International Journal of Scientific & Engineering Research, Volume 6, Issue 1, January-2015 2097

ISSN 2229-5518

OPTIMUM COMPRESSION OF A GAS BY A CYLINDRICAL SHOCK WAVE IN PRESENCE OF A MAGNETIC FIELD

BY

Dr. Awaneesh Jee Srivastava, Dr.S.N.Ojha Department of Mathematics SRMGPC, Lucknow
awaneeshcancerian@gmail.com

1. ABSTRACT

Simple analytical solutions have been obtained for cylindrical converging shock waves in an ideal perfectly conducting gas in presence of a magnetic field. These solutions are derived with the assumption that a cylindrical piston drives a shock front to the axis of the cylinder at a critical moment of cumulating. Simple particular solutions have been also derived.

Key Words Shock Waves, High Temperature, Thermonuclear, Magnetic field, Cylindrical Piston.

2. INTRODUCTION

The concentration or compression of force, energy or any other physical quantity in a small volume is an important phenomenon occurring in nature. A considerable role is played by concentric waves of cylindrical or spherical symmetry which generate high parameter physical fields about the axis or centre of cumulation. Among the other applications, it is viewed as necessary step in realizing controlled thermonuclear fusion. The converging shock waves are also employed in laboratories to maintain high temperature to study various processes which take place in gases.
The problem of compression shocks was investigated first by Guderley [1].
Guderley considered a spherical or cylindrical converging shock wave, which was created

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formally at time t = − ∞ by a spherical or cylindrical piston at a distance far from the centre or axis of cumulation, that reaches to the centre or axis at a moment t = 0. The self- similar solution of the problem was solved numerically for the situation with the initial push (piston) which does not influence on the final stages of cumulation. Further, this problem of cumulation was solved by Stanyukovich [2], Zeldovich and Raizer [3], Goldman [4] and many others by using only numerical computations. Chisnell [5], unlike previous investigations was able to get accurate analysis and numerical results and to yield a simple analytic description of flow variables at all points behind the converging shocks.
The optimum compression of an ideal gas by a cylindrical piston was considered by Zhdanov and Tribnikov [6] without forming any shock waves. In that case the sound wave fronts move with a sound velocity in front of the piston and this non-shock wave problem also needs numerical computations. Similar problem has been considered and simple analytical solutions have been derived by Farshi and Trubnikov [7] with the assumption that the piston moves with a supersonic velocity and therefore, a shock wave front exists in front of the piston.
Here, we have assumed that the piston drives a shock wave front to the axis of cylinder at the critical moment of cumulating of an ideal gas in presence of a magnetic field. Simple analytical solutions are derived and particular cases have been discussed.

3. BASIC EQUATIONS AND SHOCK CONDITIONS

Consider a cylindrical piston which instantaneously begins to converge with supersonic speed at time t = 0 and drives a shock front so that it reaches to the axis at time t = t*. In the region between the front and the piston one may consider the equations of motion in cylindrical symmetry as (Vishwakarma and Yadav [8])

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∂ρ ∂ρ  ∂ 

(3.1)


+ u + ρ


v + v  = 0,

∂t ∂r

 ∂r r 

(3.2)



∂v + v ∂v + 1 ∂p + h ∂h = 0,

∂t ∂r

ρ ∂r

ρ ∂r

(3.3)

∂(pρ− γ


) + v ∂(pρ

) = 0,

∂t ∂r

∂ ∂  ∂ 

(3.4)

h + u

h + h


v + v  = 0.

∂t ∂r

 ∂r r 

These equations govern the mass density ρ, the velocity v, the pressure p, the axial magnetic field h, all of which depend only on the time t and on the distance r from the axis of the cylinder, γ is the ratio of specific heats of the gas.
From (3.1) and (3.4), we may obtain
(3.5)

h


= constant.

ρ

Equation (3.5) holds only when the gas is perfectly conducting and magnetic field is axial in the case of cylindrically symmetric flow.
Taking pressure in the form (Golitsyn [9], Ojha and Singh [10], Vishwakarma and
Yadav [8]),

h 2


(3.6) p = β p*, p* = p +

2

= total pressure, 0 < β < 1,
and using (2.5), the effective speed of sound c is defined by
(3.7) c2 =

dp *

= a2 + b2 =

γ * p *


,

ρ

(3.8) a2 =

γp


is square of speed of sound,

ρ

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(3.9) b2 = h

ρ

is the square of the Alfven speed
and
(3.10) γ* = γ β + 2(1 − β).
Assuming β to be a constant and flow to be isentropic, we may write
(3.11)

∂(p * ρ

∂t

− γ*

) + v ∂(p * ρ

∂r

− γ*

) = 0.

The equation (3.11) is true only when (3.5) holds and β has a constant value. The assumption that β is constant, physically means that the gas pressure and the magnetic pressure are in constant ratio.
From (3.11) we may write,
(3.12) p* = A ργ*, A = constant and
(3.13) c2 = A ρ(γ*−1).
Equations (3.1) and (3.2) may be written as

2 2

=

(3.14)

c + v c

+ (γ * −1)

v + v  = 0,

∂t ∂r

 ∂r r 

and
(3.15)

∂v + v ∂v +

1 ∂c2

= 0.

∂t ∂r

(γ * −1) ∂r

We assume that at time t, in between t = 0 and t = t*, the cylindrical piston with radius Rp(t) has a supersonic velocity Vp (t) and therefore, a shock wave front moves with radius Rf(t) and velocity V f(t) in the region ahead of the piston. Again, let just ahead of the shock front (r < Rf) the mass density, pressure, temperature, magnetic field strength
and velocity of the gas be ρ0 , p0 , T0 , h0 and v0 = 0 respectively, whereas just behind of

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the shock front (Rf < r < Rp) these quantities be ρ1 , p1 , T1 , h1 and v1 correspondingly. The well known Hugoniot conditions on the shock wave front may be written as
(3.16) ρ1 (Vf − v1 ) = ρ0 Vf ,
(3.17) p1 * + ρ1 (Vf − v1 )2 = p0 * + ρ0 Vf2,
(3.18)

1 (V
v )2 +

γp1


+ h1

= 1 V 2 +

γp0


+ h0 .
2 ρ1 (γ − 1) ρ1 2

ρ0 (γ − 1) ρ0

Taking ξ = 1 − (ρ0 1 ), we may write these conditions as
(3.19) v1 = ξ Vf ,
(3.20) p1 * − p0 * = ρ0 Vf2 ξ = ρ0 Vf v1 ,
(3.21) p * = p * 
2β + ξ (γ − 1) 

 .
 2β ξ (2β + γ − 1) 
The velocity of the shock wave front Vf may be written as
(3.22) v V = 

2ξ (β + γ − 1)
p*
0 ,
 2β ξ (2β + γ − 1)  ρ0
or

*

(3.23)
V 2 =  2(β + γ −1)

p0 .
 2β ξ (2β + γ −1)  ρ0

Putting c0 2 =

γ * [γβ + 2(1 − β)]p*

0 = 0

ρ0 ρ0

in (3.23), where c0 is effective
speed of sound in front of the shock, we have
(3.24)

V 2 =


2(β + γ −1)c0
f {γβ + 2(1 − β )}{2β ξ (2β + γ −1)}
or
(3.25)

V 2 − (2β + γ −1) ξV 2 =

(β + γ −1)c0 ,

f 2β

f [γβ + 2(1 − β )]β
which gives
(3.26) Vf = G + √( C0 2+ G2),
where

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 2β + γ −1 
(3.27) G = 


v1
4β
(3.28)
and

C 2 =


(β + γ −1)c0 .

0 β (γβ + 2(1 − β )

4. SOLUTION BEHIND THE SHOCK FRONT

Substituting the relations

 r 

(4.1) v = vr = −α

*

− t 

and

 r 

(4.2) c = δ

*

− t 

into the equations (3.14) and (3.15), we observe that when
(4.3) α =

1

2 + β(γ − 2)

and

(4.4) δ =

β(γ − 2) + 1

2 (2 + β(γ − 2)

equations (3.14) and (3.15) are satisfied. Thus, we can say that (3.1) and (3.2) are simplest solutions of the flow equations with constants α and δ given by equations (3.3) and (3.4).
For such solutions the motion of the piston is described by the equations,

(4.5)

dR p

dt

= v(r = R

) = −

αR p

t* − t

 α


, Rp = R0 1 − 

 t 

and

*

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(4.6) Vp = −V0 1 − =

α −1

 t*

where R0 = Rp at t = 0, V0 = α R0 /t*. After time t = 0, the shock wave front moves
towards the axis with a constant velocity, it means Vf = − R0 /t* which is derived from

 t 

Rf = R0 1 −

. The velocity of the gas just behind the front will be given by

*

αR f

(4.7) v1 =

v(r = R ) = −

t* − t

= − α R0 /t* = − V0 = constant.
By substituting these values into the relations (3.26) we get

C 2

α (2β + γ −1) 

0

(4.8)

or

= 1 − 
f  2β

c 2 β{γβ + 2(1 − β )} 

α (2β + γ −1) 

(4.9)

and


0

2 (β + γ −1)
1 −



2β

C 2 1 

α (2β + γ − 1) 

(4.10)

or


0 = 1 −

2 α



2β

c 2 β{γβ + 2(1 − β )} 

α (2β + γ −1) 

(4.11)


0

2 α (β + γ −1)

1 −



2β

5. PARTICULAR CASES

Taking γ = 5/3, we have

(5.1) α =

and

(5.2) δ =

3

6 − β

3 − β

(6 − β) 2

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For a particular value of β = 0.5, we obtain α = 6/11 and δ = 5/11√2. In this case the law
of the piston motion has the following form,

(5.3) Rp = R0 1 − =

6 / 11

 t*

If the radius R0 and the sound velocity c0 are given, then the critical moment t* will be
given by

(5.4) t* =

R 0

c0 14

The jump in density and pressure before and behind the front are given by

(5.5)

and

ρ

1 =

ρ0

*

1

1

1 − α

= 11

5

(5.6) *

p0

= 15
Similarly, for other constant values of β (0 < β < 1), one may derive other particular solutions. These particular solutions are suitable only for the initial stages of compression. Therefore, for the final stages of compression it is necessary to use the Guderley’s numerical solutions. These solutions have only the analytical values as the simplest
solutions.

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REFERENCES

[1] Guderley, G. : Luftarhtforschung, 19, 302, (1942).
[2] Stanyukovich, K. P. : Unsteady motion of continuous media, Pergamon (1960).
[3] Goldman, E. B. : Plasma Phys. 15, 289, (1973). [4] Zeldovich, Y. B. : Physics of Shock Waves and High
and Raizer, Yu. P. : Temperature Hydrodynamic Phenomena.
Vol. II, Academic Press (1967). [5] Chisnell, R. F. : Fluid Mech. 354, 357, (1998). [6] Zhdanov, S. K. and : JETP Lett. 21, 371, (1975).
Trubnikov, B. A.
[7] Farshi, E. and : Fusion Engg. Design, 60, 99-103, (2002).
Trubnikov, B. A.
[8] Vishwakarma, J. P. : Eur. Phy. J. B, 34, 247, (2003). and Yadav, A. K.
[9] Golitsyn, G. S. : Soviet Phys., JETP, 35, 538, (1959). [10] Ojha, S. N. and : IL Nuovo Cimento, 70B, 2, 212, (1982).
Singh, S. N.

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