# Similarity method for the study of strong shock waves in magnetogasdynamics

- Rajan Arora
^{1}, - Sanjay Yadav
^{1}and - Mohd Junaid Siddiqui
^{2}Email author

**2014**:142

https://doi.org/10.1186/s13661-014-0142-2

© Arora et al.; licensee Springer 2014

**Received: **9 May 2013

**Accepted: **27 May 2014

**Published: **11 July 2014

## Abstract

In this paper, a non-dimensional unsteady adiabatic flow of a plane or cylindrical strong shock wave propagating in plasma is studied. The plasma is assumed to be an ideal gas with infinite electrical conductivity permeated by a transverse magnetic field. A self-similar solution of the problem is obtained in terms of density, velocity and pressure in the presence of magnetic field. We use the method of Lie group invariance to determine the class of self-similar solutions. The arbitrary constants, occurring in the expressions of the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. A particular case of the collapse of an imploding shock is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of flow variables. Our results are found in good agreement with the known results. All computational work is performed by using software package MATHEMATICA.

## Keywords

## Introduction

The spread of shock waves under the control of strong magnetic field is a problem of great interest to researchers in a variety of fields such as nuclear science, geophysics, plasma physics and astrophysics. Hunter [[1]], Guderley [[2]], Greifinger and Cole [[3]] studied the problem of blast wave propagation in homogeneous and inhomogeneous media. Most recently, van Dyke and Guttmann [[4]], Sharma and Radha [[5], [6]], Madhumita and Sharma [[7]], Pandey *et al.* [[8]], Sharma and Arora [[9]], Arora *et al.* [[10], [11]] presented high accuracy results and alternative approaches for the investigation of blast wave by using the self-similar solutions method. In the same decade, a number of analytical solutions for the blast wave propagations have been obtained by Sachdev [[12]], Chisnell [[13]] and Singh *et al*. [[14], [15]]. Chisnell [[13]] provided analytical solutions to the problem of converging shock waves by using the singular points method. Singh *et al*. [[14], [15]] used the method of Lie group of transformations to obtain an exact solution for unsteady equation of non-ideal gas and magnetogasdynamics.

The magnetic fields have important roles in a variety of astrophysical situations. Complex filamentary structures in molecular clouds, shapes and the shaping of planetary nebulae, synchrotron radiation from supernova remnants, magnetized stellar winds, galactic winds, gamma-ray bursts, dynamo effects in stars, galaxies and galaxy clusters as well as other interesting problems all involve magnetic fields. When the internal disturbances accompanied by an increase in pressure take place in the central region of a star, a shock wave is formed. It travels from the central region to the periphery and emerges at the surface of the star. In the present paper, we consider the problem of propagation of a one-dimensional unsteady flow of an inviscid ideal gas permeated by a transverse magnetic field with infinite electrical conductivity as it approaches the surface of a star. It is assumed that mass density distribution in the medium follows a power law of the radial distance from the point of explosion.

In flows with imploding shocks, conditions of very high temperature and pressure can be produced near the center (axis) of implosion on account of the self-amplifying nature of imploding shocks. As a result of high temperatures attained by gases in motion, the effects of nonequilibrium thermodynamics on the dynamic motion of a converging shock wave can be important.

In this paper, we use the method of Lie group invariance under infinitesimal point transformations [[16]–[18]] to study the problem of propagation of strong shock waves in a radiating and electrically conducting gas permeated by a transverse magnetic field. The arbitrary constants, occurring in the expressions for the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. The Lie symmetry approach does not necessarily take into account the boundary and initial conditions unless the same are invariant under the change of variable transformations.

## Basic equations

*p*is the gas pressure,

*ρ*is the density,

*u*is the velocity,

*γ*is the constant specific heat ratio,

*x*is the single spatial co-ordinate,

*t*is the time,

*h*is the magnetic pressure defined by $h=\mu {H}^{2}/2$ with

*μ*as magnetic permeability and

*H*is the transverse magnetic field; $m=0\text{and}1$ correspond to planar and cylindrical symmetry, respectively, and the non-numeric subscripts denote the partial differentiation with respect to the indicated variable unless stated otherwise. The equation of state is taken to be of the form

*T*is the temperature,

*R*is the gas constant, ${p}_{0}(x)$, ${h}_{0}(x)$ and ${\rho}_{0}(x)$ are some functions of

*x*. The Rankine-Hugoniot jump conditions for the strong shocks are as follows (Whitham [[19]]):

## Similarity analysis by invariance groups

*χ*,

*ψ*,

*U*,

*S*,

*P*and

*E*are to be determined in such a way that system (1) of partial differential equations together with conditions (3) and (4) is invariant with respect to transformations (5); the entity

*ε*is so small that its square and higher powers may be neglected. The existence of such a group allows the number of independent variables in the problem to be reduced by one, and thereby allowing system (1) to be replaced by a system of ordinary differential equations.

In continuation, we shall use the summation convention and introduce the notation ${x}_{1}=t$, ${x}_{2}=x$, ${u}_{1}=\rho $, ${u}_{2}=u$, ${u}_{3}=p$, ${u}_{4}=h$ and ${p}_{j}^{i}=\frac{\partial {u}_{i}}{\partial {x}_{j}}$, where $i=1,2,3,4$ and $j=1,2$.

*ψ*,

*χ*,

*S*,

*U*,

*P*and

*E*. This system, which is called the system of determining equations, is given as follows:

*a*,

*b*,

*c*, and ${k}_{1}$ are the arbitrary constants.

## Self-similar solutions

The arbitrary constants, which appear in the expressions for the infinitesimals of the invariant group of transformations, yield different cases of possible solutions as discussed below.

*ρ*,

*u*,

*p*and

*h*readily follow from the invariant surface condition which yields

*ξ*, which is determined as follows:

*A*is a dimensional constant whose dimensions are obtained by the similarity exponent

*δ*. Since the shock must be a similarity curve, it may be normalized to be at $\xi =1$. The shock path

*X*and the shock velocity

*V*are, then, given by

*ξ*is the dimensionless similarity variable, $X(t)$ is the shock location and

*V*is the shock velocity given by

*A*as a dimensional constant. Substituting (30) in the equations in system (1) and using (31), we obtain the following system of ordinary differential equations in ${S}^{\ast}$, ${U}^{\ast}$, ${P}^{\ast}$ and ${E}^{\ast}$, which on suppressing the asterisk sign becomes:

## Imploding shocks

*t*is taken to be the instant at which the shock reaches the axis so that $t\le 0$ in (28). Therefore, the definition of the similarity variable is slightly modified by setting

*x*are bounded, but with $t=0$ and finite

*x*, $\xi =\mathrm{\infty}$. In order for the quantities

*u*,

*p*,

*ρ*and

*h*to be bounded when $t=0$ and

*x*is finite, we must have the following boundary conditions at $\xi =\mathrm{\infty}$:

*C*and the column vector

*B*can be identified by observing system (28). In system (28), there is an unknown parameter

*δ*, which cannot be obtained from an energy balance or the dimensional considerations; it is computed only by solving a non-linear eigenvalue problem for a system of ordinary differential equations. The range of similarity variable is $1\le \xi <\mathrm{\infty}$ for the implosion problem, and system (44) can be solved for the derivatives ${U}^{\prime}$, ${S}^{\prime}$, ${P}^{\prime}$ and ${E}^{\prime}$ in the following form:

*C*, is given by

*k*th column by the column vector

*B*. It can be observed that $U<\xi $ in the interval $[1,\mathrm{\infty})$, while Δ is positive at $\xi =1$ and negative at $\xi =\mathrm{\infty}$ indicating that there exists a $\xi \in [1,\mathrm{\infty})$ at which Δ vanishes, and consequently the solutions become singular. In order to get a non-singular solution of (44) in the interval $[1,\mathrm{\infty})$, we select the value of the exponent

*δ*such that Δ vanishes only at the points where the determinant ${\mathrm{\Delta}}_{1}$ is zero too. It can be checked that at points where Δ and ${\mathrm{\Delta}}_{1}$ vanish, the determinants ${\mathrm{\Delta}}_{2}$, ${\mathrm{\Delta}}_{3}$ and ${\mathrm{\Delta}}_{4}$ also vanish simultaneously. To find the exponent

*δ*in such a manner, we introduce the variable

*Z*as follows:

## Numerical results and discussion

*δ*, and compute the values of

*U*,

*S*,

*P*,

*E*and ${\mathrm{\Delta}}_{1}$ at $Z=0$; the value of

*δ*is corrected by successive approximations in such a way that for these values, the determinant ${\mathrm{\Delta}}_{1}$ vanishes at $Z=0$. The values of the similarity exponent

*δ*, obtained from the numerical calculations for different values of ${C}_{0}$,

*m*and

*θ*are given in Table 1.

**Similarity exponent**
δ
**for planar and cylindrically symmetric flows and the density exponent**
θ
**with**
$\mathit{\gamma}\mathbf{=}\mathbf{1.4}$

M | θ | ${\mathit{C}}_{\mathbf{0}}$ | Computedδ | Guderley [[2]]δ | % Error |
---|---|---|---|---|---|

1 | 0.5 | 0.00 | 0.72855 | 0.74000 | 1.54% |

1 | 0.5 | 0.02 | 0.72810 | 0.75000 | 2.92% |

1 | 0.5 | 0.05 | 0.85610 | 0.81000 | 5.69% |

1 | 1 | 0.00 | 0.64108 | 0.65000 | 1.37% |

1 | 1 | 0.02 | 0.64105 | 0.65310 | 1.85% |

1 | 1 | 0.05 | 0.64098 | 0.65510 | 2.16% |

0 | 1 | 0.00 | 0.75675 | 0.71400 | 5.99% |

0 | 1 | 0.02 | 0.75748 | 0.71100 | 6.54% |

0 | 1 | 0.05 | 0.75869 | 0.75000 | 1.16% |

0 | 0.5 | 0.00 | 0.85382 | 0.82500 | 3.49% |

0 | 0.5 | 0.02 | 0.85428 | 0.80350 | 6.32% |

0 | 0.5 | 0.05 | 0.85500 | 0.86000 | 0.58% |

The typical flow profiles show that the density, pressure, temperature and velocity increase behind the shock wave with the increase in the value of *θ*; this is because a gas particle passing through the shock is subjected to a shock compression. Indeed, this increase in pressure and density behind the shock may also be attributed to the geometrical convergence or area contraction of the shock wave. Figures 1 and 2 show that the growth of the flow variables is slower in cylindrical symmetry as compared with that in planar symmetry. Figures 1 and 2 also confirm the generation of higher pressure near the axis of symmetry, *i.e.*, near $\xi =\mathrm{\infty}$. The difference between flow profiles in cylindrical waves and those in planar waves is attributed to the fact that for planar waves, the flow distribution is relatively less influenced by the interaction between the gasdynamic phenomena as compared to cylindrical waves.

## Conclusions

In the present investigation a self-similar method is used to study the flow pattern behind an exponential shock driven by a piston in ideal magnetogasdynamics. The general behavior of density, velocity and pressure profiles remains unaffected due to presence of magnetic field in ideal gas. However, there is a decrease in values of density, velocity and pressure in the case of magnetogasdynamics as compared to non-magnetic case. It may be noted that the effect of magnetic field on the flow pattern is more significant in the case of isothermal flow as compared to that of adiabatic flow.

## Declarations

## Authors’ Affiliations

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## Copyright

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