formation of shock waves in reactive magnetogasdynamic flow

Formation of shock waves in reactive magnetogasdynamic¯ow

Dheeraj Bhardwaj *

Visiting Scientist, Department of Mathematics, Eastern Illinois University, Charleston, IL 61920, USA

Received 15 September 1998; accepted 9 June 1999

Abstract

The propagation of acceleration waves in reactive magnetogasdynamic ¯ow, which is induced by themotion of a piston advancing with a ®nite acceleration in to a constant state of rest has been studied in thearticle along with the characteristic path by using the characteristics of the governing quasilinear system asthe reference co-ordinate system. A di�erential equation governing the growth and decay of an accelerationwave is derived and integrated numerically. The critical time for formation of shock wave is obtained. Thee�ect of magnetic ®eld and chemical reaction on shock formation time are studied. Ó 2000 Elsevier ScienceLtd. All rights reserved.

Keywords: Reactive shock; Magnetogasdynamic ¯ow; Acceleration waves; Critcal time; Critical amplitude

1. Introduction

A large amount of attention has recently been focused on reactive shock problem [1±4] and alsoon the phenomena associated with the propagation of acceleration waves in di�erent gaseousmedia [5]. These references contain a complete bibliography of other contributions in this ®eld.The objective of this article is to show the formation of shock waves in reactive magnetogasdy-namic ¯ow. More speci®cally, here the main emphasis is to show that how the acceleration wavesterminate into shock wave in such ¯ows. Acceleration wave is a special class of nonlinear waveprocess which admits analytic solutions. Indeed, one of the most interesting problem on thetheory of acceleration waves in continuous media is the process of formation of shock waves.Thus, in the present work the propagation of acceleration waves in reactive magnetogasdynamic

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International Journal of Engineering Science 38 (2000) 1197±1206

*Permanent address: Centre for Development of Advanced Computing, Scienti®c and Engineering Computing

Group, Pune University Campus, Ganesh Khind, Pune 411 007, India. Tel.: +91-20-5679265; fax: +91-20-5657551.

E-mail address: [email protected] (D. Bhardwaj).

0020-7225/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S0020-7225(99)00071-3

¯ow is studied by applying co-ordinate system as in [6]. Precisely the e�ects of magnetic ®eld andreaction process on the steeping and ¯attering of acceleration waves are studied.

2. Mathematical model

Under the assumption that the magnetic ®eld is azimuthal, the set of nonlinear partial di�er-ential equations governing the reactive megnetogasdynamic ¯ow is given by [4]

oqot� u

oqox� q

ouox� qu

x� 0; �1�

ouot� u

ouox� 1

qopox� B

qloBox� B2

lqx� 0; �2�

opot� u

opox� cp

ouox� cpu

x� QR�p;q;b�; �3�

oBot� u

oBox� B

ouox� 0; �4�

obot� u

obox� R�p;q;b�; �5�

p � �cÿ 1�qe; �6�

where q denotes the gas density, p the pressure, B the azimuthal magnetic ®eld, l the magneticpermeability, u the gas velocity in the direction of the motion of the piston, x the radial distancefrom the origin of symmetry, e the speci®c internal energy, c the adiabatic gas constant, Q thespeci®c energy of formation of the detonation product and QR�q; p;b� the speci®c energy input tothe system for any given p; q and b.

Recasting the above Eqs. (1)±(5) in the matrix form, we get

Ut � GUx � H � 0; �7�

where

U �

q

u

p

B

b

26666666666664

37777777777775; G �

u q 0 0 0

0 u 1=q B=ql 0

0 cp u 0 0

0 B 0 u 0

0 0 0 0 u

26666666666664

37777777777775; H �

ÿqu=x

ÿB2=lqx

ÿcpu=x� QR

0

R�p; q; b�

26666666666664

37777777777775:

A function U�x; t� satisfying (7) every where except at a characteristic curve s�t�, where U iscontinuous, but Ux and Ut may su�er from ®nite jumps, is said to be a week solution or a weak

1198 D. Bhardwaj / International Journal of Engineering Science 38 (2000) 1197±1206

discontinuity at that curve. In mechanical problems such discontinuities are called accelerationwaves [7]. These waves propagate along the characteristics. Denoting the jump of a quantity Uacross s�t� by �U �, we have along s�t�

ddt�U � � �Ut� � ds

dt�Ux�; �8�

where d=dt is the time derivative as observed from the wave front.Now, taking the jump in (7), using (8) and applying the condition of continuity �U � � 0, we

obtain

G

"ÿ ds

dtI

#�Ux� � 0: �9�

Eq. (9) shows that if ®nite discontinuities of acceleration along the characteristic curve occur, thecharacteristic speed of propagation ds=dt is an eigen value of G. It follows immediately that thereare ®ve families of characteristic curves, two of which

dxdt� u� Af �10�

represent waves propagating with �x-direction with the Alfven wave speed Af of the gas given by

Af ��B2

ql� c2

s; �11�

where c � ��cp=q

pis the local speed of sound, and remaining three form triple characteristics

dxdt� u �12�

representing the particle path.

3. Coordinate system and wave equation

The system of equations (7) is of hyperbolic type [4]. In dealing with the hyperbolic system ofpartial equations, it is convenient to use characteristic coordinates as a reference frame. Thus, weshall choose the x-axis in the direction of propagation of the wave and follow the procedure asgiven by Chu [6]; that is we introduce two characteristic variables a and w de®ned as follows:· w is a particle tag so that w is constant along a particle path

dxdt� u

D. Bhardwaj / International Journal of Engineering Science 38 (2000) 1197±1206 1199

in the �x; t� plane. If the characteristic wave front traverses a particle at time t�, this particle andits path will be labeled by w � t�.

· a is a wave tag, so that a is a constant along an outgoing characteristic

dxdt� u� Af

in the �x; t� plane. Moreover an outgoing wave generated by the piston at the instant t � t� willbe labeled as the outgoing wave a � t�. In particular, if the piston starts from rest, the wavefront will be the characteristic surface a � 0 (unless the later is transformed into or overtakenby a shock). A typical particle path and a typical trajectory of an outgoing wave in the �x; t�plane are shown in Fig. 1.Thus, we conclude that for each pair of value �a;w� there is a corresponding pair �x; t� so that

x � x�a;w�; t � t�a;w�. From the de®nition of a and w, the following partial di�erential equationswill be satis®ed

oxoa� u

otoa;

oxow� �u� Af � ot

ow: �13�

Next, denoting partial di�erentiation with respect to the characteristic variables a and w bysubscripts, we have

ut � uwxa ÿ uaxw

J; �14�

ux � uatw ÿ uwtaJ

; �15�

where

J � o�x; t�o�a;w� � ÿAf tatw:

Fig. 1. Characteristics labelling and coordinate systems.


Since `doubling up' or overlapping of ¯uid particles is prohibited from physical consideration,tw 6� 0. Consequently, J � 0 if and only if ta � 0, when two adjoining characteristics merge into ashock wave.

Now, by using (14) and (15), the system of equations (1)±(5) can be transformed into thefollowing equivalent system:

Af twqa ÿ quatw � quwta ÿ quAf tatwx�a;w� � 0; �16�

Af twua ÿ 1

qpatw � 1

qpwta ÿ BBa

qltw � BBw

qlta � B2Af tatw

lqx�a;w� � 0; �17�

Af twpa ÿ cpuatw � cpuwta � cpux�a;w�Af tatw � QRAf tatw; �18�

Af twBa ÿ Buatw � Buwta � 0; �19�ba � R�p; q; b�ta: �20�

Incorporating the Eqs. (18) and (19) into (17), we obtain

pw � qAf uw � cpux�a;w� tw �

BBw

l� B2Af

lx�a;w� tw � QRtw: �21�

Since the wave front is a characteristic surface, the ¯ow variables are continuous across thesurface. Thus, the boundary conditions at the wave front are

�q� � 0; �u� � 0; �B� � 0; �p� � 0; �b� � 0; t � w at a � 0: �22�

Further, the gas ¯ow ahead of the wave is homogeneous and at rest, we observed that Eq. (22)demands

qw � 0; uw � 0; Bw � 0; pw � 0; bw � 0; tw � 1 at a � 0: �23�

It follows from (22) and (23) that (21) is an identity when evaluated at the wave front. Eq. (17)yields

pa � q0Af0ua ÿ �B2

0ua=lAf0�; at a � 0 �24�

evaluation of (13) results in

xa � 0; xw � Af0; at a � 0; �25�

where the subscript 0 denote ¯ow variables in the undistributed region ahead of the wave front.To evaluate the amplitude a � ux of the acceleration wave at the front, we set a � uxja�0 and

invoke Eqs. (13) and (23) to obtain


a � ÿ ua

Af0ta

at a � 0: �26�

We proceed to discuss the dependence of ua and ta on time by di�erentiating (21), (24) and (13)with respect to a and w to obtain

uaw

ua� c2

0

2q0Af0

q0

w

(� B2

0

2lwc20

5

"� c2

0

A2f0

�cÿ 2�#ÿ Q

oRop

� �0

)a; �27�

taw

ta� �c� 1�

2

"� B2

0�1ÿ c=2�lq0c2

f0

#a: �28�

Now, di�erentiating (26) with respect to w and using (27) and (28), we obtain

dadw� h

1

2w

�� B2

0

4lq0wc20

�5� h�cÿ 2�� ÿ Q2

oRop

� �0

�a� �c� 1�

2

(� B2

0

lq0A2f0

�1ÿ c=2�)

a2 � 0:

�29�

This equation governs the growth and decay of the acceleration wave a � 0.In order to study the behaviour of acceleration waves, we consider the nondimensional form of

the Eq. (29) as

d/dg� A0

�� A1

1� 2g

�/� A2/

2 � 0; �30�

where

A0 � Wh; A1 � h�� 5� h�cÿ 2�

2

� ��1ÿ h�

�; A2 � C 1

�� 1ÿ h��2ÿ c�

�1� c��;

/ � a�0�a��0� ; C � �c� 1�a�w�; W � Q

oRop

� �0

w�=q0; h � c20=A2

f0; a� � ou

ox�x; t��

� �;

and w� is the value of w at t � t�.Integrating Eq. (30), we get

/ � �1n� 2g�A1=2

exp�A0g��1� A2F �g��oÿ1

; �31�

where

F �g� �Z g

0

exp�ÿA0x��1� 2x�A1=2

dx: �32�


4. Discussion

Eq. (31) shows that the characteristics will pile up at the wave front to form a shock wave,provided ta vanishes, i.e.

1� A2F �g� � 0: �33�

Since F �g�P 0, this shows that only compressive wave front C < 0 may terminate into shockwave. The expressions for the critical amplitude ac and the critical time tc for the formation of acylindrical wave are given by

ac � ��1� 2g�A1=2exp�ÿA0g�� ÿ 1

F �g��c� 1�w� 1� �1ÿ h��2ÿ c�=�1� c��

( )�34�

and

tc � t� � 2t�

A0

logA2

A0 � A2

� �; �35�Z gc

0

exp�ÿA0x��1� 2x�A1=2

dx � ÿ 1

A2

;

where gc � �tc ÿ t��=2t� .We consider c � 1:4 and an Arrhineus kind of reaction i.e.

R � exp�ÿN=e� �36�

to give clear picture to above expression. Here, N is the activation energy. Hence,

oRop

� �0

� N�cÿ 1�c2

q0c40

exp�ÿNc�cÿ 1�=c20� �

N�cÿ 1�c2

q0c40

�R0�: �37�

Now, we proceed to study the shock formation. Integral curves of Eq. (30) are depicted in Figs. 2±5 for di�erent cases. The e�ect of magnetic ®eld on the compressive waves in reactive magneto-gasdynamic ¯ow is shown in Fig. 2. It is clear from Fig. 3 that the decay rate of expansion wave isdecreased by the presence of a magnetic ®eld. The e�ect of magnetic ®eld with chemical reactionon the compressive and expansion waves which are emanating from piston movement are shownin Figs. 4 and 5. We could conclude from these ®gures that the coupling of chemical reaction andthe magnetic ®eld will cause delay in the shock formation and the point of breakdown movesoutward along the leading characteristic while on those of of expansion waves, the e�ect is toenhance the decay rate. Fig. 4 shows that the ®nite increase in the piston acceleration amounts toan early shock formation in reactive magnetogasdynamic ¯ow.


Fig. 3. E�ect of magnetic ®eld on expansion waves in reactive magnetogasdynamic ¯ow.

Fig. 2. E�ects of magnetic ®eld on the growth of acceleration waves in reactive gasdynamic ¯ow.


5. Conclusion

Thus, from above we conclude that the e�ect of the magnetic ®eld on compressive waves inreactive gasdynamic ¯ow is to cause on early shock formation, but the e�ect of magnetic ®eld on

Fig. 4. E�ect of magnetic ®eld and chemical reaction on the growth of accleration waves emanating from piston

movement.

Fig. 5. Decay of expansion waves produced by a ®nite accleration in a reactive magnetogasdynamic ¯ow.


expansion waves in reactive gasdynamic ¯ow is to decrease the decay rate. However the combinede�ect of reaction and magnetic ®eld on compressive waves which are emanating from pistonmovement is to slow down the motion of a breakdown point and thus to increase the shockformation time in such ¯ow, but the e�ect on the expansion waves is to enhance the decay rate.

Acknowledgements

Author wishes to express his gratitude to Prof. S.K. Dey, Department of Mathematics, EIU(USA) for his valuable suggestions and to the Department of Mathematics, Eastern IllinoisUniversity, Charleston (USA) for providing the facilities to carry out this work. Author alsowishes to thank Centre for Development of Advanced Computing, Pune (India) for providingcomputing facilities and permission to publish this article.

References

[1] H.M. Sternberg, Quart. J. Mech. Appl. Math. 23 (1970) 77±79.

[2] J.D. Logan, J. Peres, SIAM J. Appl. Math. 39 (1980) 512±527.

[3] R.P. Gilbert, A. Je�rey, J. Math. Anal. Appl. 89 (1982) 193±202.

[4] R. Shankar, D. Bhardwaj, J. Math. Anal. Appl. 179 (1993) 335±348.

[5] R. Shankar, L. Dixit, Fluids 30 (1987) 2194±2198.

[6] Peter P. Wegener (Ed.), BOA-TEH Chu, Nonequilibrium Flows, Part II, Marcel Dekker, New York, 1970, pp. 37±

81.

[7] P.J. Chen, Selected Topics in Wave Propagation, Noordhoo� Leyden, The Netherlands, 1976.


formation of shock waves in reactive magnetogasdynamic flow

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