5	0.329 034	2	0.919 319
				4	0.804 934
				10	0.728 979 7
				50	0.691 769 4
	$\frac{4}{3}$	1.3	0.413 258	2	0.838 811
		1.3	0.413 258	4	0.828 904
		1.35	0.433 122	2	0.810 663
		1.35	0.433 122	4	0.798 072
1	$\frac{5}{3}$	1.3	0.413 258	2	0.814 824
		1.3	0.413 258	4	0.803 618
		1.35	0.433 122	2	0.782 794
				4	0.768 757
				10	0.118 781
				50	0.082 388 21

TABLE VI.

For the ELSP, the values of β and $η_{p}^{*}$ for different values of G_e, R_e, and Γ_e, with $γ = \frac{5}{3}$ ⁠, $A_{M}^{- 2} = 0.02$ ⁠, and χ = 1.

G_e	R_e	β	Γ_e	$η_{p}^{*}$
0	0.55	0.305 259	0.5	0.890 711
	0.55	0.305 259	0.7	0.881 794
	0.58	0.336 149	0.5	0.882 916
	0.58	0.336 149	0.7	0.872 791
0.04	0.55	0.269 593	0.5	0.900 045
	0.55	0.269 593	0.7	0.892 412
	0.6	0.321 139	0.5	0.886 751
	0.6	0.321 139	0.7	0.877 251
0.06	0.55	0.252 128	0.5	0.904 725
	0.55	0.252 128	0.7	0.897 657
	0.6	0.303 003	0.5	0.893 928
			0.7	0.882 596
			10	0.858 657
			50	0.857 041
			100	0.856 84

G_e	R_e	β	Γ_e	$η_{p}^{*}$
0	0.55	0.305 259	0.5	0.890 711
	0.55	0.305 259	0.7	0.881 794
	0.58	0.336 149	0.5	0.882 916
	0.58	0.336 149	0.7	0.872 791
0.04	0.55	0.269 593	0.5	0.900 045
	0.55	0.269 593	0.7	0.892 412
	0.6	0.321 139	0.5	0.886 751
	0.6	0.321 139	0.7	0.877 251
0.06	0.55	0.252 128	0.5	0.904 725
	0.55	0.252 128	0.7	0.897 657
	0.6	0.303 003	0.5	0.893 928
			0.7	0.882 596
			10	0.858 657
			50	0.857 041
			100	0.856 84

TABLE VII.

For the ELSP, the values of β and $η_{p}^{*}$ for different values of χ, γ, and $A_{M}^{- 2}$ ⁠, with S_M = 2.3, Γ_e = 0.7, and G_e = 0.06.

χ	γ	$A_{M}^{- 2}$	β	$η_{p}^{*}$
0	$\frac{4}{3}$	0	0.283 554	0.907 305
		0.02	0.324 382	0.893 521
		0.04	0.359 229	0.882 947
	$\frac{5}{3}$	0	0.226 843	0.923 303
		0.02	0.273 417	0.904 876
		0.03	0.293 032	0.897 967
1	$\frac{4}{3}$	0	0.283 554	0.907 305
		0.035	0.350 922	0.867 267
		0.04	0.359 229	0.863 317
	$\frac{5}{3}$	0	0.226 843	0.922 330
		0.02	0.273 417	0.891 318
		0.04	0.311 126	0.871 090 1

χ	γ	$A_{M}^{- 2}$	β	$η_{p}^{*}$
0	$\frac{4}{3}$	0	0.283 554	0.907 305
		0.02	0.324 382	0.893 521
		0.04	0.359 229	0.882 947
	$\frac{5}{3}$	0	0.226 843	0.923 303
		0.02	0.273 417	0.904 876
		0.03	0.293 032	0.897 967
1	$\frac{4}{3}$	0	0.283 554	0.907 305
		0.035	0.350 922	0.867 267
		0.04	0.359 229	0.863 317
	$\frac{5}{3}$	0	0.226 843	0.922 330
		0.02	0.273 417	0.891 318
		0.04	0.311 126	0.871 090 1

FIG. 1.

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Flow variable distribution between shock front and IBS for the PLSP with azimuthal magnetic field (χ = 1), with γ = 5/3, θ = 1.4, and Γ_p = 4: (a) radial velocity; (b) density; (c) azimuthal velocity; (d) axial velocity; (e) pressure; (f) magnetic field; (g) mass; (h) radiation heat flux; (i) azimuthal vorticity component; (j) axial vorticity component; 1, $R_{p} = 0, G_{p} = 0, A_{M}^{- 2} = 0.08$ ⁠; 2, $R_{p} = 0, G_{p} = 0, A_{M}^{- 2} = 0.085$ ⁠; 3, $R_{p} = 0, G_{p} = 0.01, A_{M}^{- 2} = 0.08$ ⁠; 4, $R_{p} = 0, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; 5, $R_{p} = 0, G_{p} = 0.025, A_{M}^{- 2} = 0.08$ ⁠; 6, $R_{p} = 0.06, G_{p} = 0, A_{M}^{- 2} = 0.08$ ⁠; 7, $R_{p} = 0.06, G_{p} = 0, A_{M}^{- 2} = 0.085$ ⁠; 8, $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.08$ ⁠; 9, $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; 10, $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.08$ ⁠; 11, $R_{p} = 0.08, G_{p} = 0, A_{M}^{- 2} = 0.08$ ⁠; 12, $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 2.

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Flow variable distribution between shock front and IBS for the PLSP, with R_p = 0.08, $A_{M}^{- 2} = 0.08$ ⁠, and G_p = 0.2: (a) radial velocity; (b) density; (c) azimuthal velocity; (d) axial velocity; (e) pressure; (f) magnetic field; (g) mass; (h) radiation heat flux; (i) azimuthal vorticity component; (j) axial vorticity component; 1, $χ = 0, γ = \frac{4}{3}, θ = 1.3, Γ_{p} = 2$ ⁠; 2, $χ = 0, γ = \frac{4}{3}, θ = 1.3, Γ_{p} = 4$ ⁠; 3, $χ = 0, γ = \frac{4}{3}, θ = 1.5, Γ_{p} = 2$ ⁠; 4, $χ = 0, γ = \frac{5}{3}, θ = 1.3, Γ_{p} = 2$ ⁠; 5, $χ = 0, γ = \frac{5}{3}, θ = 1.3, Γ_{p} = 4$ ⁠; 6, $χ = 0, γ = \frac{5}{3}, θ = 1.5, Γ_{p} = 2$ ⁠; 7, $χ = 1, γ = \frac{4}{3}, θ = 1.3, Γ_{p} = 2$ ⁠; 8, $χ = 1, γ = \frac{4}{3}, θ = 1.3, Γ_{p} = 4$ ⁠; 9, $χ = 1, γ = \frac{4}{3}, θ = 1.35, Γ_{p} = 2$ ⁠; 10, $χ = 1, γ = \frac{5}{3}, θ = 1.3, Γ_{p} = 2$ ⁠; 11, $χ = 1, γ = \frac{5}{3}, θ = 1.3, Γ_{p} = 4$ ⁠; 12, $χ = 1, γ = \frac{5}{3}, θ = 1.35, Γ_{p} = 2$ ⁠.

FIG. 3.

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Flow variable distribution between shock front and IBS for the ELSP with azimuthal magnetic field (χ = 1), with γ = 5/3 and $A_{M}^{- 2} = 0.02$ ⁠: (a) radial velocity; (b) density; (c) azimuthal velocity; (d) axial velocity; (e) pressure; (f) magnetic field; (g) mass; (h) radiation flux; (i) azimuthal vorticity component; (j) axial vorticity component; 1, G_e = 0, R_e = 0.55, Γ_e = 0.5; 2, G_e = 0, R_e = 0.55, Γ_e = 0.7; 3, G_e = 0, R_e = 0.58, Γ_e = 0.5; 4, G_e = 0.04, R_e = 0.55, Γ_e = 0.5; 5, G_e = 0.04, R_e = 0.55, Γ_e = 0.7; 6, G_e = 0.04, R_e = 0.6, Γ_e = 0.7; 7, G_e = 0.06, R_e = 0.55, Γ_e = 0.5.

FIG. 4.

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Flow variable distribution between shock front and IBS for the ELSP, with S_M = 2.3, Γ_e = 0.7, and G_e = 0.06: (a) radial velocity; (b) density; (c) azimuthal velocity; (d) axial velocity; (e) pressure; (f) magnetic field; (g) mass; (h) radiation flux; (i) azimuthal vorticity component; (j) axial vorticity component; 1, $χ = 0, γ = \frac{4}{3}, A_{M}^{- 2} = 0$ ⁠; 2, $χ = 0, γ = \frac{4}{3}, A_{M}^{- 2} = 0.02$ ⁠; 3, $χ = 0, γ = \frac{4}{3}, A_{M}^{- 2} = 0.04$ ⁠; 4, $χ = 0, γ = \frac{5}{3}, A_{M}^{- 2} = 0$ ⁠; 5, $χ = 0, γ = \frac{5}{3}, A_{M}^{- 2} = 0.02$ ⁠; 6, $χ = 0, γ = \frac{5}{3}, A_{M}^{- 2} = 0.03$ ⁠; 7, $χ = 1, γ = \frac{4}{3}, A_{M}^{- 2} = 0$ ⁠; 8, $χ = 1, γ = \frac{4}{3}, A_{M}^{- 2} = 0.035$ ⁠; 9, $χ = 1, γ = \frac{4}{3}, A_{M}^{- 2} = 0.04$ ⁠; 10, $χ = 1, γ = \frac{5}{3}, A_{M}^{- 2} = 0$ ⁠; 11, $χ = 1, γ = \frac{5}{3}, A_{M}^{- 2} = 0.02$ ⁠; 12, $χ = 1, γ = \frac{5}{3}, A_{M}^{- 2} = 0.04$ ⁠.

From Figs. 1 and 2, we observe that on moving from the shock front toward the IBS, the flow variables u/u_b, ρ/ρ_b, h/h_b, J/J_b, and l_z increase, but v/v_b, w/w_b, p/p_b, m/m_b, and l_θ decrease for the PLSP. (These variables exhibit a similar flow field nature in the contour plots for the PLSP, as can be seen in Figs. 5–14.) Also, for the ELSP, Figs. 3(a), 3(c)–3(e), 3(g)–3(i), and 3(j) and Figs. 4(a)–4(e) and 4(g)–4(j) show that the variables u/u_b, p/p_b, and J/J_b increase, but v/v_b, w/w_b, m/m_b, and l_θ decrease.The variables ρ/ρ_b and l_z increase with increasing axial magnetic field, but with increasing azimuthal magnetic field they first increase and then decrease rapidly after attaining a maximum on moving from the shock front to the IBS [see panels (b) and (j) of Figs. 3 and 4]. Figures 3(f) and 4(f) show that the variable h/h_b first increases and then decreases after reaching a maximum in the flow field region behind the shock.

FIG. 5.

Contour plots of reduced radial velocity behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, and Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced radial velocity behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, and Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 6.

Contour plots of reduced density behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, and Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced density behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, and Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 7.

Contour plots of reduced azimuthal velocity behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced azimuthal velocity behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 8.

Contour plots of reduced axial velocity behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced axial velocity behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 9.

Contour plots of reduced pressure behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced pressure behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 10.

Contour plots of reduced magnetic field behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced magnetic field behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 11.

Contour plots of reduced mass behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced mass behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 12.

Contour plots of reduced radiation flux behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced radiation flux behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 13.

Contour plots of reduced axial vorticity vector behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

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Contour plots of reduced axial vorticity vector behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

FIG. 14.

Contour plots of reduced azimuthal vorticity vector behind cylindrical shock front for the PLSP with azimuthal magnetic field, with γ=53, θ = 1.4, Γp = 4: (a) Rp=0.06,Gp=0.01,AM−2=0.06; (b) Rp=0.06,Gp=0.01,AM−2=0.085; (c) Rp=0.06,Gp=0.025,AM−2=0.085; (d) Rp=0.08,Gp=0.01,AM−2=0.085.

View large Download slide

Contour plots of reduced azimuthal vorticity vector behind cylindrical shock front for the PLSP with azimuthal magnetic field, with $γ = \frac{5}{3}$ ⁠, θ = 1.4, Γ_p = 4: (a) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.06$ ⁠; (b) $R_{p} = 0.06, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠; (c) $R_{p} = 0.06, G_{p} = 0.025, A_{M}^{- 2} = 0.085$ ⁠; (d) $R_{p} = 0.08, G_{p} = 0.01, A_{M}^{- 2} = 0.085$ ⁠.

In Figs. 5–14, we present contour plots to provide a visual representation of the flow variables such as density and velocity, and to see the effect of the rotational parameter, gravitational parameter, and magnetic field parameter behind the shock front for the PLSP. In these contour plots, dark blue color represents lower values of η and light blue represents higher values. We have analyzed the distribution of flow variables from higher to lower values of η. The horizontal lines in the contour plots represent the values of the flow variables at different values of η.

Comparison of panels (a) and (b) in Figs. 5–14 illustrates the variation in the flow variables when the strength of the magnetic field is increased, comparison of panels (b) and (c) in each figure illustrates the variation in flow variables when the value of the gravitational parameter is increased, and comparison of panels (b) and (d) illustrates the variation in flow variables when the value of the rotational parameter is increased. To reveal these variations in the flow variables, we select a value of η = 0.96, and we find that when the magnetic field strength and the value of the gravitational parameter are increased, u/u_b, ρ/ρ_b, h/h_b, J/J_b, and l_θ decrease, whereas v/v_b, w/w_b, p/p_b, m/m_b, and l_z increase. The rotational parameter has the reverse effects on these flow variables compared with the effects of the magnetic field and the gravitational parameter. Similar effect of these parameters on the flow variables are also found from Fig. 1 (see Secs. VI A–VI C). Similarly, we can plot contours of the flow variables for different values of the physical parameters to compare with the results obtained from Figs. 1–4.

A. Effects of an increase in the value of the gravitational parameter G_p or G_e

The distance between the positions of the IBS and the shock front increases with increasing G_p, whereas it decreases with increasing G_e, i.e., the shock decays with G_p for the PLSP, and grows with G_e for the ELSP (see Tables IV and VI). The flow variables v/v_b, w/w_b, p/p_b, m/m_b, and l_θ increase, whereas u/u_b, ρ/ρ_b, h/h_b, J/J_b, and l_z decrease in the whole flow field region behind the shock front for the PLSP with G_p (see Fig. 1). The effects of G_p on u/u_b, ρ/ρ_b, m/m_b, p/p_b, and J/J_b are similar to the results obtained by Singh and Vishwakarma⁶⁰ in the magnetic and nonmagnetic cases. The flow variables u/u_b, v/v_b, w/w_b, m/m_b, J/J_b, and l_θ decrease in general with increasing G_e [see Figs. 3(a), 3(c), 3(d), and 3(g)–3(i)]. The variables ρ/ρ_b, h/h_b, and l_z increase near the shock front and decrease near the IBS with increasing G_e, but G_e has the reverse effects on p/p_b behind the shock front [see Figs. 3(b), 3(e), 3(f), and 3(j)].

From the above results, we observe that with an increase in the value of the gravitational parameter G_p or G_e, the flow field region between the shock front and the IBS expands for the PLSP, whereas it contracts for the ELSP. For the PLSP, this reduces the compression of the shocked region, and the mass flux m/m_b decreases; however, for the ELSP, the shocked region is highly compressed, and so the mass flux m/m_b increases.

B. Effects of an increase in the value of the rotational parameter R_p or R_e

The distance between the position of the IBS and shock front decreases with increasing R_p, whereas it increases with increasing R_e, i.e., the shock strength grows with R_p for the PLSP and decays with R_e for the ELSP (see Tables IV and VI). The flow variables u/u_b, ρ/ρ_b, h/h_b, J/J_b and l_z increase, whereas v/v_b, w/w_b, p/p_b, m/m_b, and l_θ decrease behind the shock front for the PLSP with R_p (see Fig. 1). The flow variables u/u_b, v/v_b, w/w_b, m/m_b, J/J_b, and l_θ increase in general with increasing R_e [see Figs. 3(a), 3(c), 3(d), and 3(g)–3(i)]. The variables ρ/ρ_b, h/h_b and l_z decrease near the shock front and increase near the IBS with increasing R_e [see Figs. 3(b), 3(f), and 3(j)]. The variable p/p_b increases near the shock front and decreases near the IBS with increasing R_e in the nongravitating case, but in the gravitating case, it decreases in the whole flow field with increasing R_e [see Fig. 3(e)].

Physically, this means that with an increase in the value of the rotational parameter R_p or R_e, the width of the flow field region between the shock front and the IBS is reduced owing to a decrease in compressibility for the PLSP, and so the shock become stronger. Also, the width of the flow field region between the shock front and the IBS is enlarged owing to an increase in compressibility for the ELSP, and so the shock wave decay with increasing R_e becomes weaker.

C. Effects of an increase in the magnetic field strength $A_{M}^{- 2}$

The distance between the position of the IBS and the shock front increases with increasing $A_{M}^{- 2}$ ⁠, i.e., the shock strength decays in the presence of a magnetic field for both the PLSP and ELSP (see Tables V and VII). For the PLSP, the flow variables u/u_b, ρ/ρ_b, h/h_b, and J/J_b, l_z decrease; however, l_θ, m/m_b, p/p_b, v/v_b, and w/w_b increase with increasing $A_{M}^{- 2}$ in the whole flow field region behind the shock front (see Fig. 1). The effects of $A_{M}^{- 2}$ on u/u_b, ρ/ρ_b, h/h_b, J/J_b, and p/p_b are similar to the results obtained by Vishwakarma et al.³⁹ for a nonrotating nongravitating perfect gas. For the ELSP, the variables u/u_b, v/v_b, w/w_b, m/m_b, and l_θ increase, but p/p_b decreases with increasing $A_{M}^{- 2}$ in the whole flow field region [see Figs. 4(a), 4(c)–4(e), 4(g), and 4(i)]. The variables ρ/ρ_b and l_z decrease and J/J_b and h/h_b increase in the presence of an axial magnetic field [see Figs. 4(b), 4(f), 4(h), and 4(j)]. The variables ρ/ρ_b, h/h_b, and l_z decrease near the shock front and increase near the IBS, but J/J_b decreases in the presence of an azimuthal magnetic field [see Figs. 4(b), 4(f), 4(h), and 4(j)]. For the ELSP, there has yet to be any examination of the effects of a magnetic field on the flow variables or on the strength of a shock wave in the presence of a radiation heat flux using the Lie symmetry method.

Physically, this means that an increase in the strength of the magnetic field $A_{M}^{- 2}$ has the effect of increasing the compressibility, and hence the shock wave strength decreases and the flow field region between the shock front and the IBS increases for both the PLSP and the ELSP.

D. Effects of an increase in the value of the radiative heat transfer parameter Γ_p or Γ_e

The distance between the positions of the IBS and the shock front increases with Γ_p and Γ_e, i.e., the shock wave decays with an increase in the value of the radiation parameter for both the PLSP and ELSP (see Tables V and VI). For the PLSP, the flow variables u/u_b, ρ/ρ_b, h/h_b, J/J_b, and l_z decrease with increasing Γ_p, whereas p/p_b, v/v_b, w/w_b, and m/m_b, l_θ increase (see Fig. 2). In this case, the effects of Γ_p on u/u_b, u/u_b, ρ/ρ_b, m/m_b, p/p_b, and J/J_b are similar to the results obtained by Singh and Vishwakarma⁶⁰ in both the nonrotating and nonmagnetic and the magnetic and rotating cases. For the ELSP, u/u_b and J/J_b, decrease with increasing Γ_e, but v/v_b, w/w_b, m/m_b and l_θ increase [see Figs. 3(a), 3(c), 3(d), and 3(g)–3(i)]. The variables ρ/ρ_b, h/h_b, and l_z decrease near the shock front and increase near the IBS with increasing Γ_e, but Γ_e has the reverse effect on p/p_b behind the shock front [see Figs. 3(b), 3(e), 3(f), and 3(j)].

Physically, this means that with an increase in the value of the radiation parameter Γ_p or Γ_e, the shock strength decreases, because the shocked region between the shock front and IBS expands for both the PLSP and ELSP. Owing to the reduced transfer of heat energy, the radiation flux J/J_b decreases in the whole shocked region for both the PLSP and ELSP. For the PLSP, the shock strength is strongly influenced by small values of the radiation parameter in the presence of an axial magnetic field in comparison with the case of an azimuthal magnetic field. At higher values of the radiation parameter, the shock strength is influenced by the presence of an azimuthal magnetic field (see Table V). For the ELSP, the shocked region is more compressed by an increase in the value of Γ_e in a self-gravitating gas than in a nongravitating gas (see Table VI).

E. Effects of an increase in the value of the adiabatic exponent γ

The distance between the positions of the IBS and shock front increases with γ for the PLSP but decreases for the ELSP, i.e., with increasing γ, the shock strength decays for the PLSP but grows for the ELSP (see Tables V and VII). With increasing γ, the flow variables ρ/ρ_b, u/u_b, J/J_b, h/h_b, and l_z decrease, whereas v/v_b, w/w_b, m/m_b, p/p_b, and l_θ increase in the whole flow field region for PLSP (see Fig. 2). For the ELSP, u/u_b, v/v_b, w/w_b, m/m_b, J/J_b, and l_θ decrease with increasing γ, whereas p/p_b increase in general behind the shock front [see Figs. 4(a), 4(c)–4(e), and 4(g)–4(i)]. In the presence of an azimuthal magnetic field, ρ/ρ_b and h/h_b decrease, but ρ/ρ_b increases in the nonmagnetic case [see Figs. 4(b) and 4(f)]. With increasing γ, h/h_b decreases, whereas ρ/ρ_b increases near the shock front and decreases near the IBS in the presence of an axial magnetic field [see Figs. 4(b) and 4(j)]. The variable l_z increases with increasing γ in the absence of magnetic field or in the presence of an axial magnetic field, but in the presence of an azimuthal magnetic field, it increases near the shock front but decreases near the IBS [see Fig. 4(j)].

F. Effects of an increase in the value of the similarity index θ for the PLSP

The shock strength decreases with increasing θ (see Table V). The flow variables ρ/ρ_b, J/J_b, w/w_b, m/m_b, h/h_b, and l_z decrease with increasing θ, but v/v_b and p/p_b increase [see Figs. 2(b)–2(h) and 2(j)]. The variables u/u_b and l_θ decrease throughout in the presence of an axial magnetic field. With increasing θ, in the presence of an azimuthal magnetic field, u/u_b increases near the shock front and decreases near the IBS, but θ has the reverse effect on l_θ [see Figs. 2(a) and 2(i)].

VII. CONCLUSIONS

The propagation of a shock wave in a self-gravitating ideal gas in the presence of a magnetic field, radiation heat flux, and rotation of the medium has been studied here using the Lie symmetry method. The advantage of this method is that it provides all possible similarity solutions (optimal solutions) of the problem under consideration. We have discussed the effects on the shock wave and the flow variables between the shock front and IBS of different values of physical parameters, such as those associated with gravitation, rotation, radiation, ambient magnetic field strength, adiabatic index, and similarity exponent. Understanding the propagation of a magnetogasdynamic shock wave in self-gravitating gas with radiative heat flux is essential for the study of a variety of astrophysical phenomena. Explosions in astrophysical and geophysical environments generate shock waves in which the shock may follow a power law shock path (PLSP) or an exponential law shock path (ELSP).^78,83 Gravity is the dominant force involved in interactions at small scales in the interstellar medium and governs the formation, shape, and trajectory of astronomical objects, such as in the formation of stars and planetary systems and in the creation of supernova remnants.^28,29,45,81 The present study has also included an analysis of data from exploding wire experiments in a rotating electrically conducting medium, and cylindrically symmetric hypersonic flow problems associated with meteors or reentry vehicles.^9,37

The following conclusions of this study are summarized from the results shown in Tables IV–VII and Figs. 1–4:

The shock strength decays in the presence of a radiation heat flux and of a magnetic field in for both a PLSP and an ELSP. Physically, energy transformation through shock waves is less in the presence of a magnetic field.
With an increase in the value of the rotational parameter, the shock strength is increased, whereas it is reduced with an increase in the value of the adiabatic index or the gravitational parameter for a PLSP. For an ELSP, the rotation and gravitational parameters and the adiabatic exponent have effects on the shock strength that are the reverse of those for a PLSP.
The effects of the rotational parameter on shock strength are the reverse of those of the adiabatic index and the gravitational parameter.
When the magnetic field is changed from axial to azimuthal, the flow field region between the shock front and the IBS increases for both PLSP and ELSP cases, i.e., the shock strength is decreased in the presence of an azimuthal magnetic field compared with an axial magnetic field.
When the value of the rotational parameter increases, the pressure decreases near the shock front and increases near the IBS in the nongravitating case, but it decreases everywhere in the flow field in the gravitating case for a PLSP. Also, with an increase in the value of the rotational parameter, the pressure decreases in general for an ELSP.

The results of the current study may be of particular interest in the following contexts.

A. Role of shock waves in star formation

The interstellar medium, which consists of dust, gas, and charged particles, makes up around 10%–15% of the galactic disk’s total mass.⁸⁴ It is widely believed that new stars and planetary systems form primarily within interstellar molecular clouds. An anomalous dissipation mechanism generates shock waves in these clouds. These shock waves propagate in the clouds and trigger transportation phenomena, leading to regions of increased density, enabling gravitational self-compression (or Jeans instability).⁸⁵ Observations reveal a strong correlation between the presence of molecular clouds and star formation. The main question here is to determine which factor plays a more critical role: condensation processes influenced by the strength of shock propagation or gravitational self-compression driven by the shock waves.

B. Shock wave formation in supernova explosions

Stellar explosions leads to supernovae, after which about 10¹⁵ erg of kinetic energy is dispersed in the ambient medium. The interaction of the supersonically (∼5000–10 000 km/s) moving ejecta (stellar material) with the circumstellar/interstellar medium (CSM/ISM) results in the formation of a shock wave, which creates a shell of shock-heated plasma. The interaction of the outermost ejecta with the CSM/ISM marks the beginning of the formation of a supernova remnant.^86,87 During a supernova explosion, high amounts of heat radiation are emitted in the ISM, which influences the propagation of shock waves and ejected stellar material.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

G. Nath: Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). P. Upadhyay: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

APPENDIX: DERIVATION OF OPTIMAL SYSTEM FOR SUBALGEBRAS

For interested readers, a complete derivation of the optimal system for the subalgebras are given as follows:

As

V_{1}

⁠,

V_{2}

⁠,

V_{3}

⁠,

V_{4}

⁠, and

V_{5}

are independent vector fields of a five-dimensional Lie algebra, every vector of that Lie algebra can be expressed as a linear combination of independent vectors

V_{α}

⁠, α = 1, 2, …, 5. Thus, a general vector

V

of the Lie algebra can be written as

V = c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5},

(A1)

where c₁ = c, c₂ = σ₂₂, c₃ = σ₄₄, c₄ = k, and c₅ = a.

The adjoint action of

V_{1}

on

V

can be expressed with the help of Table II as

\begin{array}{l} A d \exp (ϵ_{1} V_{1}) (c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5}) \\ = c_{1} A d \exp (ϵ_{1} V_{1}) V_{1} + c_{2} A d \exp (ϵ_{1} V_{1}) V_{2} \\ + c_{3} A d \exp (ϵ_{1} V_{1}) V_{3} + c_{4} A d \exp (ϵ_{1} V_{1}) V_{4} \\ + c_{5} A d \exp (ϵ_{1} V_{1}) V_{5}, \\ = [\begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \end{matrix}] A_{1} {[\begin{matrix} V_{1} & V_{2} & V_{3} & V_{4} & V_{5} \end{matrix}]}^{Tr}, \end{array}

where superscript Tr indicates the transpose of the matrix, and

A_{1} = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ - ϵ_{1} & 0 & 0 & 0 & 1 \end{matrix}) .

(A2)

The adjoint action of

V_{2}

on

V

can be expressed with the help of Table II as

\begin{array}{l} A d \exp (ϵ_{2} V_{2}) (c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5}) \\ = [\begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \end{matrix}] A_{2} {[\begin{matrix} V_{1} & V_{2} & V_{3} & V_{4} & V_{5} \end{matrix}]}^{Tr}, \end{array}

where the matrix

A_{2} = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) .

(A3)

The adjoint action of

V_{3}

on

V

can be expressed with the help of Table II as

\begin{array}{l} A d \exp (ϵ_{3} V_{3}) (c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5}) \\ = [\begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \end{matrix}] A_{3} {[\begin{matrix} V_{1} & V_{2} & V_{3} & V_{4} & V_{5} \end{matrix}]}^{Tr}, \end{array}

where the matrix

A_{3} = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & e^{ϵ_{3}} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) .

(A4)

The adjoint action of

V_{4}

on

V

can be expressed with the help of Table II as

\begin{array}{l} A d \exp (ϵ_{4} V_{4}) (c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5}) \\ = [\begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \end{matrix}] A_{4} {[\begin{matrix} V_{1} & V_{2} & V_{3} & V_{4} & V_{5} \end{matrix}]}^{Tr}, \end{array}

where the matrix

A_{4} = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & - ϵ_{4} & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - ϵ_{4} & 1 \end{matrix}) .

(A5)

The adjoint action of

V_{5}

on

V

can be expressed with the help of Table II as

\begin{array}{l} A d \exp (ϵ_{5} V_{5}) (c_{1} V_{1} + c_{2} V_{2} + c_{3} V_{3} + c_{4} V_{4} + c_{5} V_{5}) \\ = [\begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \end{matrix}] A_{5} {[\begin{matrix} V_{1} & V_{2} & V_{3} & V_{4} & V_{5} \end{matrix}]}^{Tr}, \end{array}

where the matrix

A_{5} = (\begin{matrix} e^{ϵ_{1}} & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & e^{ϵ_{5}} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) .

(A6)

We can then express the general adjoint matrix D with the help of the matrices A₁, A₂, A₃, A₄, and A₅ from Eqs. – as

D = A_{1} A_{2} A_{3} A_{4} A_{5} = (\begin{matrix} e^{ϵ_{1}} & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & - ϵ_{4} e^{ϵ_{5}} & 0 \\ 0 & 0 & 0 & e^{ϵ_{3} + ϵ_{5}} & 0 \\ - ϵ_{1} e^{ϵ_{1}} & 0 & 0 & - ϵ_{4} e^{ϵ_{5}} & 1 \end{matrix}) .

(A7)

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This work is licensed under a Creative Commons Attribution 4.0 International License .

Similarity solution via one-dimensional Lie subalgebras for shock waves in a self-gravitating and rotating gas with magnetic field and radiation heat flux

I. INTRODUCTION

II. GOVERNING EQUATIONS AND SHOCK JUMP CONDITIONS

III. LIE SYMMETRY ANALYSIS

IV. LIE BRACKETS AND OPTIMAL SYSTEM

A. Formation of one-dimensional subalgebra for optimal solution

V. SIMILARITY SOLUTION AND BOUNDARY CONDITION

A. Case I

B. Case II

C. Case III

D. Case IV

E. Case V

VI. NUMERICAL SOLUTION AND DISCUSSION OF RESULTS

A. Effects of an increase in the value of the gravitational parameter G_p or G_e

B. Effects of an increase in the value of the rotational parameter R_p or R_e

C. Effects of an increase in the magnetic field strength $A_{M}^{- 2}$

D. Effects of an increase in the value of the radiative heat transfer parameter Γ_p or Γ_e

E. Effects of an increase in the value of the adiabatic exponent γ

F. Effects of an increase in the value of the similarity index θ for the PLSP

VII. CONCLUSIONS

A. Role of shock waves in star formation

B. Shock wave formation in supernova explosions

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX: DERIVATION OF OPTIMAL SYSTEM FOR SUBALGEBRAS

REFERENCES

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Similarity solution via one-dimensional Lie subalgebras for shock waves in a self-gravitating and rotating gas with magnetic field and radiation heat flux

I. INTRODUCTION

II. GOVERNING EQUATIONS AND SHOCK JUMP CONDITIONS

III. LIE SYMMETRY ANALYSIS

IV. LIE BRACKETS AND OPTIMAL SYSTEM

A. Formation of one-dimensional subalgebra for optimal solution

V. SIMILARITY SOLUTION AND BOUNDARY CONDITION

A. Case I

B. Case II

C. Case III

D. Case IV

E. Case V

VI. NUMERICAL SOLUTION AND DISCUSSION OF RESULTS

A. Effects of an increase in the value of the gravitational parameter Gp or Ge

B. Effects of an increase in the value of the rotational parameter Rp or Re

C. Effects of an increase in the magnetic field strength AM−2

D. Effects of an increase in the value of the radiative heat transfer parameter Γp or Γe

E. Effects of an increase in the value of the adiabatic exponent γ

F. Effects of an increase in the value of the similarity index θ for the PLSP

VII. CONCLUSIONS

A. Role of shock waves in star formation

B. Shock wave formation in supernova explosions

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX: DERIVATION OF OPTIMAL SYSTEM FOR SUBALGEBRAS

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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pubs.aip.org

Connect with AIP Publishing

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A. Effects of an increase in the value of the gravitational parameter G_p or G_e

B. Effects of an increase in the value of the rotational parameter R_p or R_e

C. Effects of an increase in the magnetic field strength $A_{M}^{- 2}$

D. Effects of an increase in the value of the radiative heat transfer parameter Γ_p or Γ_e