Self-similar Cylindrical Ionizing Shock Waves in a Rotational ...

Universal Journal of Applied Mathematics 1(2): 44-54, 2013DOI: 10.13189/ujam.2013.010203 http://www.hrpub.org

Self-similar Cylindrical Ionizing Shock Waves ina Rotational Axisymmetric Non-ideal Gas with

Radiation Heat Flux

K. K. Singh∗, Bineeta Nath

Department of Mathematics, North-Eastern Hill University, Shillong-793022, India

∗Corresponding Author: [email protected]

Copyright c⃝2013 Horizon Research Publishing All rights reserved.

Abstract Similarity solutions are obtained forone-dimensional adiabatic flow behind a gas-ionizingcylindrical shock wave propagating in a rotationalaxisymetric non-ideal gas with radiation heat flux, inpresence of an azimuthal magnetic field. The electricalconductivity in the medium ahead of the shock isassumed to be negligible, which becomes infinitely largeafter passage of the shock . The ambient medium isconsidered to have variable axial velocity component inaddition to the variable azimuthal velocity. In orderto obtain the similarity solutions, the initial density ofthe medium is assumed to be constant and the initialangular velocity to be obeying a power law and tobe decreasing as the distance from the axis increases. Effects of an increase in the value of the parameterof non-idealness of the gas, in the value of radiationparameter and in the value of the index for variation ofazimuthal velocity of the ambient medium ( or in thevalue of the index for variation of the ambient magneticfield ) on the shock propagation are investigated. It isobserved that the non-idealness of the gas or radiationheat-flux has decaying effect on the shock wave.

Keywords Ionizing Shock Wave, Non-Ideal Gas,Similarity Solutions, Radiation Heat-Flux, RotatingMedium

1 Introduction

At low temperature such as T < 104 0K and moderatepressure such as one atmosphere, the thermal radiationeffects are negligible in the analysis of fluid dynamics.However, in this space age, we are concerned with manytechnological developments in space hypersonic flight,fission and fusion reactions etc.in which the tempera-ture is very high and density of the gas is very low. As aresult, the thermal radiation effects are no longer negligi-ble. In gasdynamics, if the effects of radiation are takeninto account, the basic non-linear equations are of a verycomplicated form and therefore it is essential to estab-lish approximations which are physically accurate and

can afford considerable simplifications. The problemof interaction of radiation with gasdynamics has beenstudied by many authors by using the method of self-similarity developed by Sedov[1]. Marshak [2] studiedthe effect of radiation on the shock propagation by intro-ducing the radiation diffusion approximation. He solvedboth the cases of constant density and constant pres-sure fields without invoking conditions of self-similarity.Using the same mode of radiation, Elliott [3] discussedthe conditions leading to self-similarity with a specifiedfunctional form of the mean free-path of radiation andobtained a solution for self-similar explosion. Wang [4],Helliwell [5] and Nicastro [6] treated the problems ofradiating walls, either stationary or moving, generatingshock at the head of self-similar flow-fields. Ghoniem etal. [7] obtained a self-similar solution for spherical explo-sion taking into account the effects of both conductionand radiation in the two limits of Rosseland radiativediffusion and Plank radiative emission.

Since at high temperatures that prevail in the prob-lems associated with shock waves a gas is partially ion-ized, electromagnetic effects may also be significant. Acomplete analysis of such a problem should thereforeconsist of the study of the gasdynamic flow and the elec-tromagnetic and radiation fields simultaneously. A num-ber of problems relating to shock wave propagation ina perfect gas with radiative and magnetohydrodynamiceffects have been studied (see, for example, Singh andMishra [8], Vishwakarma et al. [9], Singh et al. [10],Ganguly and Jana [11], Nath et al. [12]).

The formulation of self-similar problems and exam-ples describing the adiabatic motion of non-rotating gasmodels of stars, are considered by Sedov[1], Zel’dovichand Raizer[13], Lee and Chen[14] and Summers[15]. Ro-tation of stars significantly affects the process takingplace in their outer layers. Therefore question connectedwith the explosions in rotating gas atmospheres are ofdefinite astrophysical interest. Chaturani[16] studiedthe propagation of cylindrical shock waves through a gashaving solid body rotation, and obtained the solutionsby a similarity method adopted by Sakurai[17]. Nath etal.[12] obtained the similarity solutions for the flow be-hind spherical shock waves propagating in a non-uniform

Universal Journal of Applied Mathematics 1(2): 44-54, 2013 45

rotating interplanetary atmosphere with increasing en-ergy. Ganguly and Jana[11] studied the propagation of amagnetogasdynamic spherical shock wave (with infiniteshock - Mach number) in a self-gravitating and radiativeatmosphere. They, also, considered the medium behindthe shock to be rotating, but neglected the rotation ofthe undisturbed medium.

At extreme conditions that prevail in most of the prob-lems associated with shock waves, the assumption thatthe gas is ideal is no longer valid. In recent years, sev-eral studies have been performed concerning the problemof strong shock waves in non-ideal gases, in particular,by Anisimov and Spiner [18], Ranga Rao and Purohit[19], Roberts and Wu [20, 21], Vishwakarma and Mau-rya [22], Vishwakarma and Nath [23], Vishwakarma et al.[24], Singh et al. [25] and Vishwakarma and Pandey [26]among others. Anisimov and Spiner [18] investigatedthe problem of a point explosion in a non-ideal gas bytaking the equation of state in a simplified form, whichdescribes the behaviour of the medium satisfactorily atlow densities. Ranga Rao and Purohit [19] studied theself-similar flow of a non-ideal gas driven by an expand-ing piston and obtained solutions by taking the equationof state suggested by Anisimov and Spiner [18]. Robertsand Wu [20, 21] studied the problem of a spherical im-plosion by assuming that the gas obeys a simplified vander Waals equation of state. Vishwakarma and Mau-rya [22] analysed the propagation of a spherical shockwave in a non-ideal gas with the equation of state givenby Anisimov and Spiner [18], taking into account theeffects of radiation heat flux. Vishwakarma and Nath[23] obtained the similarity solutions for the flow be-hind an exponential shock in a non-ideal gas in boththe cases, when the flow behind the shock is isothermalor adiabatic. Vishwakarma et al. [24] adopted simpli-fied van der Waals model of a non-ideal gas to obtainthe self-similar solutions for the flow behind a magne-togasdynamic cylindrical shock wave propagating in arotating gas in the presence of an azimuthal magneticfield. Singh et al. [25] used self-similar method to studynumerically the flow pattern behind a strong cylindricalshock wave driven by a piston moving with time ac-cording to an exponential law in a non-ideal gas in thepresence of an axial magnetic field. Vishwakarma andPandey [26] adopted van der Waals model of a non-idealgas to obtain the self-similar solutions for the flow be-hind a magnetogasdynamic spherical shock wave propa-gating in a radiating gas in presence of an azimuthalmagnetic field. In all the last three papers, the au-thors have taken the electrical conductivity of the ini-tial medium and the medium behind the shock to beinfinite. But, in many practical cases the medium maybe of low conductivity which becomes highly conduct-ing due to passage of a strong shock. Such a shock waveis called a gas-ionizing shock or, simply ionizing shock.The propagation of a ionizing shock has been studiedby Greenspan[27], Greifinger and Cole[28], Christer[29],Rangarao and Ramana[30], Singh[31], Vishwakarma andSingh[32, 33] and Singh and Nath[34] in a perfect or non-ideal gas.

We, in the present work, extended the problem treatedby Vishwakarma and Singh [32] by taking the mediumto be a rotational axisymmetric non-ideal gas instead

of simply a non-ideal gas. The problem of line explo-sion with time dependent energy release and radiationin the presence of an azimuthal magnetic field is con-sidered due to its relevence to the experiments on pincheffect and exploding wires. A gas-ionizing cylindricalshock wave generated by line explosion, is propagatedin a rotational axisymmetric non-ideal gas which has avariable azimuthal fluid velocity together with a vari-able axial fluid velocity (Levin and Skopina [35]). Thecounter pressure (the pressure ahead of the shock) istaken into account. The radiation pressure and radia-tion energy are considered very small in comparison tomaterial pressure and energy, respectively, and there-fore only radiation flux is taken into account. A diffu-sion model for an optically thick grey gas is assumed.The fluid velocities and azimuthal magnetic field in theambient medium are assumed to obey power laws. It isexpected that such fluid velocity and magnetic field mayoccur in the atmospheres of rotating planets and stars.In order to obtain similarity solutions of the problemit is necessary to take the density of the ambient non-ideal gas to be a constant. Effects of an increase in thevalue of the index for variation of azimuthal velocity ofthe ambient medium (or index for variation of ambientmagnetic field), in the value of the parameter of the non-idealness of the gas and in the value of non-dimensionalradiation parameter on the flow-field behind the shockare investigated.

2 Basic Equations and BoundaryConditions

The fundamental equations governing the unsteady,adiabatic axisymmetric rotational flow of a perfectlyconducting and non-ideal gas in which the effects of ra-diation flux and azimuthal magnetic field may be sig-nificant, can be written as (Christer and Helliwell[36],Vishwakarma and Pandey[37], Gretler and Wehle[38],Levin and Skopina[35], Whitham[39])

∂ρ

∂t+ u

∂ρ

∂r+ ρ

∂u

∂r+

ρu

r= 0, (1)

∂u

∂t+ u

∂u

∂r+

1

ρ

(∂p∂r

+ µh∂h

∂r+

µh2

r

)− v2

r= 0, (2)

∂h

∂t+ u

∂h

∂r+ h

∂u

∂r= 0, (3)

∂v

∂t+ u

∂v

∂r+

uv

r= 0, (4)

∂w

∂t+ u

∂w

∂r= 0, (5)

∂e

∂t+ u

∂e

∂r− p

ρ2(∂ρ∂t

+ u∂ρ

∂r

)+

1

ρr

∂

∂r(Fr) = 0, (6)

where ρ, p, h are the density, pressure and azimuthalmagnetic field, respectively, u, v and w are the radial,azimuthal and axial components of the fluid velocity q⃗in the cylindrical coordinates (r, θ, z), µ is the magneticpermeability, r and t are the distance and time, F is theradiation heat flux, and e is the internal energy per unitmass. Also,

v = Ar, (7)

46Self-similar Cylindrical Ionizing Shock Waves in a Rotational Axisymmetric

Non-ideal Gas with Radiation Heat Flux

where A is the angular velocity of the medium at radialdistance r from the axis of symmetry. In this case thevorticity vector ζ⃗ = 1

2curl q⃗ has the components

ζr = 0, ζθ = −1

2

∂w

∂r, ζz =

1

2r

∂

∂r(rv). (8)

In most of the cases the propagation of shock wavesarises in extreme conditions under which the assump-tion that the gas is ideal is not a sufficiently accuratedescription. To discover how deviations from the idealgas can affect the solutions, we adopt a simple model.We assume that the gas obeys a simplified van der Waalsequation of state of the form (Roberts and Wu[20,21])

p =ΓρT

1− bρ, e = CvT =

p(1− bρ)

ρ(γ − 1), (9)

where Γ is the gas constant, Cv = Γγ−1 is the specific

heat at constant volume and γ is the ratio of specificheats. The constant b is the ‘ van der Waals excludedvolume’; it places a limit ρmax = 1

b , on the density ofthe gas.Assuming local thermodynamic equilibrium and a dif-

fusion model for an optically thick grey gas (Pomran-ing[40]), the differential approximation of the radiationtransport equation can be written in the following form

F = −cα

3

∂

∂r(σT 4), (10)

where 14σc is the Stefan-Boltzman constant, c the ve-

locity of light and α the Rosseland mean free path forradiation. The assumption of an optically thick greygas is physically consistent with the neglect of radiationpressure and radiation energy in the equation system(1)-(6)(Nicastro[6]).Following Wang[4], we take

α = α0ρνTκ, (11)

where α0, ν and κ are constants. It will be seen laterthat the exponents ν and κ must satisfy the similarityrequirements. The self-similarity condition puts no con-straints on specification of the density dependence onα.We assume that a cylindrical shock wave is propagat-

ing outwards from the axis of symmetry in a rotatingnon-ideal gas with constant initial density and negligi-ble electrical conductivity in presence of an azimuthalmagnetic field. The flow variables immediately ahead ofthe shock front are

u = 0, ρ = ρ1 = constant

h = h1 = h0Rn(n < 0) ( Rosenau[41] )

v = v1 = BRd, w = w1 = GRa

F = F1 = 0 ( Laumbach and Probstein[42] ),(12a− f)

where R is the shock radius; a, d, n, h0, B and G areconstants.From (7) and (12), we find that the initial angular

velocity varies as

A1 = BRd−1. (13)

It decreases as the distance from the axis increases ifd− 1 < 0 i.e.d < 1.The momentum equation (2) in the undisturbed state

of the gas, gives

p1 =ρ1B

2R2d

2d− (n+ 1)µh2

0R2n

2n, (14)

where d = n.Ahead of the shock, the components of the vorticity

vector, therefore vary as

ζr1 = 0, (15)

ζθ1 = −a GRa−1

2, (16)

ζz1 =(d+ 1)BRd−1

2. (17)

Due to passage of the shock, the gas is highly ionizedand its electrical conductivity becomes infinitely large.The conditions across such a gas-ionizing shock are (c.f. Singh and Srivastva [43], Vishwakarma and Pandey[37], Vishwakarma and Singh[32] )

ρ2(V − u2) = ρ1V = ms,

p2 − p1 = msu2,

e2 +p2ρ2

+1

2(V − u2)

2 − F2

ms= e1 +

p1ρ1

+1

2V 2,

h2 = h1, v2 = v1, w2 = w1, T2 = T1, (18a− g)

where the subscripts 1 and 2 refer to the values justahead and just behind the shock, respectively, and Vdenotes the shock velocity. The shock front is assumedto be opaque and it does not receive any heat flux fromexternal sources . Therefore, the heat flux F2 is the heatflux exchanged between the flow- field and the shockfront. Here we have assumed that the shock is isother-mal (the formation of the isothermal shock is a result ofmathematical approximation in which the flux is takento be proportional to the temperature gradient.This ex-cludes the possibility of temperature jump, see for ex-ample Zel’dovich and Raizer[13], Rosenau and Franken-thal[44], Singh and Mishra[8]).From ( 18 a - g ), we get

ρ2 = ρ1/β,

u2 = (1− β)V,

p2 =[ 1

γM2+ (1− β)

]ρ1V

2,

h2 = h1, v2 = v1, w2 = w1,

F2 = ρ1V3(γ+1)

2(γ−1)

[2βγ+1

(1

M2 + γ + b̄)− β2

− 2b̄γ+1 − 2

M2(γ+1) −γ−1γ+1

], (19a− g)

where

β = b̄+1

γM2, (20)

and b̄ = bρ1 is the parameter of non-idealness of thegas. Here, M is the shock-Mach number referred to the

frozen speed of sound(γp1

ρ1

)1/2, and MA is the Alfven-

Mach number. The shock-Mach number and Alfven-Mach number are given by M = V

(γp1/ρ1)1/2and MA =

V(µh2

1/ρ1)1/2, respectively.


Following Levin and Skopina [35] and Nath [45,46],we obtain the jump conditions for the components ofvorticity vector across the shock as

ζθ2 =1

βζθ1 , ζz2 =

1

βζz1 . (21)

The total energy E of the flow-field behind the shockis not constant, but assumed to be time dependent andvarying as (Rogers [47], Freeman [48])

E = E0ts, s ≥ 0, (22)

where E0 and s are constants. The positive values ofs correspond to the class in which the total energy in-creases with time. This increase can be achieved by thepressure exerted on the fluid by expanding surface (acontact surface or a piston).

3 Similarity Solutions

We introduce the following similarity transformationsto reduce the equations of motion into ordinary differen-tial equations (Abdel-Raouf and Gretler [49], Ghoniemet al. [7], Vishwakarma and Singh [32], Singh and Nath[34])

u = V U(η), v = V ϕ(η), w = VW (η), ρ = ρ1D(η),

p = ρ1V2P (η),

√µh =

√ρ1V H(η), F = ρ1V

3Q(η),(23a− g)

where U, ϕ,W,D,P,H and Q are functions of the non-dimensional variable ( similarity variable ) η only, and

η =r

R, R = R(t). (24)

At the shock front η = 1. The total energy of the non-ideal gas behind the shock is given by

E = 2π∫ R

rp

[12ρ(u

2 + v2 + w2) + p(1−bρ)γ−1 + µh2

2

]rdr

= E0ts, (25)

where rp is the radius of the inner expanding surface.Applying the similarity transformations (23) to the re-lation (25), we find that the motion of the shock frontis given by the equation

E0ts

2πρ1J= R2V 2, (26)

where

J =

∫ 1

ηp

[12D(U2 + ϕ2 +W 2) +

P (1− b̄D)

γ − 1+

H2

2

]ηdη,

(27)ηp being the value of ‘η’ at the inner expanding surface.

For similarity solutions, the shock-Mach number Mand the Alfven-Mach number MA which occur in theshock conditions (33) must be constant parameters.Therefore, we have

V = LRd, (28)

where ‘L’ is a constant.

From (28), on integration, we get

R = [(1− d)L]1/(1−d)t1/(1−d). (29)

Also, from (26), we get

R =( E0

2πρ1JL2

)1/(2d+2)ts/(2d+2). (30)

On comparing (29) and (30), we obtain

s =2(1 + d)

1− d. (31)

This shows that for s > 0,−1 < d < 0.From (13), (14) and (28), we get

B

L=

√2d

γM2+

1 + d

M2A

. (32)

Using the similarity transformations (23), the shockconditions (19) transform into

U(1) = 1− β, D(1) =1

β, ϕ(1) =

√2d

γM2+

1 + d

M2A

,

W (1) =G

L, H(1) = M−1

A , P (1) =1

γM2+ (1− β),

Q(1) =β

γ − 1

(γ + b̄+

1

M2

)− γ + 1

2(γ − 1)β2

− b̄

γ − 1− 1

M2(γ − 1)− 1

2, (33a− g)

where a = d is taken in order to make the shock condi-tions consistent for similarity solutions.In addition to the shock conditions (33), the condition

to be satisfied at the inner boundary surface is that thevelocity of the fluid is equal to the velocity of the innerboundary itself. This kinematic condition, from (23)and (24), can be written as

U(ηp) = ηp. (34)

Using transformations (23), the equations of motion(1)-(6) take the form

(U − η)dD

dη+D

dU

dη+

DU

η= 0, (35)

(U−η)dU

dη+

1

D

dP

dη+

H

D

dH

dη+

H2

ηD+dU− ϕ2

η= 0, (36)

(U − η)dH

dη+H

dU

dη+ dH = 0, (37)

(U − η)dϕ

dη+(Uη

+ d)ϕ = 0, (38)

(U − η)dW

dη+ dW = 0, (39)

(U − η)dP

dη− γP

(1− b̄D)D(U − η)

dD

dη+

γ − 1

1− b̄D

dϕ

dη+

(γ − 1)ϕ

(1− b̄D)η+ 2dP = 0. (40)



By using (9) and (11) in (10), we get

F = −4

3cσα0ρ

ν[p(1− bρ)

Γρ

]3+κ ∂T

∂r. (41)

Again, using (9) in (41), and then the similarity trans-formations (23), we get

Q = −4

3

cσα0ρν−11 L5+2κ

Γ4+κ(1− b̄D)3+κDν−κ−4 ×

P 4+κR5d+2κd−1[1− b̄D

P

dP

dη− 1

D

dD

dη

](42)

Equation (42) shows that the similarity solution of thepresent problem exists only when

5d+ 2κd− 1 = 0.

Therefore,

κ =1− 5d

2d. (43)

Therefore, (42) becomes

Q = −N(1− b̄D)3+κDν−κ−4P 4+κ ×[1− b̄D

P

dP

dη− 1

D

dD

dη

], (44)

where

N =4

3

cσα0ρν−11 L5+2κ

Γ4+κ(45)

is a non-dimensional radiation parameter. N dependson the mean free path of radiation.From (35)-(40) and (44), we get

dD

dη= − D

U − η

[dUdη

+U

η

], (46)

dH

dη= − H

U − η

[dUdη

+ d], (47)

dϕ

dη= − ϕ

U − η

[Uη

+ d], (48)

dW

dη= − dW

U − η, (49)

dP

dη=

[ H2

U − η− (U − η)D

]dUdη

+

dH2

U − η− dUD − H2

η+

ϕ2D

η, (50)

dϕ

dη=

1− b̄D

γ − 1

[(U − η)2D −H2 − γP

1− b̄D

]dUdη

+

dDU(U − η)(1− b̄D)

(γ − 1)− H2(1− b̄D)d

γ − 1+

H2(U − η)(1− b̄D)

η(γ − 1)− 2dP (1− b̄D)

γ − 1−

ϕ2D(U − η)(1− b̄D)

η(γ − 1)− γPU

η(γ − 1)

−Q

η(51)

and

dU

dη=

1[(H2

(U−η)P − (U − η)DP

)(1− b̄D) + 1

U−η

] ×

[− U

η(U − η)+

dUD(1− b̄D)

P

+H2(1− b̄D)

P

(1η− d

U − η

)− ϕ2D(1− b̄D)

ηP

− Q

N(1− b̄D)3+κDν−κ−4P 4+κ

]. (52)

Also, applying the similarity transformations on (8), weobtain the non-dimensional components of the vorticityvector lr = ζr

V/R , lθ = ζθV/R , lz = ζz

V/R , in the flow-field

behind the shock as

lr = 0, lθ =dW

2(U − η),

lz =ϕ

2

[1η−

d+ Uη

U − η

]. (53a− c)

Now, the ordinary differential equations (46) - (52)may be integrated, numerically, with boundary condi-tions (33) and the appropriate values of the constantparameters γ, b̄, N, d, G

L ,M−2 and M−2

A , to obtain thesolution for the flow behind the shock surface.For exhibiting the numerical solutions, it is conve-

nient to write the flow-variables in the following non-dimensional form as

u

u2=

U(η)

U(1),

v

v2=

ϕ(η)

ϕ(1),

w

w2=

W (η)

W (1),

ρ

ρ2=

D(η)

D(1),

p

p2=

P (η)

P (1),

h

h2=

H(η)

H(1),F

F2=

Q(η)

Q(1). (54a− g)

4 Results and Discussion

Similarity considerations lead to the following rela-tions among the constants a, d, n, s and κ :

a = d = n, κ =1− 5d

2d, s =

2(1 + d)

1− d.

The distribution of the flow variables between theshock front (η = 1) and the inner expanding surface(η = ηp) is obtained by numerical integration of (46) -(52) with the boundary conditions (33) by Runge-Kuttamethod of fourth order. For the purpose of numericalintegration, the values of the constant parameters aretaken to be (Roberts and Wu [20], Elliott[3], Singh andMishra [8], Rosenau [41], Vishwakarma and Singh [32], Singh and Nath [34]) γ = 1.4; b̄ = 0, 0.05, 0.1; ν =−2;M2 = 20;M2

A = 10; GL = 0.1; d = −0.2,−0.4 and

N = 10, 100. The value b̄ = 0 corresponds to the case ofa perfect gas.Figures 1 to 9 show the variation of the flow vari-

ables uu2, ρ

ρ2, h

h2, v

v2, w

w2, p

p2, F

F2, the non-dimensional

azimuthal component of vorticity vector lθ and the non-dimensional axial component of vorticity vector lz withη at various values of the parameter b̄, d and N with


Table 1. Position of inner expanding surface ηp for γ = 1.4, M2 = 20, MA2 = 10, ν = −2, G

L= 0.1, and various values of b̄, N , and d

b̄ N d ηp10 -0.2 0.976613

-0.4 0.9820280.0 100 -0.2 0.976445

-0.4 0.97329110 -0.2 0.946630

-0.4 0.9430700.05 100 -0.2 0.946556

-0.4 0.93802310 -0.2 0.916432

-0.4 0.9063780.1 100 -0.2 0.916387

-0.4 0.903018

Table 2. Density ratio β across the shock front for b̄ = 0 , 0.05 , 0.1, M2 = 20 and γ = 1.4

b̄ β0.0 0.03571430.05 0.08571430.1 0.135714


γ = 1.4, ν = −2,M2 = 20,M2A = 10 and G

L = 0.1. Theposition of the inner expanding surface ηp and the den-sity ratio across the shock front β = ρ1

ρ2are shown in

tables 1 and 2 for different cases. It is shown that, as wemove inward from the shock front towards the inner con-tact surface, the reduced magnetic field h

h2, the reduced

axial component of fluid velocity ww2

, the reduced radi-

ation heat flux FF2

, the non-dimensional azimuthal com-ponent of vorticity vector lθ and the non-dimensionalaxial component of vorticity vector lz increase; whereasthe reduced pressure p

p2and the reduced azimuthal com-

ponent of fluid velocity vv2

decrease. Also, as we moveinwards from the shock surface, the reduced radial com-ponent of fluid velocity u

u2increases, and starts to de-

crease after attaining a maximum near the inner con-tact surface. The reduced density ρ

ρ2decreases from the

shock front to the inner contact surface in all the casesexcept in the case b̄ = 0, N = 10, d = −0.4, where it in-creases and starts to decrease after attaining a maxium.The behaviour of the radial component of fluid velocity,density, pressure, magnetic field and radiation heat fluxprofiles are similar to those obtained by Vishwakarmaand Singh [32].

The effects of an increase in the value of the parameterof non-idealness of the gas b̄ are

(i) to increase the value of β ( i.e. to decrease theshock strength, see Table 2 );

(ii) to increase the distance of the inner expandingsurface from the shock front ( see Table 1 ) i.e. the flow-field behind the shock become somewhat rarefied. Thisshows the same result as in ( i ), i.e. there is a decreasein the shock strength;

(iii) to increase the reduced radial component of fluidvelocity u

u2and the reduced density ρ

ρ2, in general;



(iv) to decrease the reduced magnetic field hh2, the

reduced radiation heat flux FF2

, the reduced axial com-ponent of fluid velocity w

w2and the non-dimensional az-

imuthal and axial components of vorticity vector lθ andlz;(v) to increase the reduced pressure p

p2and the re-

duced azimuthal component of fluid velocity vv2.

The effects of an increase in the value of radiationparameter N are(i) to increase the distance of the inner expanding sur-

face from the shock front, i.e. to decrease the shockstrength. This decrease in the shock strength is due tothe fact that the transport of energy through radiationis faster at higher values of N ;(ii) to decrease the reduced radial component of ve-

locity uu2

, the reduced azimuthal magnetic field hh2, the

reduced axial component of fluid velocity ww2

, the re-

duced radiation heat flux FF2

and the non-dimensionalazimuthal and axial components of vorticity vector lθand lz; but the effect is small when d = −0.2 (see fig-ures 1, 3, 5, 7, 8 and 9);(iii)to increase the reduced azimuthal component of

fluid velocity vv2

and the reduced pressure pp2; but the

effect is small when d = −0.2 (see figures 4 and 6).Thus the effects of radiation parameter on the flow

variables are negligible when d = −0.2.The effects of an increase in the value of the index for

variation of ambient fluid velocities d or a (or the indexfor variation of ambient azimuthal magnetic field n) are(i) to increase the value of ηp (see Table 1), in general,

i.e. to decrease the distance of the inner expanding sur-face from the shock front. This shows that an increasein the value of d increases the shock strength;(ii)to decrease the reduced radial component of fluid

velocity uu2

and the non-dimensional axial component ofvorticity vector lz when N = 10; but to increase themwhen N = 100 (see figures 1 and 9);(iii) to decrease the reduced magnetic field h

h2, the

reduced axial component of fluid velocity ww2

and the

reduced radiation heat flux FF2

(see figures 3,5 and7);(iv) to decrease the reduced azimuthal component of

fluid velocity vv2, in general (see figure 5);

(v) to increase the reduced pressure pp2

(see figure 6):

(vi) to increase the shock velocity (see equation (28)).Present self-similar model may be used to describe

some of the overall features of a ”driven” shock waveproduced by a flare energy release E(= E0t

s) that istime dependent. For s > 0 the energy E increases withtime and the solutions then correspond to a blast waveproduced by intense, prolong flare activity in a rotatingstar when the wave is driven by fresh erupting plasmafor some time, and its energy tends to increase as itpropagates from star.

5 Conclusions

In the present paper, similarity solutions are obtainedfor the flow-field behind a gas-ionizing cylindrical shockwave propagating in a rotational axisymmetric non-idealgas with radiation heat flux in presence of an azimuthalmagnetic field. On the basis of this study one may drawthe following conclusions:

(i) the shock velocity varies as the ambient-fluid ve-locities ( i.e. as the ambient magnetic field );(ii) the non-idealness of the gas or the presence of

radiation heat flux has decaying effect on the ionizingshock wave;(iii) an increase in the parameter of non-idealness of

the gas b̄, or in the index for variation of ambient fluidvelocities d or a , or in the index for variation of ambientazimuthal magnetic field n, or in the radiation param-eter N modifies the distribution of the flow variablesbehind the shock.

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