Shock Waves (1994) 4:163--165

�9 Springer-Verlag 1994

Technical note

Effect of overtaking disturbances on strong cylindrical shock in a self-gravitating gas R.P. Yadav, M. Gupta, S. Tripathi

Department of Physics, Govt. P.G. College, Uttarkashi-249193, India

Received September 9, 1993; accepted August 8, 1994

Abstract. The effect of overtaking disturbances on the prop- agation of strong cylindrical shock in a self-gravitating gas has been studied by an approximate technique developed by Yadav (1992). Assuming an initial density distribution law as e0 = e'r -w, where e' is the density at the axis of symmetry and w is a constant, the analytical expressions for shock velocity and shock strengths modified by over- taking waves have been obtained. The results accomplished here have been compared with those for freely propagating shock.

The conclusions arrived here agreed with experimental results.

Finally, the modified expressions for the pressure, the density and the particle velocity immediately behind the shock have also been derived.

Key words: Shock wave propagation, Overtaking distur- bances

1. Introduction

Study of propagation of shock waves in a self-gravitating gas has astrophysical significance. In recent years many techniques have been proposed to study the propagation of shock waves. An approximate method for solving problems of shock waves has been developed by Chester (1954), Chis- nell (1955) and Whitham (1958). This approximation is that of a freely propagating shock, i.e. in CCW approximation, a shock is not affected by the disturbances in the flow behind the shock. Recently, Yadav (1992) has developed a method for investigating the effect of overtaking disturbances on the freely propagating shock.

This paper considers the effect of overtaking distur- bances on the propagation of strong cylindrical shock in self-gravitating gas. Technique developed by Yadav (1992) has been used to obtain analytical expressions for modified shock velocity and shock strength for an initial density dis- tribation eo = e'r -w where e ~ is the density at the axis of symmetry and w is a constant. The results accomplished

here have been compared with those for freely propagating shock.

It is observed that the conclusions arrived here are in agreement with experimental results (Terao 1983; Terao and Wagner 1991).

Finally, the modified expressions for the pressure, the density and the particle velocity immediately behind the shock have also been derived.

2. Theory

The equations governing the cylindrically symmetrical flow of the gas enclosed by the shock front under the influence of its own gravitation are:

Ou Ou 1 0t) Gm o - 7 + ~ + ~ + - 7 - = 0

Oe+u-~r+e + = 0 o-7

op ~op_~2(Oe & ) 0-7 + ~ 57 + ~ = o (1)

Orn = 2~rre

Or

where rn(r, t), u(r, t), p(r, t) and e(r, ~) denote respectively the mass inside a cylinder of radius r and unit length, the velocity, the pressure and the density at a distance r at time t.

Let P0 and eo denote the undisturbed value of pressure and density in front of the shock and p, e be the values of the respective quantities at any point immediately, after passage of the shock, the Rankine-Hugoniot relations, for strong shock reduce to:

2eoU 2 ( ' y + 1~ - - e = e 0

v -y + 1 ' \ 7 -2 - f J

u = 7 + 1 , a = S U 7 - 1 (2)

164

where U is the shock velocity, a is sound velocity behind the shock and S[= {23`/(3' - 1)} I/z] is a constant.

First, ignoring the effect of overtaking disturbances i.e. freely propagating description of the problem as obtained by Kumar and Saxena (1983), has been given briefly.

For freely propagating shock, the characteristic form of system of (1), that is, the form in which equations contain derivatives in only one direction in (r, t) plane is:

ea2u dr aGm dr dp + ea du + - - -~ - 0 (3)

u w a r u + a T 2

a relation valid for C+ disturbances. Using (2) and (3) we have:

U 2 B G m dU2 + A + 0 (4) dr r r 2

where

A = 4 3 ` / { 2 + V/23`(3` - 1)} {2 + X/27/(3`- 1)}

B : (3`+ 1) 2 {23`/( 7 - 1)}1/2} / {2 + X/27(3`- 1)}

x {2+ X / 2 7 / ( 7 - 1 ) }

on integration (4) yields

U 2 = r - A { K - B G / m r A - 2 d r } (5)

where K is a constant of integration. Now assuming the initial density distribution law as

eo = e'r -w �9 (6)

The mass inside a cylinder of radius r and unit length, is written as

2~ce t m = - - r 2 . w . (7)

2 - w

Using (7) we have from (5)

{ 27re'BG rl+A_w} (8) U 2 r -A K - ( 2 _ w ) ( l + A _ w )

This expression represents the shock velocity for freely propagating shock.

The condition of hydrostatic equilibrium prevailing in front of shock is written as

1 dpo Gm eo d-"r-- + - 7 = 0 (9)

which gives:

P----2- ~ = 2~rrl-2~ (10) enG (2 - w)(2w - 1)

and

27retTG a~ = (2 - w)(2w - 1) r0 -~ ) " (11)

The expression for freely propagating shock strength can be written as:

( U ) 2 - r ( w - l - A ) ~ G K 2

27re'BG r l+A-w } (12) x K - ( 2 - w ) ( l + A - w )

where K~ = 27re'7/(2 - w)(2w - 1). For C+ disturbances generated by the shock, the fluid

velocity increment by use of (2) and (8), may be expressed as

du+ 1 [Audr B G m d r ] - + - - ( 1 3 )

7 + 1 r U ~ "

To estimate the strength of overtaking disturbances, an independent C_ characteristic is considered. The differential equation valid across a C_ disturbance is taken as

eaZu dr aeGm dr dp - eadu + - - - 0 (14)

u - - a 1" u - - a r 2 "

Now using conditions (2) and substituting into (14) and remembering (6) and (7) also, we get:

a m dr (15) dU = - A 1 U ; + B 1 2U r 2

where

A l = 4 7 / { 2 - V / 2 7 ( 7 - I ) } { 2 - V / 2 7 / ( 7 - 1 ) }

B 1 = (3` + 1) 2 {2"//(")' - 1)}1/2 / { 2 - V/23`("[ - 1)}

x {2 - ~/23`/(3` - 1)}

using ( 2 ) , ( 1 5 ) gives:

d u _ - 1 [ - A 1 u d r + B , G m d r 1 3' + 1 r y ~ (16)

which determines the strength of the overtaking distur- bances.

The expressions (8) and (12) for freely propagating shock is obtained by considering that the pressure and fluid velocity increments behind the shock is supported solely by C+ disturbances. In presence of overtaking disturbances, the fluid velocity increments behind the shock will be related as:

2 du+ + du_ = dU . (17)

3,'+1

Substituting values from (13) and (16), we have from (17):

dU2 A* U2 = B* Gm - - + - - ( 1 8 ) dr r r 2

where

A* = 43`('7 - 1)('y + 2)/('), - 2)(2 + 3, - , . , /2)

B * = 2")'2(') ' + 1)2/(3` - 2)(2 + 7 - ,./2)

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Table 1. Comparison of freely propagating shock velocity ( U / ~ ) and shock strength (U/ao) with modified values. Variation of these values with propagation distance arealso given for w = 1.5

Propagation Freely propagating Modified values distance r U(x,/eTG U/ao U(V"e~ U/ao

1.10 56.4541 19.4983 86.0687 29.7269 1.14 56.0541 19.5338 100.6782 35.0846 1.18 55.5706 19.5682 117.1525 41.1792 1.22 55.3024 19.6015 135.6516 48.0806 1.26 54.9483 19.6337 156.3426 55.8632 1.30 54.6077 19.6649 179.4034 64.6055 1.34 54.2792 19.6953 205.0146 74.3904 1.38 53.9623 19.7250 233.3718 85.3048 1.42 53.6564 19.7537 264.6785 97.4409 1.46 53.3604 19.7817 299.1337 100.8944 1.50 53.0742 19.8089 336.9655 125.7663

which yields:

B*Ge'2rr rl_~o U 2 = K r -A* + . (19) (2 - w)(1 + A* - w)

This equation represents expression for shock velocity modified by overtaking disturbances. The modified expres- sion for shock strength can be written as:

(=o): 1) x 2 1 r e ' ~ + (1 + A* - w)

(20)

3. Discussions

Equations (8) and (18) respectively represent shock velocity for freely propagation and under the influence of overtaking waves. General consequence of (8) and (18) is that strength- ening or weakening of shocks depend on w. The important case w = 1 + A (for freely propagation) or w = 1 + A* (for modified case), corresponds to the shock moving with constant strength, cf. (12) and (20), that is to a shock whose radius increases uniformly for all times. Any such case is, however, astrophysically important because of the fact that the radial velocity observations of most noval in the course of their outbursts indicate velocity of expansion which dif- ferent as they may be from star to star remains more or less constant within periods of the order of several days.

Initially taking U/ao = 20 (unmodified) at r = 1.0 for 7 = 1.4, w = 1.2, the variations of shock velocity and shock strength with propagation distance have given in the Table 1. It is found that freely propagating shock velocity decrease while modified shock velocity and shock strength both increase with propagation distance.

It is evident from Table 2 that shock velocities increase while shock strength decrease with w.

Table2. Variation of shock velocity and shock strength with w for 3' = 1.4 a t r = 1.1

W Freely propagating Modified values

U ( ~ U/ao U(x/e~ U/ao

1.2 54,9834 19.8133 86.0032 30.9898 1.5 56,4541 19.4982 86.0687 29.7269 1.76 56,0020 15.4749 86.2446 23.8319 1.95 57,3333 7.7039 87.6428 11.7765

In the present case, it is important to note that shock is strengthened if the effect of overtaking disturbances is con- sidered (cf. Table 1). The modified shock velocity and shock strength increase with propagation distance more rapidly than predicted for freely propagating shock. Terao (1983), Terao and Wagner (1991) experimentally observed that pres- sure, temperature and propagation velocity of detonation wave increase more rapidly than estimated for freely propa- gating shock (Whitham 1958). Thus the results accomplished here are in good agreement with experimental conclusions (Terao 1983; Terao and Wagner 1991).

Finally, the modified expressions for the pressure, the density and the particle velocity immediately behind the shock can be written as:

2e t [ * p= ~ r-(A +w) +

e - - -

and

U =

27re'B*G ] (2 - w)(1 + A* - w) rl-2w

e ' (7+ 1) r - w (21) 7 - 1

2 I 2rce'B*G -] 1/2

7 + i K r - A * + (2 - w)(1 + A* - w) I a

Acknowledgement. First author (RPY) thanks the University Grants Com- mission of India for financial support.

References

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