Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas

Wave Motion 44 (2007) 346–354

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Interaction of a characteristic shock with a weakdiscontinuity in a non-ideal gas

Manoj Pandey, V.D. Sharma *

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India

Received 21 August 2006; received in revised form 1 December 2006; accepted 6 December 2006Available online 21 December 2006

Abstract

The Lie group of point transformations, which leave the equations for plane and radially symmetric flows of a non-idealgas invariant, are used to obtain an exact solution that exhibits space-time dependence. We consider the propagation of aweak discontinuity through a state, characterized by this solution. Further, the evolution of a characteristic shock and itsinteraction with the weak discontinuity are studied. The properties of reflected and transmitted waves and the jump inshock acceleration, influenced by the incident wave amplitude and the van der Waals excluded volume, are completelycharacterized, and certain observations are noted in respect of their contrasting behaviour.� 2006 Elsevier B.V. All rights reserved.

Keywords: Characteristic shock; van der Waals gas; Weak discontinuity; Group theoretic method

1. Introduction

A detailed study towards gaining a better understanding of the wave interaction problem within the contextof hyperbolic system has been carried out by Jeffery [1] and Brun [2], the applications of which to elasticity andmagneto-fluiddynamics have been carried out by Morro [3,4]. A further contributions to the study of charac-teristic shocks (see, Boillat and Ruggeri [5]) and wave interactions which enable the evaluation of reflected andtransmitted wave amplitudes when the discontinuity wave encounters a characteristic shock may be found inthe paper of Boillat and Ruggeri [6]. The application of this work to interaction with a contact shock has beencarried out by Ruggeri [7], Virgopia and Ferraioli [8] and Jena and Sharma [9]. It is a known fact that a shockundergoes an acceleration jump as a consequence of its interaction with a weak wave [10]; this fact wasaccounted for in the work Brun [2] and Boillat and Ruggeri [6] and Radha et al. [10].

In this paper, we consider the unsteady equations that govern a planar, cylindrically symmetric and spher-ically symmetric flows in a van der Waals gas, and use the method of Lie group invariance to determine aparticular exact solution that exhibits space-time dependence [11]; this enables us to study the non-linear wave

165-2125/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

* Corresponding author. Tel.: +91 25768482; fax: +91 25764450.E-mail address: [email protected] (V.D. Sharma).

0
doi:10.1016/j.wavemoti.2006.12.002
mailto:[email protected]

M. Pandey, V.D. Sharma / Wave Motion 44 (2007) 346–354 347

interaction phenomenon influenced by the real gas effects, such as the van der Waals excluded volume, anddescribe a complete history of the evolutionary behaviour of a characteristic shock together with the incident,reflected and transmitted waves.

2. Basic equations and their infinitesimal operator

The unsteady one-dimensional axi-symmetric motion of a van der Waals gas obeying the equation of statep(1 � bq) = qRT can be written as

qt þ qux þ uqx þ mqux¼ 0;

qðut þ uuxÞ þ px ¼ 0;

pt þ upx þ qa2 ux þ mux

� �¼ 0;

ð1Þ

where x is the spatial coordinate being either axial in flows with planar (m = 0) geometry or radial in cylin-

drically symmetric (m = 1) and spherically symmetric (m = 2) flows, t the time, q the density, p the pressure,

R the gas constant, T the temperature; the entity a ¼ cpqð1�bqÞ

� �1=2

is the equilibrium speed of sound with c as the

specific heats ratio lying in the region 1 < c < 2 and b the van der Waals excluded volume, which lies in therange 0.9 · 10�3

6 b 6 1.1 · 10�3. It may be noticed that the case b = 0 corresponds to the ideal gas. The lettersubscripts denote the partial differentiation with respect to the indicated variable unless stated otherwise.

Using a straight forward analysis it is found that for m 5 0, a one parameter Lie group of infinitesimaltransformations that leave the system (1) invariant constitute a Lie algebra generated by the following infin-itesimal operators (see, [12], [13] and [14,15,16]):

X 1 ¼ 2xox þ tot þ uou þ 2pop;

X 2 ¼ xox þ uou þ 2pop;

X 3 ¼ ot:

However, for m = 0, in addition to the above X1, X2, X3, we also have the infinitesimal operators

X 4 ¼ ox and X 5 ¼ tox þ ou:

2.1. Similarity analysis and autonomous system

Following Donato and Oliveri [11], one may look for the introduction of suitable invertible transforma-tions, which are built by considering the canonical variables associated with the commuting infinitesimal oper-ators, allowing one to map the given system(1) to an equivalent autonomous form; this autonomous systemmay admit simple solutions which provide nontrivial solutions when expressed in terms of the originalvariables.

It can be verified that the infinitesimal generators X1 and X2 commute, that is,

½X 1; X 2� ¼ X 1X 2 � X 2X 1 ¼ 0;

which means that the operators X1 and X2 generate a 2-dimensional Abelian sub-algebra, and hence, the sys-tem (1) can be transformed by the invertible point transformation (see, Conforto [13]):

s ¼ ln t; g ¼ lnxt2; q ¼ Sðg; sÞ; u ¼ x

tV ðg; sÞ; p ¼ x2

t2Qðg; sÞ: ð2Þ

Then, using (2) in (1), and taking into account that

ot ¼1

tos �

2

tog; ox ¼

1

xog;

system (1) takes the following autonomous form:

348 M. Pandey, V.D. Sharma / Wave Motion 44 (2007) 346–354

Ss þ ðV � 2ÞSg þ SV g þ ðmþ 1ÞSV ¼ 0;

SV s þ SðV � 2ÞV g þ Qg þ 2Qþ SV ðV � 1Þ ¼ 0;

Qs þ ðV � 2ÞQg þcQ

1� bSV g þ ðmþ 1ÞV� �

þ 2QðV � 1Þ ¼ 0:

ð3Þ

It may be noticed that unlike the original system (1), the autonomous system (3) admits simple solutions; forinstance, the system (3) has a particular exact solution given by

V ¼ 1; S ¼ S0 expf�ðmþ 1Þsg; Q ¼ Q0 expf�2ðgþ sÞgfexpðmþ 1Þs� bS0gc

; ð4Þ

which, via (2), yields the following particular exact solution to the original system (1):

q ¼ q̂ðt=t0Þ�ðmþ1Þ; u ¼ x=t; p ¼ p̂ ðt=t0Þðmþ1Þ � bq̂

� �c.; ð5Þ

where q̂ and p̂ are some reference constant values; notice that at t = t0, q ¼ q̂, u = x0/t0 and p ¼ p̂=ð1� bq̂Þcwith bq̂ < 1. In the above particular solution, the particle velocity exhibits linear dependence on the spatialcoordinate; indeed, such a state can be visualized in terms of an atmosphere filled with a gas which has spa-tially uniform pressure and density variations on account of the particle motion. This class of solutions of thegoverning system has been discussed by Pert [17], Sharma et al. [18] and Clarke [19]; Pert showed that such aform of the velocity distribution is useful in modelling the free expansion of polytropic gases, and it is attainedin the limit of large time.

3. Evolution of a characteristic shock

The governing system (1) can be written in the form

U t þ AU x ¼ B; ð6Þ
where U = (q,u,p)tr, B = (�mqu/x, 0,�mqa2u/x)tr and
A ¼u q 0

0 u q�1

0 qa2 u

0B@

1CA:

The matrix A has eigenvalues

kð1Þ ¼ ðuþ aÞ; kð2Þ ¼ u; kð3Þ ¼ ðu� aÞ; ð7Þ
and the corresponding left and right eigenvectors may be written as follows:
Lð1Þ ¼ ð0; 1; 1=qaÞ; Rð1Þ ¼ ðq=a; 1; qaÞtr;Lð2Þ ¼ ð1; 0; �1=a2Þ; Rð2Þ ¼ ð1; 0; 0Þtr;Lð3Þ ¼ ð0; 1; �1=qaÞ; Rð3Þ ¼ ð�q=a; 1; �qaÞtr:

ð8Þ

For a characteristic shock, the shock curve coincides with a characteristic curve and its velocity coincides withan eigenvalue of the system, both ahead and behind the shock (see, Jeffrey [1], Boillat [20] and Anile et al. [21]).If the corresponding eigenvalue is simple, then the shock is a characteristic shock if, and only if, the wave isexceptional, namely $k.R = 0, where k is an eigenvalue of A, R is the corresponding right eigenvector, and $ isthe gradient operator with respect to U. The condition for the existence of a characteristic shock(i.e., $ k.R = 0) is satisfied corresponding to the eigen value k(2) = u, which implies that there exists a charac-teristic shock propagating with the speed V = u originating from a point, say, (x1, t0). Across a characteristicshock, the Rankine-Hugoniot conditions are given by [q] = f, [u] = 0 and [p] = 0, where f is an unknown func-tion of t, which needs to be determined; the square brackets enclosing an entity denote jump in that entityacross the characteristic shock, i.e., [U] = U � U*, where U* and U on the right hand side denote the valuesof U just ahead of the shock and behind the shock, respectively.

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

τ–1

ζ

b = 0

b = 0.2

b = 0.4

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

τ–1ζ

b = 0

b = 0.2

b = 0.4

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

τ–1

ζ

b = 0

b = 0.2

b = 0.4

a b c

Fig. 1. (a–c): Variation of f (Jump in Mass Density) with s�1 influenced by the van der Waals excluded volume b for Fig. 1(a) plane(m = 0), Fig. 1(b) cylindrical (m = 1), and Fig. 1(c) spherical (m = 2) flows.


The evolutionary law for f can be obtained by multiplying (6) by L(2), the left eigenvector of A correspond-ing to the eigenvalue k(2) = u, and then taking the jumps in the usual manner across the characteristic shock,i.e.,

Lð2Þd½U �

dtþ ½Lð2Þ�dU �

dt¼ Lð2Þ½B� þ ½Lð2Þ�B�; ð9Þ

where d/dt = o/ot + uo/ox denotes the material time derivative following the shock.Now, using (6) and (8) in (9), we obtain the following transport equation for f

dfdt¼ ff1� bð2q� fÞg

cpdpdt: ð10Þ

When the flow behind the characteristic shock is characterized by (5), the flow just ahead of the shock isU* = (q � f,u,p), where f satisfies the transport equation:

dfdt¼ �

ð1þ bfÞðt=t0Þðmþ1Þ � 2bq̂n o

ðt=t0Þðmþ1Þ � bq̂n o

tðmþ 1Þf; ð11Þ

which on integration yields

f ¼f0 sðmþ1Þ � bq̂� �

sðmþ1Þ sðmþ1Þ 1� bq̂þ bf0ð Þ � bf0f g ; ð12Þ

where s = t/t0 and f0 is the value of f at s = 1.Eq. (12) shows that f! 0 as s!1, i.e., the characteristic shock eventually decays to zero. However for a

given s, a small increase in b causes f to increase; indeed, the real gas effects (b > 0) result in slowing down thedecay rate of f relative to what it would be in a corresponding ideal gas (b = 0). Curves in Fig. 1 illustrate theevolutionary behaviour of the characteristic shock in a planar (m = 0), cylindrically symmetric (m = 1) andspherically symmetric (m = 2) flows.

4. Evolution of weak discontinuity

Let us suppose that the first derivative of the Cauchy data U (x, t0) has jump at x0 < x1; this amounts tosupposing that at t = t0, there are both a C1 discontinuity wave at x0 and a shock wave at x1. Let us considerthat the C1 discontinuity is propagating along the characteristic curve determined by dx

dt ¼ kð1Þ originating fromthe point (x0, t0) in the region U = (q,u,p)tr swept by the characteristic shock. Then the transport equation forthe C1 discontinuity is given by [22,23]


Lð1ÞdKdtþ ðU x þ KÞðrkð1ÞÞK

� �þ ðrLð1ÞÞK� �tr dU

dtþ ðLð1ÞKÞ ðrkð1ÞÞUx þ kð1Þx

� �� rðLð1Þf Þ� �

K ¼ 0; ð13Þ

where K, which denotes the jump in Ux across the C1 discontinuity, is collinear to the right eigenvector R(1),i.e., K ¼ ~pðtÞRð1Þ with ~pðtÞ as the amplitude of the C1 wave. Using (6), (7), (8), and (5) in (13), we obtain thefollowing transport equation for the wave amplitude:

dpdsþH1ðsÞpþH2ðsÞp2 ¼ 0; ð14Þ

where p,s,H1, and H2 are dimensionless quantities defined as p ¼ ~pt0, s = t/t0, H1 ¼ 52s 1þ ðmþ1Þðc�1Þbq̂

5ðsðmþ1Þ�bq̂Þ

n oþ ðmþ1Þðc�1Þsm

4ðsðmþ1Þ�bq̂Þ mþ 1

kðsðmþ1Þ�bq̂Þc�1

2 �1

� �and H2 ¼ ðcþ1Þsðmþ1Þ

2ðsðmþ1Þ�bq̂Þ with k ¼ 12ðmþ 1Þðc� 1Þk1ðcp̂=q̂Þ�1=2 as a positive

constant.

Eq. (14), on integration, yields the wave amplitude p as

pðsÞ ¼ p0wðsÞ1þ p0IðsÞ ; ð15Þ

where IðsÞ ¼R s

1H2ðsÞwðsÞds and

wðsÞ ¼ s�ð4�mÞ=2 sðmþ1Þ � bq̂1� bq̂

� ðc�1Þðm�1Þ�24 1� kðsðmþ1Þ � bq̂Þðc�1Þ=2

1� kð1� bq̂Þðc�1Þ=2

!�ðm2Þ:

It may be noticed that for values of m, c, and bq̂ considered here, the function w(s) is non-zero, finite andcontinuous on [1,s), and it approaches zero as s!1 with I(1) <1. Thus, it follows that for p0 > 0, (i.e.,an expansion wave), p(s)! 0 as s!1, implying thereby that the wave decays and dies out eventually;the corresponding situation is illustrated by the curves in Fig. 2(a–c), which show that the real gas effects serveto enhance the decaying of an expansion wave. However, for p0 < 0 (i.e., a compression wave), it follows from(15) that there are three possibilities:

(i) Let jp0j < pc, where pc = 1/{I(1)}. Then p is finite, non-zero, and continuous over [1,1) and p! 0 ass!1, since lim

s!1wðsÞ ¼ 0. Thus, there exists a critical value pc of the initial discontinuity such that if

jp0j < pc, then the wave decays; the corresponding situation is illustrated by the curves in Fig. 2(d–f) with�pc 6 p0 < 0.

(ii) Let jp0j > pc. Then there exists a finite time sc > 1, given by I(sc) = 1/jp0j such that p is finite, non-zero,and continuous on [1,sc) and jpj ! 1 as s! sc. This signifies the appearance of a shock wave at aninstant sc; indeed, a compression wave culminates into a shock in a finite time only when the initial dis-continuity associated with the wave exceeds a critical value. The corresponding situation is illustrated bythe curves in Fig. 2(d–f) with p0 < �pc < 0, showing that an increase in b serves to hasten the onset of ashock wave; however, the wavefront curvature has an opposite effect in the sense that the shock forma-tion time in a spherically symmetric flow is larger then for the cylindrically symmetric or a plane flow(see, Fig. 2(d–f)).

(iii) Let jp0j = pc. Then p is finite, non-zero, and continuous over [1,1); further, since H1

H2! 0 as s!1, it

follows that p! 0 as s!1, implying thereby that the wave ultimately decays; an evolutionary behav-iour of p is depicted in Fig. 2(d–f).

5. Collision of the weak discontinuity with the characteristic shock

The weak discontinuity originating from (x0, t0) is the fastest of those generated at x0, and it has a velocitygreater than that of the shock which originates from (x1, t0). We now envisage the situation when this C1 dis-continuity wave encounters the shock wave at time s = sp < sc so that there is no secondary shock formation inthe region behind the characteristic shock. At this time, there are reflected and transmitted discontinuity waves

1.5 2 2.5 3τ

0.25

0.5

0.75

1

1.25

1.5

1.75

ππ0

b=0.0b=0.2b=0.4

π0 >0, m=0

1.5 2 2.5 3τ

1

2

3

4

5b=0.0b=0.2b=0.4

π0 <−πc <0, m=0

−πc ≤π0 <0, m=0

1.5 2 2.5 3τ

0.25

0.5

0.75

1

1.25

1.5

1.75

2

b=0.0b=0.2b=0.4

π0 >0, m=1

1.5 2 2.5 3τ

1

2

3

4

5

b=0.0b=0.2b=0.4

π0 <−πc <0, m=1

−πc ≤π0 <0, m=1

1.5 2 2.5 3τ

0.25

0.5

0.75

1

1.25

1.5

1.75b=0.0b=0.2b=0.4

π0 >0, m=2

1.5 2 2.5 3τ

1

2

3

4

5

b=0.0b=0.2b=0.4

π0 <−πc <0,m=2

−πc ≤π0 <0,m=2

ππ0

ππ0

ππ0

ππ0

ππ0

a

b

c

d

e

f

Fig. 2. (a–f): Evolution of C1 wave influenced by the van der Waals excluded volume b for plane (m = 0), cylindrical (m = 1), and spherical(m = 2) flows.


traveling along the characteristics issued from the collision point (xp, tp). In order to study the amplitudes ofthe reflected and transmitted weak discontinuities, we consider the generalized conservation system which is adirect consequence of the original system (6), and have the following forms in the regions behind and ahead ofthe shock dx/dt = V, propagating with the speed V = u:

Gtðx; t; UÞ þ F xðx; t; UÞ ¼ Hðx; t; UÞ;Gtðx; t; U �Þ þ F xðx; t; U �Þ ¼ Hðx; t; U �Þ;

ð16Þ

where U = (q,u,p)tr and U* = (q � f,u,p)tr are the solution vectors to the left and just to the right of the shockcurve, and G, F, and H are given by

G ¼ q; qu;ð1� bqÞpðc� 1Þ þ

qu2

2

� tr

;

F ¼ qu; qu2 þ p;ð1� bqÞpðc� 1Þ þ

qu2

2þ p

� u

� tr

;

H ¼ �mqux;�mqu2

x;mpu

xbq� cðc� 1Þ

� � mqu3

2x

� tr

:

ð17Þ


Let P(xp, tp) be the point at which the fastest C1 discontinuity of (16)1, moving along the character-istic dx/dt = k(1) and originating from the point (x0, t0) intersects the discontinuity line dx

dt ¼ V . As in

[7], the amplitudes of the incident, reflected and transmitted waves on the discontinuity line are givenby the relations

KðP Þ ¼ pðtpÞRð1Þp ; KðRÞðPÞ ¼ aðtpÞRð1Þp ;

KðT ÞðP Þ ¼ bðtpÞRð1Þ�p ;ð18Þ

where the subscript p refers to the values evaluated at P on the shock. The evolutionary equations which, afterinteraction, determine the jump in the shock acceleration j½ _V �j ¼ _V tþp � _V t�p , and the amplitudes a and b ofreflected and transmitted waves, respectively, are the following:

j½ _V �jðG� G�Þp þ ðrGÞpaðV � kð3Þp Þ2Rð3Þp � ðr�G�pÞbðV � kð1Þ�p Þ

2Rð1Þ�p ¼ ðrGÞppðV � kð1Þp Þ2Rð1Þp ; ð19Þ

which is a system of three inhomogeneous algebraic equations for the unknowns j½ _V �j; a and b.We may note that at s = sp,

kð1Þ ¼ uþ cpqð1� bqÞ

� 1=2

; kð1Þ� ¼ uþ cpðq� fÞð1� bðq� fÞÞ

� 1=2

;

kð2Þ ¼ u; kð2Þ� ¼ u; ð20Þ

kð3Þ ¼ u� cpqð1� bqÞ

� 1=2

; kð3Þ� ¼ u� cpðq� fÞð1� bðq� fÞÞ

� 1=2

;

and thus, in order that the discontinuity V ¼ kð2Þ ¼ kð2Þ� is a physical shock, the following Lax evolutionaryconditions must hold [6]:

kð2Þ� ¼ V < kð1Þ� ; or equivalently kð3Þ < V ¼ kð2Þ ¼ kð2Þ� < kð1Þ:

This, in effect, asserts that when the incident wave with velocity k(1) at s = sp encounters the characteristicshock, it gives rise to one reflected wave with velocity k(3) and one transmitted wave with velocity kð1Þ� alongthe characteristics issuing from the collision point s = sp. The reflection and transmission coefficients a and band the jump in the shock acceleration j½ _V �j at the collision time s = sp can be determined from the algebraicsystem of Eqs. (19), i.e.,

ðG� G�Þpj½ _V �j þ ðrGÞpRð3Þp ðV � kð3Þp Þ2a� ðr�G�ÞpR�ð1Þp ðV � kð1Þ�p Þ

2b ¼ �ðrGÞpRð1Þp ðV � kð1Þp Þ2p: ð21Þ

In view of relations (7) and (8), the system (21) can be written as the following system of algebraic equations inthe unknowns j½ _V �j, a and b:

fj½ _V �j � aqa� a�ðq� fÞb ¼ �aqp;

fuj½ _V �j � aqðu� aÞa� a�ðq� fÞðuþ a�Þb ¼ �aqðuþ aÞp;

fu2

2� bp

c� 1

� j½ _V �j � aq

u2

2� bp

c� 1

� � a2uqþ a3qð1� bqÞ

c� 1

� �a

� a�ðq� fÞ u2

2� bp

c� 1

� þ a2

�uðq� fÞ þ a3�ðq� fÞð1� bðq� fÞÞ

c� 1

� �b

¼ � aqu2

2� bp

c� 1

� þ a2uqþ a3qð1� bqÞ

c� 1

� �p:

ð22Þ

The above algebraic system on solving, yields

Table 1(a–c): Amplitudes a and b of transmitted and reflected waves, and the shock acceleration j½ _V �j influenced by the van der Waals excludedvolume b in planar and radially symmetric flows

b a, m = 0 a, m = 1 a, m = 2

p0 > 0 p0 < 0 p0 > 0 p0 < 0 p0 > 0 p0 < 0

(a)0 �0.010269 0.032486 �0.0071446 0.024055 �0.0012399 0.00433720.2 �0.010658 0.042809 �0.010294 0.034873 �0.0019897 0.00814830.4 �0.0109064 0.050307 �0.016350 0.0550858 �0.035083 0.018594

b b, m = 0 b, m = 1 b, m = 2

p0 > 0 p0 < 0 p0 > 0 p0 < 0 p0 > 0 p0 < 0

(b)0 0.010276 �0.032511 0.0071586 �0.034894 0.0012412 �0.00434150.2 0.010665 �0.042841 0.0103053 �0.034910 0.0019921 �0.00815830.4 0.010911 �0.050327 0.016366 �0.0551558 0.035139 �0.018624

b j½ _V �j, m = 0 j½ _V �j, m = 1 j½ _V �j, m = 2

p0 > 0 p0 < 0 p0 > 0 p0 < 0 p0 > 0 p0 < 0

(c)0 �8.22897 25.9088 �4.2763 14.3979 �0.496111 1.735380.2 �11.6702 31.1226 �5.56354 18.8438 �0.714957 2.927940.4 �14.1886 38.4045 �8.06283 27.1728 �1.1003 5.8317


b ¼ aa�

1þ a�a

1� fq

� � �1

pðspÞ;

a ¼ � 1þ a�a

1� fq

� � �1

pðspÞ;

j½ _V �j ¼ � qf

1� fq

� a� þ 1þ f

q

� a

� �1þ a�

a1� f

q

� � �1

pðspÞ:

ð23Þ

The coefficients a and b determine the amplitude vectors K(R) = aR(3)(sp) and KðT Þ ¼ bRð1Þ� ðspÞ of the reflectedand transmitted waves propagating along the characteristic fronts with velocities k(3) and kð1Þ� , respectively.Moreover, an increase either in b or in the magnitude of the initial discontinuity p0 associated with the incidentwave, both cause the amplitudes a, b and the shock acceleration to increase in magnitude (see Table 1). It maybe noticed that if the incident wave is compressive (respectively, expansive), then the reflected wave is expan-sive (respectively, compressive), while the transmitted wave is compressive (respectively expansive). Eq. (23)demonstrate, as would be expected, that in the absence of incident wave (i.e., p0 = 0), the shock accelerationvanishes and there are no reflected or transmitted waves. Also, it is clear from (23)3 that after the impact, theshock will either accelerate or decelerate depending on whether the incident wave is compressive or expansive;the result is in agreement with the observation made by Courant and Friedrich [24], that if the shock front isovertaken by a compression (respectively, expansion) wave, it is accelerated (respectively, decelerated), andconsequently the strength of the shock increases (respectively, decreases).

6. Results and conclusion

Lie group analysis is used to obtain an exact solution of the equations that describe an unsteady planar andradially symmetric flows of a non-ideal gas. The evolution of the characteristic shock and that of a weak dis-continuity in a state characterized by the exact solution is studied. It is shown that the characteristic shockeventually decays; however the real gas effects (b > 0) result in slowing down the decay rate relative to whatit would be in a corresponding ideal gas (b = 0). The effects of van der Waals excluded volume on the decay ofcharacteristic shock for planar, cylindrical, and spherical motions are shown in Fig. 1(a–c). Evolutionary


behaviour of a weak discontinuity, influenced by the real gas effects (b > 0), is exhibited in Fig. 2(a–f). It isshown that a compression wave culminates into a shock after a finite time, only if the initial discontinuity asso-ciated with it exceeds a critical value i.e., p0 < �pc < 0; and an increase in the van der Waals excluded volume b

serves to hasten the onset of a shock wave. However, when �pc 6 p0 < 0 or p0 > 0, in both the cases the wavedecays eventually and an increase in b enhances the decaying of the weak discontinuity (see, Fig. 2(d–f)); how-ever, the wavefront curvature has an opposite effect in the sense that the shock formation time in a sphericallysymmetric flow is larger then for the cylindrically symmetric or a plane flow (see, Fig. 2(d–f)). It is shown thatwhen this weak discontinuity encounters the characteristic shock, it gives rise to a reflected wave and a trans-mitted wave with amplitudes a and b, respectively; these amplitudes are determined along with the jump inshock acceleration at the collision time. It is shown that an increase, either in b or in p0 (the magnitude ofthe initial discontinuity associated with the incident wave), both cause a, b, and j½ _V �j to increase in magnitude.It is observed that if the incident wave is compressive (respectively, expansive), then the reflected wave isexpansive (respectively, compressive), while the transmitted wave is compressive (respectively, expansive);also, it is noticed that after the impact, the shock will either accelerate or decelerate depending on whetherthe incident wave is compressive or expansive.

Acknowledgement

Authors are thankful to one of the referees for making certain point more explicit. Research support fromISRO-IIT Bombay, Space Technology Cell (Ref. No. 05-IS001) is gratefully acknowledged.

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Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas

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