PROPAGATION OF IONIZING SHOCK WAVE IN A SELF-GRAVITATING NON-IDEAL GAS

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79 | P a g e

PROPAGATION OF IONIZING SHOCK WAVE IN A SELF-GRAVITATING NON-IDEAL GAS

B. Nath

1

, S. Talukdar

2

1,2Assistant Professor, Department of Mathematics, GIMT-Tezpur (India)

ABSTRACT

Propagation of a gas ionizing shock wave in a self-gravitating non-ideal gas in presence of a spatially decreasing azimuthal magnetic field is investigated. The electrical conductivity of the medium ahead of the shock is assumed to be negligible, which becomes infinitely large due to the passage of the shock. The initial density of the gas is assumed to obey a power law. Distribution of the flow variables are obtained in the flow- field behind the shock. Similarity solutions are obtained and effects of variation of the non-idealness parameter, shock Mach number and the variation of initial density exponent are investigated.

Keywords: Magnetogasdynamics, Non-Ideal Gas, Self-Gravitational Effects, Self-Similar Flow, Shock Wave.

I. INTRODUCTION

Shock waves are common in the interstellar medium because of a great variety in supersonic motions and energetic events, such as cloud-cloud collision, bipolar outflow from young prostellar objects, powerful mass losses by massive stars in the late stage of their evolution (stellar winds), super nova explosions, etc. Shock waves are also associated with spiral density waves, radio galaxies and quasars. The explanation and analysis for the internal motion in stars is one of the basic problems in astrophysics. According to observational data, the unsteady motion of a large mass of gas followed by sudden release of energy results in flare-ups in novae and supernovae. A qualitative behavior of the gaseous mass may be discussed with the help of the equations of motion and equilibrium taking gravitational forces into account. Carrus et al.[1] and Sedov [2] have obtained numerical solutions for self-similar adiabatic flows in self gravitating gas, independently. Purohit [3] and Singh and Vishwakarma [4] have discussed homothermal flows behind a spherical shock wave in a self-gravitating gas using similarity method. Shock waves through a variable density medium have been treated by several authors eg. Sakurai [5], Rogers [6], Sedov [2], Rosenau and Frankenthal [7], Nath et. al [8], Vishwakarma and Singh [4]

and many others. Their results are more applicable to shocks formed in the deep interior of stars. Recently Nath et. al [10] studied the propagation of magnetogasdynamics shock wave in a self gravitating gas with exponentially varying density. In all of the works mentioned above the medium is taken to be perfect gas.

The assumption that the gas is ideal is no longer valid when the flow takes place at high temperature. The popular alternative to the ideal gas is a simplified van der Waals model. Robert and Wu [11, 12] studied the problem of a spherical implosion by adopting the same van der Waals model. Vishwakarma et. al [13] studied one dimensional unsteady self similar flow behind a strong shock driven out by a cylindrical (or spherical) piston moving in a medium which is assumed to be a non ideal gas obeying simplified van der Waals equation

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of state. They have taken the electrical conductivity medium to be infinite. But in many practical cases the medium may be of low conductivity which becomes highly conductive due to the passage of a shock. Such a shock is called a gas-ionizing shock or simply ionizing shock. The propagation of an ionizing shock has been studied by Greenspan [14], Greifinger and Cole [15], Christer [16], Ranga Rao and Ramana [17], Singh [18], Vishwakarma and Singh [19], Singh and Nath [20] in perfect or non-ideal gas.

In the present work we obtain self similar solution for the flow behind a gas ionizing shock wave propagating in a self gravitating non ideal gas in presence of an azimuthal magnetic field. The medium is supposed to have low conductivity which becomes highly conducting due to passage of a strong shock. The initial density of the gas and the initial azimuthal magnetic field are assumed to vary as some power of distance. The total energy of the flow field behind the shock is assumed to be increasing with time due to pressure exerted by a piston or inner expanding surface. The gas ahead of the shock is assumed to be at rest. Effects of variation of shock Mach number, non-idealness parameter and the variation of initial density exponent are investigated.

II. FUNDAMENTAL EQUATIONS AND BOUNDARY CONDITIONS

The fundamental equations governing the unsteady adiabatic spherically symmetric flow of perfectly conducting and self-gravitating gas in which an azimuthal magnetic field is permeated and heat conduction and viscous stress are negligible [10, 21, 22] are:

u 2 u

u 0

t r r r

     

   (1)

2 2

u u 1 p h h Gm

u h 0

t r r r r r

 

    

       

      (2)

h h u hu

u h 0

t r r r

  

   

   (3)

m 2

4 r 0

r

   

 (4)

2

e e p

u u 0

t r t r

        (5)

where r and t are independent space and time co-ordinates, respectively, u is the fluid velocity,

the density, p the pressure, h the azimuthal magnetic field, e the internal energy per unit mass,

the magnetic permeability, m the mass contained in the sphere of radius r and G is the gravitational constant.

The above system of equations should be supplemented with an equation of state. We assume that the gas obeys a simplified van der Waals equation of state of the form [11, 12, 23, 24]

 

 

V

p 1 b p T , e C T

1 b 1

  

  

     (6)

where

is the gas constant, C TV 1

 

  is the specific heat at constant volume and

is the ratio of specific heats. The constant b is the van der Waals excluded volume; it places a limit max 1

  b, on the density of the gas.

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81 | P a g e A spherical shock is supposed to be propagating in the undisturbed non-ideal gas with variable density,

Ar, 0 3,

     where A and

are constants. The azimuthal magnetic field in the undisturbed gas is assumed to vary as

h  Br , 0

k

  k 1.5

, where B and k are constants (Rosenau [25]). Also, the gas has negligible electrical conductivity in presence of an azimuthal magnetic field. But due to the passage of the shock, the gas is highly ionized and the electrical conductivity is infinitely large.

The flow variables immediately ahead of the shock front are

k

1 1 1

u 0,  AR , h BR

 

2

2 2k

 

1

2 GA 2 B

p R , k 1

3 1 1 2

      

         (7)

3 1

m 4 AR 3



  

where R is the shock radius and the subscript ‘1’ denotes the conditions immediately ahead of the shock.

The jump conditions at the gas-ionizing shock wave are given by the principles of conservation of mass, momentum, magnetic field and energy [26, 27, 28] are:

1V 2(V u )2 ms1

    

2 1 s 2

p p m u (8)

2 2

1 2

1 2 2

1 2

p 1 p 1

e V e (V u )

2 2

     

 

1 2 1 2

h h , m m

where subscript ‘2’ denotes the conditions immediately behind the shock front and dR

V dt

 

 

  denotes the velocity of the shock front.

From equation (8), we get

u2 (1 )V, 21,

  

2

2 2 1

p 1 (1 ) V ,

M

 

   

3

2 2 1

m 4 AR , h h ,

3



 

  (9)

where

2 12 1

1

M V p

 

    is the shock Mach number referred to the frozen speed of sound

12 1 1

p 

 

  . The quantity

0    1

is obtained by the relation

1

2b 2M2

1 .

   

    (10)

The total energy E of the flow-field behind the shock is not constant, but assumed to be time dependent and varying as [29, 30, 31]

s

EE t , s0 0 (11)

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where

E

0 and s are constants. This increase in energy can be achieved by the pressure exerted on the fluid by an expanding surface, which may be, physically, the surface of a stellar corona or condensed explosives or a diaphragm containing a very high-pressure driver gas.

III. SELF-SIMILAR TRANSFORMATIONS

Following the general similarity analysis we define the two characteristic parameters ‘a’ and ‘d’ with independent dimensions as [2]

   

a A (12)

and

 

d E0

A

 

    (13)

The non-dimensional independent variable

in this case will be

15

E0

A rt





 

    (14)

where 2 s

5

  

  (15)

and

is a constant such that

assume the value ‘1’ at the shock surface. This gives the shock propagation law in the explicit form as:

1 0 5

R E t

A



 

   (16)

From this relation, we have

1

0 0

V R

V R



 

  

  (17)

where V0and R0are the velocity and radius of the shock at the instant of its generation.

We express the fluid variables as

 

uVU  ,  1D

 

 , pV21P

 

 ,m 1R N3

 

 ,   h1 1VH

 

 (18) where U, D, P, N and H are functions of the non-dimensional variable

only.

For the existence of similarity solutions, the shock-Mach number and the Alfven Mach number should be constants, therefore

 2

and 10

 4 s

(19)

where 1  2 1 s

 6

or 2  2.5 0

 s 1 .

The equations of motion (1) to (5) can be transformed into a system of ordinary differential equations.

U

dD DdU 2U D 0

d d

 

        (20)

 

02

G N

dU 1 1 dP H dH H

U U 0

d D d D d

 

  

           (21)

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83 | P a g e

U

dH 1 H HdU HU 0

d 2 d

  

 

          (22)

dN 2

d   4 D

 (23)

1 bD U

 

dPd(U )D dP dD2 1

1 bD P

 

   1 bD

 P 0 (24) where

  

 

2

0 2 A

3 1 1 2

G M

2 M 2 1

   

   

    (25)

and

2 12 1

A 2

1

M V

h

 

    is the Alfven Mach number.

From equations (20) to (24), we get

dD D dU 2U

d U d

   

      

        (26)

dH H dU U 1

d U d 2

   

      

       (27)

     

2 2 2

0 2

G N

dP H dU 1 H U 1 H

D U U

d D U d D U 2 D

         

               

                

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   

   

 

 

2 2

0 2

2 2

1 H U 1 H G N

U D U 2 D

1 bD D U dU

d H 1 bD P D 1 bD U 2 PU 2 1 P

D 1 bD U D U

         

    

               

 

     

                      

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dN 2

d   4 D

 (30)

Using similarity transformations (18), the shock conditions (9) are transformed into U(1) (1 ),

D(1)1

 , H(1)MA1, 12

P(1) (1 ) ,

M

 

   N(1)

34 

(31) In addition to the shock conditions (31), the kinematic condition to be satisfied at the inner expanding surface is given by

 

p p

U   

(32) where pis the value of  at the inner expanding surface.

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IV. RESULTS AND DISCUSSION

The distribution of the flow variables in the flow-field behind the shock front are obtained by numerical integration of the equations (26) to (30) with the boundary conditions (31) by using Runge-Kutta method of fourth order. For the purpose of numerical integration, the values of the constant parameters are taken to be

2

5 ; M 5, 10, 15; MA 0.005, 0.01; b 0, 0.05, 0.1;

3

      1.1, 1.2, 1.3.The value b0 corresponds to

the ideal gas case.

The expression for gravitational parameter G0 is given by equation (25), the exponent in the shock propagation

law  is given by 2

 where ω is the exponent of law of variation of the initial density. Also the exponent in

the law of variation of the total energy behind the shock s is related with ω by 10 4 s  .

Fig 1(a)-1(e) shows the variation of the non-dimensional flow variables U, D, P, H and N with  for different values of b and

and Table 1 shows the density ratio and the position of the inner expanding surface in different cases. Fig 1(a), 1(b), 1(d) shows that the non-dimensional velocity U, density D and mass N increases as we move from the shock front towards the inner expanding surface. Also fig 1(c) shows that the non- dimensional pressure P increases attains a maximum and then decreases rapidly near the inner surface. Fig 1(e) shows that the non-dimensional mass decreases as we proceed towards the inner expanding surface.

Fig 2(a)-2(e) shows the profiles of the non-dimensional flow variables for different values of  and M2 and Table 2 shows the variation of the density ratio, the gravitational parameter and the position of the inner expanding surface for different cases.

V. TABLES AND FIGURES

Table 1: Density Ratio

and Position of the Inner Expanding Surface

p

for Different Values of

and

b

and

5

 3

,

M25

,

MA20.01

.

b  p

1.1

0 0.05

0.1

0.4000 0.4375 0.4750

0.857427 0.841868 0.825170

1.2

0 0.05

0.1

0.4000 0.4375 0.4750

0.849694 0.835003 0.819260

1.3

0 0.05

0.1

0.4000 0.4375 0.4750

0.841013 0.827580 0.813081

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85 | P a g e Table 2: Density Ratio, Position of the Inner Expanding Surface p and Gravitational Parameter

G0for Different Values of

and M2and 5

 3, b0.1,MA2 0.005

M 2  G0p

1.1

5 10 15

0.475 0.400 0.375

0.00294835 0.00113398 0.00052919

0.825261 0.855662 0.865493

1.2

5 10 15

0.475 0.400 0.375

0.00630254 0.00286497 0.00171887

0.819299 0.851303 0.861644

1.3

5 10 15

0.475 0.400 0.375

0.00926680 0.00439666 0.00277327

0.813063 0.846797 0.857691

Fig 1: Variation of non dimensional flow variables for different values of (a) velocity, (b) density, (c) pressure, (d) magnetic field and (e) mass

1. 1.1, b0; 2. 1.1, b0.05; 3. 1.1, b0.1; 4. 1.2, b0; 5. 1.2, b0.05; 6. 1.2, b0.1;

7. 1.3, b0; 8. 1.3, b0.05; 9.  1.3, b0.1;

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It is found that:

The effects of an increase in the value of the non-idealness parameter

b

are:

1. To increase the value of ; to decrease the shock strength (see Table 1)

2. To increase the distance of the inner expanding surface from the shock front. Physically it means that the gas behind the shock is less compressed. (see Table 1)

3. To decrease the non-dimensional velocity U, density D, magnetic field H and pressure P. (Fig 1(a)—1(d)) 4. To increase the non-dimensional mass N. (Fig 1(e))

The effects of an increase in the exponent of the density variation ω are:

Fig 2: Variation of non dimensional flow variables for different values of (a) velocity, (b) density, (c) pressure, (d) magnetic field and (e) mass

1. 1.1, M25; 2. 1.1, M210; 3. 1.1, M215; 4. 1.2, M25; 5. 1.2, M210;

6. 1.2, M215; 7. 1.3, M25; 8. 1.3, M210; 9.  1.3, M215;

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87 | P a g e 1. To increase the distance of the inner expanding surface from the shock front (see table 1 and 2).

2. To decrease the velocity U, density D, magnetic field H, pressure P and to increase the mass N at any point in the flow-field behind the shock. (see Fig 1(a)-(e), 2(a)-(e))

3. To increase the value of the parameter of gravitation G0 (see Table 2).

The effects of an increase in the value of the shock Mach number M, are:

1. To decrease the value of  (see Table 2).

2. To decrease the distance of the inner expanding surface from the shock front (see Table 2).

3. To increase the non-dimensional velocity U, density D, pressure P, magnetic field and to increase the non- dimensional mass N. (see Fig 2(a)—2(e))

4. To decrease the value of the parameter of gravitation G0 (see Table 2)

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