research article an exact analytical solution of the...

Hindawi Publishing CorporationJournal of FluidsVolume 2013, Article ID 810206, 4 pageshttp://dx.doi.org/10.1155/2013/810206

Research ArticleAn Exact Analytical Solution of the Strong ShockWave Problem in Nonideal Magnetogasdynamics

S. D. Ram,1 R. Singh,2 and L. P. Singh2

1 Department of Mathematics, Mata Sundri College, University of Delhi, Delhi 110002, India2Department of Applied Mathematics, Indian Institute of Technology (BHU), Varanasi 221005, India

Correspondence should be addressed to S. D. Ram; [email protected]

Received 27 May 2013; Accepted 11 September 2013

Academic Editor: Miguel Onorato

Copyright © 2013 S. D. Ram et al.This is an open access article distributed under theCreativeCommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We construct the solutions to the strong shock wave problem with generalized geometries in nonideal magnetogasdynamics. Here,it is assumed that the density ahead of the shock front varies according to a power of distance from the source of the disturbance.Also, an analytical expression for the total energy carried by the wave motion in nonideal medium under the influence of magneticfield is derived.

1. Introduction

Thepropagation of shock waves, generated by a strong explo-sion in earth’s atmosphere is of great interest both frommath-ematical and physical point of view due to its numerous appli-cations in various fields. They result from a sudden releaseof a relatively large amount of energy; typical examples arelightening and chemical or nuclear explosions. Assume thatwe have an explosion, following which there may exist for awhile a very small region filled with hot matter at high pres-sure, which starts to expand outwards with its front headedby a strong shock. The process generally takes place in a veryshort time after which a forward-moving shock wave devel-ops, which continuously assimilates the ambient air into theblast wave.The study of strong shockwave problems has beenof long interest for researchers in fields ranging from con-densed matter to fluid dynamics due to its theoretical andpractical importance. Practically, it is recognized that strongshock waves are excellent means for generating very high-pressure, high temperature plasma at the center of explosion.Many authors, for example, Arora and Sharma [1], Sakurai[2, 3] and Rogers [4] have presented exact solutions for theproblem of strong shock wave with spherical geometry, sincethe study of spherically symmetric motion is important forthe theory of explosion in various gasdynamic regimes.Recently, Singh et al. [5, 6] presented an approximate ana-lytical solution to the system of first order quasilinear partial

differential equations that govern a one dimensional unsteadyplanar and nonplanar motion in ideal and nonideal gases,involving discontinuities.

The study of interaction between gasdynamic motion ofan electrically conducting medium and a magnetic field hasbeen of great interest to scientists and engineers due to itsapplication in astrophysics, geophysics, and interstellar gasmasses. Taylor [7] obtained exact solution of the equationsgoverning themotion in a gas generated by a point explosion.Singh et al. [8] studied the influence of the magnetic fieldupon the collapse of the cylindrical shock wave problem.Menon and Sharma [9] investigated the influence ofmagneticfield on the process of steepening and flattening of the char-acteristic wave front in a plane and cylindrically symmetricmotion of ideal plasma. Oliveri and Speciale [10] used thesubstitution principle to obtain an exact solution for unsteadyequation of perfect gas. Murata [11] obtained the exactsolution for the one dimensional blast wave problem withgeneralized geometry. Singh et al. [12] have used quasisimilartheory to construct an analytical solution for the strong shockwave problem with generalized geometries in a nonideal gassatisfying the equation of state of the Van der Waals type.

The present paper aims to construct the closed form solu-tion of the basic equations governing the one dimensionalunsteady flows of a nonideal gas involving strong shockwavesunder the influence of transverse magnetic field. The basicconfiguration investigated here is that which arises when

2 Journal of Fluids

a transverse magnetic field is generated by a current of finite,constant strength passing along a straight wire of infinitelength and either a shock or detonation wave propagates withuniform speed outwards from the wire into the ambientundisturbed gas at rest. Also, an expression for the total en-ergy carried by the wave motion is obtained.

2. Problem Formulation

The basic equations for unsteady flow of a one dimensionalgasdynamic motion may be written as [8, 12–14]

𝜕𝜌

𝜕𝑡+ 𝑢𝜕𝜌

𝜕𝑟+ 𝜌(

𝜕𝑢

𝜕𝑟+𝑚

𝑟𝑢) = 0, (1)

𝜕𝑢

𝜕𝑡+ 𝑢𝜕𝑢

𝜕𝑟+1

𝜌(𝜕𝑝

𝜕𝑟+𝜕ℎ

𝜕𝑟) = 0, (2)

𝜕𝑝

𝜕𝑡+ 𝑢𝜕𝑝

𝜕𝑟+ 𝑎2𝜌(𝜕𝑢

𝜕𝑟+𝑚

𝑟𝑢) = 0, (3)

𝜕ℎ

𝜕𝑡+ 𝑢𝜕ℎ

𝜕𝑟+ 2ℎ(

𝜕𝑢

𝜕𝑟+𝑚

𝑟𝑢) = 0, (4)

where 𝜌 is the gas density, 𝑢 is the velocity along the 𝑥-axis,𝑝 is the pressure, ℎ = 𝜇𝐻2/2 is the magnetic pressure with𝐻as the magnetic field strength, 𝜇 the magnetic permeability, 𝑡is the time, 𝑎 = (𝛾𝑝/𝜌(1 − 𝑏𝜌))1/2 is the speed of sound, 𝑟 isthe single spatial coordinate being either axial in flows withplaner (𝑚 = 0) geometry or radial in cylindrically symmetric(𝑚 = 1) and spherically symmetric (𝑚 = 2) flows, and 𝛾 isthe constant specific heat ratio.

The system of (1)–(4) is supplemented with a Van derWaals equation of state of the form:

𝑝 (1 − 𝑏𝜌) = 𝜌R𝑇, (5)

where 𝑏 is the Van der Waals excluded volume which isknown in terms of themolecular interaction potential in hightemperature gases,𝑇 is the absolute temperature, andR is thegas constant. It may be noted that the case 𝑏 = 0 correspondsto an ideal gas (ideal in the sense that the particle interactionsare absent).

The propagation velocity of the shock front 𝑈 may begiven as

𝑑𝑅

𝑑𝑡= 𝑈, (6)

where 𝑅 is the position of the shock front which is a functionof time 𝑡.

3. Rankine-Hugoniot Conditions

The Rankine-Hugoniot relations, given by the principle ofconservation of mass, momentum, and energy across theshock front may be expressed as [5, 6, 15]

𝜌 =𝛾 + 1

(𝛾 − 1 + 2𝑏)𝜌𝑎, (7a)

𝑢 =2 (1 − 𝑏)

𝛾 + 1𝑈, (7b)

𝑝 ={

{

{

2 (1 − 𝑏)

𝛾 + 1+2𝐶0((𝛾 − 1) 𝑏 − 𝛾)

(𝛾 − 1 + 2𝑏)2

}

}

}

𝜌𝑎𝑈2, (7c)

ℎ =𝐶0

2(

(𝛾 + 1)

(𝛾 − 1 + 2𝑏))

2

𝜌𝑎𝑈2, (7d)

where 𝑏 = 𝑏𝜌𝑎and 𝐶

0= 2ℎ𝑎/𝜌𝑎𝑈2 is the cowling number.

In the present problem the density 𝜌𝑎is taken to vary

according to the power law of radius of the shock front 𝑅,given as

𝜌𝑎= 𝜌0𝑅𝜆, (8)

where 𝜌0and 𝜆 are constants. The constant 𝜆 is determined

in the subsequent analysis.

4. Solution of the Strong Shock Wave Problem

We construct a relation for the pressure in the flow field sat-isfying the Rankine-Hugoniot relations (7a)–(7d) as

𝑝 ={

{

{

𝐶0(𝛾 + 1) (𝑏 + 𝛾) − (𝛾 − 1 + 2𝑏)

2

(𝑏 − 1) (𝛾 − 1 + 2𝑏)

}

}

}

𝜌𝑢2. (9)

Inserting (9) into (2) and (3) yields

𝜕𝑢


𝜕𝑟+{𝐶0(𝛾 + 1) + 4 (1 − 𝑏)} (𝛾 − 1 + 2𝑏)

8(1 − 𝑏)2

× (𝑢2

𝜌

𝜕𝜌

𝜕𝑟+ 2𝑢

𝜕𝑢

𝜕𝑟) = 0,

(10)

𝜕𝑢


𝜕𝑟+(𝛾 − 1)

2

+ (3𝛾 − 1) 𝑏

2 (𝛾 − 1) (1 − 𝑏)𝑢(𝜕𝑢

𝜕𝑟+𝑚

𝑟𝑢) = 0.

(11)Combining (10) and (4) and integrating the resulting equa-tion we get

𝑆 (𝑡) = 𝜌𝑢2−𝛼𝑟−𝑚𝛼, (12)

where 𝑆(𝑡) is a arbitrary function of integration. Also, 𝛼 isconstants given as

𝛼 =4 (1 − 𝑏) {(𝛾 − 1)

2

+ (3𝛾 − 1) 𝑏}

(𝛾 − 1) {(𝐶0(𝛾 + 1) + 4 (1 − 𝑏)) (𝛾 − 1 + 2𝑏)}

. (13)

Journal of Fluids 3

Using the solution (12), (1) takes the form

(2 − 𝛼)

𝑢

𝜕𝑢

𝜕𝑡+ (1 − 𝛼)

𝜕𝑢

𝜕𝑟− (1 + 𝛼)

𝑚𝑢

𝑟−1

𝑆

𝑑𝑆

𝑑𝑡= 0. (14)

Solving (4) and (14), we have

𝑢 = − Ω𝑟

𝑆

𝑑𝑆

𝑑𝑡, (15)

whereΩ = 1/{𝛿 + 𝑚(𝛼 + 𝛿)} is a constant and 𝛿 given as

𝛿 = 1 + (1 −𝛼

2)((𝛾 − 1)

2

+ 𝑏 (3𝛾 − 1)

(𝛾 − 1) (1 − 𝑏)) . (16)

Plugging the value of 𝑢 from (15) into (14), yields

𝑆 (𝑡) = 𝑆0𝑡−𝜂. (17)

Here, 𝑆0is arbitrary constant and constant 𝜂 is given as

𝜂 =(𝛼 − 2)

(𝛼 − 1)Ω + (𝛼 + 1)𝑚Ω − 1. (18)

Using the Rankine-Hugoniot relations (7a)–(7d) and (15)in (6) yield the analytical expression for the radius of theshock front as

𝑅 (𝑡) = 𝑡((𝛾+1)/2)(Ω𝜂/(1−𝑏))

. (19)

With the help of Rankine-Hugoniot condition (7a)–(7d),we may determine the value of the constant 𝜆 as

𝜆 =(𝛾 + 1) (𝑚 − 1) − 2 (1 − 𝑏) [(𝛼 + 1) (𝑚 + 1) − 2]

(𝛾 + 1). (20)

Consequently, the analytical solution of the blast waveproblem described in the prior section is given as

𝑢 = Ω𝜂𝑟

𝑡,

𝜌 =1

(Ω𝜂)(2−𝛼)

𝑆0𝑟(𝑚+1)𝛼−2

𝑡(2−𝛼−𝜂)

,

𝑝 ={

{

{

𝐶0(𝛾 + 1) (𝑏 + 𝛾) − (𝛾 − 1 + 2𝑏)

2

(𝑏 − 1) (𝛾 − 1 + 2𝑏)

}

}

}

×1

(Ω𝜂)−𝛼𝑆0𝑟(𝑚+1)𝛼

𝑡−(𝛼+𝜂)

,

ℎ ={

{

{

(𝐶0(𝛾 + 1) + 4 (1 − 𝑏)) (𝛾 − 1 + 2𝑏)

4(1 − 𝑏)2

−𝐶0(𝛾 + 1) (𝑏 + 𝛾) − (𝛾 − 1 + 2𝑏)

2

(𝑏 − 1) (𝛾 − 1 + 2𝑏)

}

}

}

×1

(Ω𝜂)−𝛼𝑆0𝑟(𝑚+1)𝛼

𝑡−(𝛼+𝜂)

.

(21)

For a spherical motion of an ideal gas, the solutions (21) re-cover the results presented by some authors [2, 4, 7, 11].

5. Behaviour of the Energy

The total energy carried by thewavemotion in nonidealmed-ium under the influence of transverse magnetic field is ex-pressed as [15]

𝐸 = 4𝜋∫

𝑅

0

{1

2𝜌𝑢2+(1 − 𝑏𝜌)

𝛾 − 1𝑝 + ℎ} 𝑟

𝑚𝑑𝑟, (22)

where 𝐸 is the function of time and represents the sum of thekinetic and internal energy of the gas.

Using solutions (21) into (22), we have

𝐸 (𝑡) = Π𝑡−(𝛼+𝜂)+((𝑚+1)(𝛾+1)(𝛼+1)Ω𝜂/2(1−𝑏))

. (23)

The constant Π appearing in the above relation may beexpressed as

Π =4𝜋

(𝑚 + 1) (𝛼 + 1)

×{

{

{

1 +𝐶0(𝛾 + 1)

3

4(1 − 𝑏)2

+ (4𝑏2

− 𝑏 (4 + 𝐶0+ (𝐶0− 4) 𝛾)

+ (𝛾 − 1)2

− 𝐶0(𝛾 + 1)) ((𝛾 − 1 + 2𝑏)

2

)

−1}

}

}

×𝑆0

(Ω𝜂)−𝛼.

(24)

6. Result and Discussion

The solution (21), which is an exact solution for nonidealmagnetogasdynamic problems, obtained in the form of apower in the distance and time, is same as provided by [6–11]for strong andweak shockwaves in ideal gases. However, for aplanar and nonplanar nonideal magnetogasdynamic motion,the solutions (21) are a new exact solution in case of strongshock waves for arbitrary values of adiabatic index. From thesolution equations (21), it is also clear that in the absence ofmagnetic field solutions coincides with the solution given bySingh et al. [5].

It may be noted here that the total energy carried by thewave remains constant in planar case of ideal gas, whereas incase of nonideal medium it varies with respect time 𝑡 and isgiven by (23). Also, due to increase in magnetic field strengthtotal energy monotonically decreases with time 𝑡 and thisprocess is slowed down due to the increase in Van der Waalsexcluded volume.

7. Conclusion

In the present paper, a new exact solution is derived for astrong shockwave problem in nonidealmagnetogasdynamics

4 Journal of Fluids

with the density ahead of the shock which is varied as a powerof the distance from the origin of the shock wave. The effectof coupling between the nonideal effects and magnetogasdy-namics phenomena on the flow field is analyzed. The exactsolution presented in (21) becomes the same results, for theabsence of magnetic field, presented by Singh et al. [5]. Thebehavior of the total energy is also presented by (23).

Acknowledgment

R. Singh acknowledges the financial support from the UGC,New Delhi, India, under the SRF scheme.

References

[1] R. Arora and V. D. Sharma, “Convergence of strong shock in aVan der Waals gas,” SIAM Journal on Applied Mathematics, vol.66, no. 5, pp. 1825–1837, 2006.

[2] A. Sakurai, “On the propagation and structure of the Blast wave,I,” Journal of the Physical Society of Japan, vol. 8, no. 5, pp. 662–665, 1953.

[3] A. Sakurai, “On the propagation and structure of a Blast wave,II,” Journal of the Physical Society of Japan, vol. 9, no. 2, pp. 256–266, 1954.

[4] M. H. Rogers, “Analytic solutions for the Blast-waves problemwith an atmosphere of varying density,” Astrophysical Journal,vol. 125, pp. 478–493, 1957.

[5] L. P. Singh, S. D. Ram, and D. B. Singh, “Analytical solution ofthe Blast wave problem in a non-ideal gas,” Chinese PhysicsLetters, vol. 28, no. 11, Article ID 114303, 2011.

[6] L. P. Singh, S. D. Ram, andD. B. Singh, “Exact solution of planarand nonplanar weak shock wave problem in gasdynamics,”Chaos, Solitons & Fractals, vol. 44, no. 11, pp. 964–967, 2011.

[7] J. L. Taylor, “An exact solution of the spherical Blast waveproblem,” Philosophical Magazine, vol. 46, pp. 317–320, 1955.

[8] L. P. Singh, S. D. Ram, and D. B. Singh, “The influence of mag-netic field upon the collapse of a cylindrical shock,”Meccanica,vol. 48, no. 4, pp. 841–850, 2013.

[9] V. V. Menon and V. D. Sharma, “Characteristic wave fronts inmagnetohydrodynamics,” Journal of Mathematical Analysis andApplications, vol. 81, no. 1, pp. 189–203, 1981.

[10] F. Oliveri and M. P. Speciale, “Exact solutions to the unsteadyequations of perfect gases through Lie group analysis and sub-stitution principles,” International Journal of Non-Linear Me-chanics, vol. 37, no. 2, pp. 257–274, 2002.

[11] S. Murata, “New exact solution of the Blast wave problem in gasdynamics,” Chaos, Solitons and Fractals, vol. 28, no. 2, pp. 327–330, 2006.

[12] L. P. Singh, S. D. Ram, and D. B. Singh, “Quasi-similar solutionof the strong shock wave problem in non-ideal gas dynamics,”Astrophysics and Space Science, vol. 337, no. 2, pp. 597–604, 2012.

[13] R. Courant and K. O. Friedrichs, Supersonic Flow and ShockWave, Wiley, New York, NY, USA, 1948.

[14] G. B.Whitham, Linear andNon-LinearWaves,Wiley, NewYork,NY, USA, 1974.

[15] C. C. Wu and P. H. Roberts, “Structure and stability of a spher-ical shock wave in a van der Waals gas,” Quarterly Journal ofMechanics and Applied Mathematics, vol. 49, no. 4, pp. 501–543,1996.

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