# Flow behind an exponential shock wave in a rotational axisymmetric perfect gas with magnetic field and variable density

- G. Nath
^{1}Email author and - P. K. Sahu
^{1}

**Received: **19 May 2016

**Accepted: **22 August 2016

**Published: **8 September 2016

## Abstract

A self-similar model for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential shock wave driven out by a cylindrical piston moving with time according to an exponential law in an ideal gas in the presence of azimuthal magnetic field and variable density is discussed in a rotating atmosphere. The ambient medium is assumed to possess radial, axial and azimuthal component of fluid velocities. The initial density, the fluid velocities and magnetic field of the ambient medium are assumed to be varying with time according to an exponential law. The gas is taken to be non-viscous having infinite electrical conductivity. Solutions are obtained, in both the cases, when the flow between the shock and the piston is isothermal or adiabatic by taking into account the components of vorticity vector. The effects of the variation of the initial density index, adiabatic exponent of the gas and the Alfven-Mach number on the flow-field behind the shock wave are investigated. It is found that the presence of the magnetic field have decaying effects on the shock wave. Also, it is observed that the effect of an increase in the magnetic field strength is more impressive in the case of adiabatic flow than in the case of isothermal flow. The assumption of zero temperature gradient brings a profound change in the density, non-dimensional azimuthal and axial components of vorticity vector distributions in comparison to those in the case of adiabatic flow. A comparison is made between isothermal and adiabatic flows. It is obtained that an increase in the initial density variation index, adiabatic exponent and strength of the magnetic field decrease the shock strength.

## Keywords

## Background

The formulation of self-similar problems and examples describing adiabatic motion of non-rotating gas models of stars are discussed by Sedov (1959), Zel’Dovich and Raizer (1967), Lee and Chen (1968) and Summers (1975). The problem of propagation of magneto-gasdynamic shock waves in a rotating interplanetary atmosphere assumes special significance in the study of astrophysical phenomena. The experimental studies and astrophysical observations show that the outer atmosphere of the planets or stars rotates due to rotation of the planets or stars. Macroscopic motion with supersonic speed occurs in an interplanetary atmosphere with rotation and shock waves are generated. Further, the interplanetary magnetic field is connected with the rotation of the sun which implies that a large scale of magnetic field might appear in the rapidly rotating stars. Therefore, the rotation of planets or stars considerably affects the process happening in their outer layers, thus question connected with the explosions in rotating gas atmospheres are of definite astrophysical interest. Chaturani (1971) obtained the solutions for the propagation of cylindrical shock wave through a gas having solid body rotation by a similarity method adopted by Sakurai (1956). Nath et al. (1999) obtained the similarity solutions for the flow behind the spherical shock waves propagating in a non-uniform rotating interplanetary atmosphere with increasing energy. A theoretical model of propagation of strong spherical shock waves in a self-gravitating atmosphere with radiation flux in presence of a magnetic field and considering the medium behind the shock to be rotating but neglecting the rotation of the undisturbed medium was studied by Ganguly and Jana (1998). The self-similar solution for adiabatic flow headed by a magnetogasdynamic cylindrical shock wave in a rotating non-ideal gas is obtained by Vishwakarma et al. (2007).

Sedov (1959) (see Rao and Ramana 1976) indicated that a limiting case of a self-similar flow-field with a power-law shock is the flow-field formed with an exponential shock. Rao and Ramana (1976) obtained approximate analytical solutions for the problem of unsteady self-similar motion of a perfect gas displaced by a piston according to an exponential law.

The purpose of present work is to obtain the self-similar solutions for the flow behind the strong cylindrical shock wave generated by a moving piston in a rotational axisymmetric flow of a gas with variable density, variable azimuthal and axial fluid velocities under isothermal and adiabatic flow conditions (Levin and Skopina 2004; Nath 2010, 2011).

Rao and Ramana (1976), Vishwakarma and Nath (2007) and Nath (2014, 2015) have studied the problem which we have considered in the present study by taking initial density constant without considering the effect of magnetic field in rotating or non-rotating medium. Singh et al. (2011) have considered same problem by taking initial magnetic field and initial density constant with the assumption that the gas to be non-ideal and medium to be non-rotating, whereas we have considered the medium to be rotating and the initial magnetic field and initial density decreasing exponentially. Shock waves through a variable-density medium have been treated by Sakurai (1956), Rogers (1957), Sedov (1959), Rosenau and Frankenthal (1976), Nath et al. (1999), Vishwakarma and Yadav (2003), Nath (2011) and others. Their results are more applicable to the shock formed in the deep interior of stars. Also, the material within star occurs within a strong magnetic field and the interplanetary magnetic field is connected with the rotation of the sun which implies that a large scale of magnetic field might appear in the rapidly rotating stars. Thus our problem is more realistic than the previous works corresponding to the physical phenomenon.

The analysis of the flow field in the region between the shock and the piston are presented for both the cases of adiabatic and isothermal flows. The isothermal flow assumption is physically realistic, when radiation heat transfer effects are implicitly present. The temperature behind the shock, as the shock propagates, increases and becomes very large so that there is intense transfer of energy by radiation and when intense heat exchange between particles of gas takes place, we may assume that there is no temperature gradient throughout the flow field, i.e., \(\dfrac{\partial T}{\partial r}\rightarrow 0\) . Therefore, the temperature in the flow field depends only on time t and not on the distance r from the center of the explosion, i.e., \(T=T(t)\), and the flow is isothermal as describe by Sedov (1959), Laumbach and Probstein (1970), Sachdev and Ashraf (1971) and Zhuravskaya and Levin (1996). This assumption on the character of the flow corresponds to the beginning of a very strong explosion (for example: underground, volcanic and cosmic explosions, coal-mine blasts) when the gas temperature is extremely high. A detailed mathematical theory of one-dimensional isothermal blast waves in a magnetic field was developed by Lerche (1979, 1981). With this assumption, we obtain the solutions in “Equations of motion and boundary conditions–isothermal flow” and “Self-similarity transformations” sections. In “Adiabatic flow” section, we present the solutions for the flow taken to be adiabatic.

The effects of variation of the Alfven-Mach number, the initial density variation index and the ratio of the specific heat of the gas on the shock strength and flow variables are investigated. It is found that the assumption of zero temperature gradient brings a profound change in the distribution of density, non-dimensional azimuthal and axial components of vorticity vectors as compared to those of the adiabatic case. A comparison between the obtained solutions and the existing solutions of Rao and Ramana (1976) is made in non-magnetic case. Also, a comparison between the solutions in the case of isothermal and adiabatic flows is made. Further, it is shown that the consideration of zero temperature gradient and an increase in the strength of ambient magnetic field, the initial density variation index or adiabatic exponent of the gas decrease the shock strength and widen the disturbed region between the shock and the piston. Effects of gravitation and viscosity are not taken into account.

## Equations of motion and boundary conditions–isothermal flow

*p*, \(\rho\),

*h*and

*T*are the pressure, the density, the azimuthal magnetic field and the temperature; \(\mu\) is the magnetic permeability. Here the electrical conductivity of the gas is assumed to be infinite.

*R*is the gas constant. The gas constant

*R*and the temperature

*T*are assumed to obey the thermodynamic relations \(R = C_{p} - C_{v}\) and \(e_{m} = C_{v} T\), where \(C_{v} = \dfrac{R}{\gamma - 1}\) is the specific heat at constant volume and \(e_{m}\) being the internal energy per unit mass of the gas can be written as

*C*,

*E*, \(h_{0}\), \(\sigma\), \(\delta\), \(\alpha\) and \(\lambda\) are the dimensional constants, and the subscript ‘a’ refers to the conditions immediately ahead of the shock front.

*V*(Laumbach and Probstein 1970; Vishwakarma and Nath 2007; Nath 2011).

## Self-similarity transformations

*U*, \(\phi\) ,

*W*,

*P*,

*G*and

*H*are function of \(\xi\) only.

*B*and \(\eta\) can be obtained from Eqs. (1), (2) and (29) as

## Adiabatic flow

In this section, we present the self similar solution for the adiabatic flow behind a strong shock driven out by a cylindrical piston moving according to the exponential law (1), in the case of ideal gas with magnetic field. The strong shock conditions, which serve as the boundary conditions for the problem will be same as the shock conditions (25) in the case of isothermal flow.

*h*with their respective values at the shock, we obtain

## Results and discussion

Variation of the density ratio \(\beta \left( =\dfrac{\rho _{a}}{\rho _{n}}\right)\) across the shock front and the position of the piston surface \(\xi _{p}\) for different values of \(M_{A}^{-2}\), \(\gamma\) and \(\dfrac{\sigma }{i}\)

\(M_{A}^{-2}\) | \(\gamma\) | \(\beta\) | Position of the piston surface \(\xi _{p}\) | |||
---|---|---|---|---|---|---|

Isothermal flow | Adiabatic flow | |||||

\(\dfrac{\sigma }{i} = 1\) | \(\dfrac{\sigma }{i} = 1.5\) | \(\dfrac{\sigma }{i} = 1\) | \(\dfrac{\sigma }{i} = 1.5\) | |||

0 | \(\dfrac{4}{3}\) | 0.142857 | 0.899886 | 0.834074 | 0.956562 | 0.951338 |

\(\dfrac{5}{3}\) | 0.25000 | 0.822819 | 0.690086 | 0.917366 | 0.908795 | |

0.01 | \(\dfrac{4}{3}\) | 0.165804 | 0.892594 | 0.835098 | 0.934578 | 0.925728 |

\(\dfrac{5}{3}\) | 0.261039 | 0.823930 | 0.705373 | 0.904250 | 0.892942 | |

0.1 | \(\dfrac{4}{3}\) | 0.296396 | 0.840903 | 0.800793 | 0.856906 | 0.836709 |

\(\dfrac{5}{3}\) | 0.34838 | 0.808796 | 0.744513 | 0.842153 | 0.820695 |

Figures 1 and 2 show the variation of the flow variables \(\dfrac{u}{u_{n}}, \dfrac{v}{v_{n}},\dfrac{w}{w_{n}} , \dfrac{\rho }{\rho _{n}}, \dfrac{p}{p_{n}} ,\dfrac{h}{h_{n}}\), the non-dimensional azimuthal component of vorticity vector \(l_{\theta }\) and the non-dimensional axial component of vorticity vector \(l_{z}\) against the similarity variable \(\xi\) at various values of the parameters \(M_{A}^{-2}\), \(\gamma\) and \(\dfrac{\sigma }{i}\) in the isothermal and adiabatic cases respectively.

Figures 1e and 2f show that the reduced azimuthal magnetic field \(\dfrac{h}{h_{n}}\) increases but in the case of isothermal flow it decreases after attaining the maximum value. Figures 1f and 2g show that the reduced azimuthal component of vorticity vector \(l_{\theta }\) decreases; whereas it increases after attaining a minima for \(\dfrac{\sigma }{i} = 1.5\) in case of isothermal flow.

- (i)
to increase the value of \(\beta\) i.e. to decrease the shock strength (see Table 1);

- (ii)
to decrease \(\xi _{p}\) ingeneral (except the case when \(\dfrac{\sigma }{i} = 1.5\), \(\gamma = \dfrac{5}{3}\) for isothermal flow), i.e. to increase the distance of the piston from the shock front. Physically it means that the flow-field behind the shock become somewhat rarefied which is same as in (i) above (see Table 1);

- (iii)
the flow variables \(\dfrac{u}{u_{n}}\), \(\dfrac{\rho }{\rho _{n}}\), \(\dfrac{p}{p_{n}}\) and \(\dfrac{h}{h_{n}}\) increase in case of isothermal flow, but these flow variables decrease ingeneral in case of adiabatic flow (see Figs. 1a, d, e, 2a, d–f)

- (iv)
to increase the flow variables \(\dfrac{v}{v_{n}}\) and \(\dfrac{w}{w_{n}}\) ingeneral (see Figs. 1b, c, 2b, c);

- (v)
the non-dimensional azimuthal component of vorticity vector \(l_{\theta }\) increases near shock and decreases near piston, but it increases at any point in the flow field behind the shock in the case of isothermal flow when \(\dfrac{\sigma }{i} = 1\), \(\gamma = \dfrac{4}{3}\); and in the case of adiabatic flow for all values of the parameters. (see Figs. 1f, 2g);

- (vi)
the non-dimensional axial component of vorticity vector \(l_{z}\) decreases ingeneral; whereas it decreases near the shock and increases near the piston in the case of isothermal flow (see Figs. 1g, 2h).

- (i)
the value of \(\beta\) increased i.e. the shock strength is decreased (see Table 1);

- (ii)
the distance of the piston from the shock front is increased. This shows the same result as given in (i) above, i.e. there is a decrease in the shock strength (see Table 1);

- (iii)
to increase the flow variables \(\dfrac{u}{u_{n}}\), \(\dfrac{v}{v_{n}}\) and \(\dfrac{w}{w_{n}}\), but to decrease the flow variables \(\dfrac{\rho }{\rho _{n}}\) at any point in the flow-field behind the shock front (see Figs. 1a–d, 2a–d);

- (iv)
to decrease the reduced pressure \(\dfrac{p}{p_{n}}\); whereas it increases in the case of adiabatic flow with magnetic field (i.e. \(M_{A}^{-2}\ne 0\)) (see Figs. 1d, 2e);

- (v)
to decrease the reduced azimuthal magnetic field \(\dfrac{h}{h_{n}}\) for \(M_{A}^{-2}=0.01\) and to increase it for \(M_{A}^{-2} = 0.1\) in case of adiabatic flow; whereas it decreases near shock and increases near piston in case of isothermal flow (see Figs. 1e, 2f);

- (vi)
to increase the non-dimensional azimuthal component of vorticity vector \(l_{\theta }\); whereas in the case of isothermal flow it increases near shock and decreases near piston when \(\dfrac{\sigma }{i} = 1.5\) (see Figs. 1f, 2g);

- (vii)
to decrease the non-dimensional axial component of vorticity vector \(l_{z}\) near shock and increase near piston; but it decreases at any point in the flow field behind the shock in the case of adiabatic flow, when \(M_{A}^{-2} = 0\) (see Figs. 1g, 2h).

- (i)
to decrease \(\xi _{p}\) i.e. to decrease the shock strength (see Table 1);

- (ii)
to decrease the flow variables \(\dfrac{u}{u_{n}}\), but to increase the flow variables \(\dfrac{v}{v_{n}}\), \(\dfrac{w}{w_{n}}\) and \(l_{\theta }\) at any point in the flow-field behind the shock front (see Figs. 1a–c, f, 2a–c, g;

- (iii)
to decrease \(\dfrac{\rho }{\rho _{n}}\); whereas in the case of isothermal flow it increases near shock and decreases near piston when \(M_{A}^{-2}=0.1\) (see Figs. 1d, 2d);

- (iv)
to increase \(\dfrac{p}{p_{n}}\) when \(M_{A}^{-2} \ne 0\) for adiabatic flow and when \(M_{A}^{-2} = 0.1\) for isothermal flow; whereas it decreases when \(M_{A}^{-2}= 0, 0.01\) for isothermal flow and when \(M_{A}^{-2} = 0\) for adibatic flow (see Figs. 1d, 2e);

- (v)
to decrease \(\dfrac{h}{h_{n}}\) in the case of adiabatic flow, but in the case of isothermal flow it decreases near shock and increases near piston (see Figs. 1e, 2f);

- (vi)
to decrease \(l_{z}\) near shock and increases near piston, but in the case of adiabatic flow it decreases at any point in the flow field in the absence of magnetic field (i.e. when \(M_{A}^{-2} = 0\)) (see Figs. 1g, 2h).

## Conclusion

- (i)
The distance between shock and piston increases (i.e. shock strength decreases) with an increase in the strength of the ambient magnetic field \(M_{A}^{-2}\), the adiabatic exponent \(\gamma\) or the initial density variation index \(\dfrac{\sigma }{i}\).

- (ii)
An increase in the value of \(\dfrac{\sigma }{i}\) decrease the flow variables \(\dfrac{u}{u_{n}}\), \(\dfrac{\rho }{\rho _{n}}\), \(\dfrac{h}{h_{n}}\), \(l_{z}\) ingeneral; whereas in the case of the flow variables \(\dfrac{v}{v_{n}}\), \(\dfrac{w}{w_{n}}\) and \(l_{\theta }\) the reverse behaviour is observed.

- (iii)
An increase in the value of initial density variation index \(\dfrac{\sigma }{i}\) or adiabatic exponent of the gas \(\gamma\) have same behaviour on the flow variables \(\dfrac{v}{v_{n}} , \dfrac{w}{w_{n}} , l_{z}\) and the shock strength; whereas these parameters have opposite behaviour on the flow variable \(\dfrac{u}{u_{n}}\). Also, by increasing the value of \(\dfrac{\sigma }{i}\) or \(\gamma\) the flow variables \(\dfrac{\rho }{\rho _{n}}\), \(\dfrac{p}{p_{n}}\) and \(\dfrac{h}{h_{n}}\) show the same behaviour (except the cases when \(M_{A}^{-2}= 0.1\) ).

- (iv)
An increase in \(\gamma\) or \(M_{A}^{-2}\) increases the radial velocity \(\dfrac{u}{u_{n}}\) in case of isothermal flow; whereas in the case of adiabatic flow it decreases with an increase in \(M_{A}^{-2}\) and increases with an increase in \(\gamma\). Also, the flow variables \(\dfrac{v}{v_{n}}\) and \(\dfrac{w}{w_{n}}\) increase with an increase in \(M_{A}^{-2}\) or \(\gamma\), when \(\dfrac{\sigma }{i} = 1\).

- (v)
The novel applications of this study include analysis of data from exploding wire experiments in conducting medium and cylindrically symmetric hypersonic flow problems associated with meteors or reentry vehicles (Hutchens 1995). Also, the solutions obtained can be used to interpret measurements carried out by space craft in the solar wind and in neighbourhood of the Earths magnetosphere.

## Declarations

### Authors’ contributions

GN has suggested this research problem. PKS has formulated the problem and did all the calculations part and found out results. GN and PKS were involved in drafting manuscript. Both authors read and approved the final manuscript.

### Competing interests

Both authors declare that they have no competing interests.

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## Authors’ Affiliations

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