Evolution of weak discontinuity in a van der Waals gas

Ain Shams Engineering Journal (2015) 6, 1129–1132

Ain Shams University

Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

Evolution of weak discontinuity in a van der Waals gas B. Bira a b

a,* ,

T. Raja Sekhar

b

Department of Mathematics, National Institute of Science and Technology, Palur Hills, Berhampur-8, India Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-2, India

Received 1 August 2014; revised 6 February 2015; accepted 1 March 2015 Available online 18 April 2015

KEYWORDS Van der Waals gas; Lie symmetry analysis; Exact solution; Weak discontinuities

Abstract In this article, the Lie symmetries analysis that leaves the system of partial differential equations (PDEs), governed by the one dimensional unsteady flow of an isentropic, inviscid and perfectly conducting compressible fluid obeying the van der Waals equation of state invariant, is presented. Using these symmetries the governing system of PDEs is reduced into system of ordinary differential equations (ODEs). Then the reduced system of ODEs is solved analytically which in turn produces the exact solution for the governing PDEs. Further, the influence of the van der Waals excluded volume in the behavior of evolution of weak discontinuity is studied extensively. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Many physical phenomena in this universe are modeled by systems of nonlinear hyperbolic PDEs. The explicit determination of exact solutions to such system of nonlinear PDEs of physical interest is an important task. To solve such system of nonlinear PDEs, no general theory is available there and it is also very difficult to systematically construct their exact solutions. Lie group analysis is one of the systematic and most powerful techniques to obtain the exact solutions for such nonlinear system of PDEs (see, [1–4]). This technique has been applied * Corresponding author. E-mail addresses: [email protected], birabibekananda@gmail. com (B. Bira). q Peer review under responsibility of Ain Shams University.

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by many researchers to solve different flow phenomena over different geometries. Similarity solutions for three dimensional Euler equations using Lie group analysis is found in [5], whereas the use of same technique to obtain some exact solutions to the ideal magnetogasdynamic equations is described in [6]. Exact solution to axisymmetric flow of shallow water equations by Lie group point transformation is found in [7] whereas in [8], the author derived the self similar solutions for system of PDEs describing a plasma with axial magnetic field (h-pinch). The work in [9,10] accounts symmetry reductions, group invariant solutions and some exact solutions of (2 + 1)-dimensional Jaulent–Miodek equation. Propagation of weak discontinuities in binary mixture of ideal gases in one-dimensional ideal isentropic magnetogasdynamics has been studied in [11,12] while the interaction of a weak discontinuity wave with the elementary waves for the Euler equations governing the flow of ideal polytropic gases is investigated in [13]. In [14], they found the extensive study on evolution of weak discontinuities in a two-dimensional steady supersonic flow of a non-ideal radiating gas.

http://dx.doi.org/10.1016/j.asej.2015.03.003 2090-4479 Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1130

B. Bira, T. Raja Sekhar

In the present work, Lie group transformations have been used to reduce the governing system of PDEs to a system of ODEs. We solve the reduced system of ODEs and a particular exact solution to the governing system of PDEs is obtained. Further, we discuss the evolution of weak discontinuity in the presence of van der Waals excluded volume. 2. Group analysis The one dimensional unsteady flow of an isentropic, inviscid and perfectly conducting compressible fluid obeying the  c q equation of state p ¼ k 1aq can be written as [15] qt þ uqx þ qux ¼ 0; ut þ uux þ

B kcqc2 Bx þ qx ¼ 0; q ð1  aqÞcþ1

where q; u and B are density, velocity and magnetic field respectively. Here a is the van der Waals excluded volume, k is a positive constant, c is the adiabatic exponent and the independent variables t and x denote time and space respectively. Now, we consider Lie group of transformations with the independent variables x and t, and with the dependent variables q; u; B for the current problem as t ¼ t þ /1 ðx;t;q;u;BÞ þ Oð2 Þ; x ¼ x þ /2 ðx;t;q;u;BÞ þ Oð2 Þ; q ¼ q þ w1 ðx;t; q;u; BÞ þ Oð2 Þ; u ¼ u þ w2 ðx; t;q; u;BÞ þ Oð2 Þ; B ¼ B þ w3 ðx; t; q; u; BÞ þ Oð2 Þ; ð2Þ

where /1 ; /2 ; w1 ; w2 and w3 are the generators to be determined such that the system of PDEs (1) remains invariant with respect to the transformations (2) and  is very small group parameter. A straightforward analysis [1], provides us the following infinitesimal transformations:

w2 ¼ a4 ;

/2 ¼ a3 þ a4 t þ a2 x;

w1 ¼ 0;

w3 ¼ 0;

ð3Þ

where a1 ; a2 ; a3 and a4 are arbitrary constants. The similarity variables can be obtained from the characteristic equations given as below: dt dx dq du dB ¼ ¼ ¼ ¼ ; /1 /2 w1 w2 w3

ð4Þ

ð7Þ Substituting the variables from (7) in (1) we obtained   a3 dR dU U þR ¼ 0; dn a1 dn   a3 dU P dP kcRc2 dR þ þ ¼ 0; U a1 dn R dn ð1  aRÞcþ1 dn   a3 dP dU þP ¼ 0: U dn a1 dn

dt dx dq du dB ¼ ¼ ¼ ¼ : a1 þ a2 t a3 þ a4 t þ a2 x 0 a4 0

ð8Þ

The system of ODEs (8) can be solved numerically. Case C: a1 ¼ 0 and a2 ¼ 0. This case yields the similarity and dependent variables as follows: a4 x n ¼ t; q ¼ R; u ¼ þ U; B ¼ P; ð9Þ a3 þ a4 t which reduces (1) to system of ODEs as dR a4 þ R ¼ 0; dn a3 þ a4 n dU a4 þ U ¼ 0; dn a3 þ a4 n dP a4 þ P ¼ 0: dn a3 þ a4 n

ð10Þ

Further, we obtain the solution of (10) as R¼

C1 ; a3 þ a4 n

U¼

C2 ; a3 þ a4 n

P¼

C3 ; a3 þ a4 n

ð11Þ

where C1 ; C2 and C3 are arbitrary integration constants. The corresponding solution of (1) is q¼

i.e.,

ð6Þ

which can be solved numerically. Case B: a1 – 0 and a2 ¼ 0. For this case we obtained the similarity and dependent variables as follows:   a4 2 a3 a4 n¼x t þ t ; q ¼ R; u ¼ t þ U; B ¼ P: 2a1 a1 a1

ð1Þ

Bt þ uBx þ Bux ¼ 0;

/1 ¼ a1 þ a2 t;

  a4 dR dU U þR ¼ 0; dn a1 dn   a4 dU P dP kcRc2 dR þ þ ¼ 0; U a1 dn R dn ð1  aRÞcþ1 dn   a4 dP dU þP ¼ 0; U a1 dn dn

C1 ; a3 þ a4 t

u¼

a4 x þ C2 ; a3 þ a4 t

P¼

C3 : a3 þ a4 t

ð12Þ

3. Evolution of weak discontinuities

Now we consider different cases to obtain the solutions for the governing system of PDEs (1). Case A: a1 ¼ 0 and a2 – 0. The similarity variable and new dependent variables are   a4 a3 a4  t ln ta2 ; q ¼ R; u ¼ ln t þ U; B ¼ P: n¼ xþ a2 a2 ð5Þ Using (5) in (1), we obtain the following reduced system of ODEs:

The matrix form of the governing hyperbolic system is Wt þ HWx ¼ 0;

ð13Þ

where W ¼ ðq; u; BÞT is a column vector with superscript T denoting transposition, while H is a matrix with elements 2 H11 ¼ H22 ¼ H33 ¼ u; H12 ¼ q; H21 ¼ cq ; H13 ¼ H31 ¼ 0; H23 ¼ B ; H32 q

c1

kcq ¼ B, where c2 ¼ ð1aqÞ The matrix H has the cþ1 .

eigenvalues

Evolution of weak discontinuity

1131

k1 ¼ u  w;

k2 ¼ u; k3 ¼ u þ w; ð14Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where w ¼ c2 þ b and b ¼ Bq with the corresponding left and right eigenvectors l1 ¼ ðc2 ; wq; BÞ;

r1 ¼ ðq; w; BÞT ;

l2 ¼ ðB; 0; qÞ;

r2 ¼ ðB; 0; c2 Þ ;

2

l3 ¼ ðc ; wq; BÞ;

T

ð15Þ

T

r3 ¼ ðq; w; BÞ :

The transport equation for the weak discontinuity across the third characteristic of a hyperbolic system of equations is given by [16]:   dK dW l3 þ ðWx þ KÞðrk3 ÞK þ ððrl3 ÞKÞT dt dt þ ðl3 KÞððrk3 ÞWx þ ðk3 Þx Þ ¼ 0;

ð16Þ

@ @ @ where K ¼ br3 and r ¼ ð@q ; @u ; @B Þ; K is the jump in Wx across the weak discontinuity wave with amplitude b, propagating along the curve determined by dx ¼ k3 originating from the dt point ðx0 ; t0 Þ. Substitution of (12) together with (14) and (15) in (16) leads to the following Bernoulli type of equation for the amplitude b:

db þ H1 ðx; tÞb2 þ H2 ðx; tÞb ¼ 0; dt

dx ¼uþw dt

Figure 1 The behavior of b with t for b0 > 0, here a ¼ 0 (solid line) and a ¼ 0:02 (dashed line).

ð17Þ

where   3C2 ða3 þa4 taC1 Þcþ2 a4 ða3 þ a4 tÞ kcðc þ 1ÞCc1 þ 3 ða 4 þa tÞ 3 4 H1 ðx; tÞ ¼  1 ; cþ3 pffiffiffiffiffiffi C23 ða3 þa4 taC1 Þcþ1 2 c 2 2 C1 ða3 þ a4 t  aC1 Þ kcC1 þ 3 ða þa tÞ 3

4

  a C2 ða þa taC Þcþ2 4ka4 cðð1  cÞða3 þ a4 tÞ  2aC1 ÞCc1 þ kcðc þ 1ÞCc1 þ 4 3 ða3 þa4 tÞ3 1 3 4   H2 ðx;tÞ ¼ 3C2 ða3 þa4 taC1 Þcþ1 4ða3 þ a4 tÞða3 þ a4 t  aC1 Þ kcCc1 þ 3 ða þa tÞ3 3

4

2a4 : þ ða3 þ a4 tÞ

Figure 2 The behavior of b with t for b0 < 0 and jb0 j P bc , here a ¼ 0 (solid line) and a ¼ 0:02 (dashed line).

The solution of (17) can be written in quadrature form as R  t b0 IðtÞ bðtÞ ¼ 1þb where IðtÞ ¼ exp H ðxðsÞ; sÞds and 2 t 0 0 JðtÞ R  Rt t 0 0 0 JðtÞ ¼ t0 H1 ðxðt Þ; t Þ exp t0 H2 ðxðsÞ; sÞds dt . In the functions H1 and H2 , we find that both integrals IðtÞ and JðtÞ are finite and continuous on ½t0 ; 1Þ. With the initial conditions b ¼ b0 and x ¼ x0 at t ¼ t0 , we studied the behavior of the weak discontinuity which is well observed in the Fig. 1–3. For b0 > 0 and t ! 1, it is clear that IðtÞ ! 0 whereas Jð1Þ < 1, which gives rise to an expansion wave and the wave decays and dies out eventually, and the corresponding situation is shown in Fig. 1. However, when b0 < 0, which corresponds to a compressive wave, there exists a positive quantity bc > 0, for a finite time tc given by the solution of Jðtc Þ ¼ jb10 j such that, when jb0 j P bc ; bðtÞ increases from b0 and terminates into a shock; the corresponding situation is illustrated by the curve in Fig. 2. In Fig. 3, it can be observed that, for jb0 j < bc ; bðtÞ initially decreases from b0 and reaches to minimum at finite time. From the Figs. 1 and 3, it is noticed that increase in the van der Waals excluded volume enhances the decay rate of weak discontinuity whereas Fig. 2 shows that an increase in van der Waals excluded volume serves to hasten onset of a shock wave.

Figure 3 The behavior of b with t for b0 < 0 and jb0 j < bc , here a ¼ 0 (solid line) and a ¼ 0:02 (dashed line).

4. Conclusion The present work devoted to the Lie symmetry analysis of the system of PDEs governed by one dimensional unsteady flow of

1132 an isentropic, inviscid and perfectly conducting compressible fluid obeying van der Waals equation of state. The infinitesimal transformations that leave the system of governing PDEs invariant and consequently reduce the system of PDEs to system of ODEs are identified. The reduced system of ODEs is solved and particular exact solution for the governing system of PDEs is derived. This exact solution to mathematical equations plays an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. Furthermore, the behavior of weak discontinuity has been discussed across the solution curve which is well illustrated by the Figs. 1–3. For b0 > 0 and b0 < 0 or jb0 j < bc , in both the cases the wave decays and dies out eventually, which has been well observed in Figs. 1 and 3 whereas the Fig. 2 clearly shows the appearance of shock for the case b0 < 0 and jb0 j P bc . Moreover, in Figs. 1 and 3, it is noticed that the van der Waals excluded volume enhances the decay rate of weak discontinuity while in Fig. 2, it is observed that the presence of van der Waals excluded volume reduces the shock formation time as compared to what it would be in a corresponding ideal gas ða ¼ 0Þ.

B. Bira, T. Raja Sekhar [8] Jena J. Self-similar solutions in a plasma with axial magnetic field (h-pinch). Meccanica 2012;47(5):1209–15. [9] Zhang Y, Liu X, Wang G. Symmetry reductions and exact solutions of the (2 + 1)-dimensional JaulentMiodek equation. Appl Math Comput 2012;219(3):911–6. [10] Bira B, Raja Sekhar T. Symmetry group analysis and exact solutions of isentropic magnetogasdynamics. Indian J Pure Appl Math 2013;44(2):153–65. [11] Barbera E, Giambo S. Propagation of weak discontinuities in binary mixtures of ideal gases. Rend Circ Mat Palermo (2) 2012;61(2):167–78. [12] Bira B, Raja Sekhar T. Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics. Appl Anal 2014;93(12):2598–607. [13] Radha R, Sharma VD. Interaction of a weak discontinuity with elementary waves of Riemann problem. J Math Phys 2012;53(1):013506, 12 pp. [14] Singh LP, Husain A, Singh M. Evolution of weak discontinuities in a non-ideal radiating gas. Commun Nonlinear Sci Numer Simul 2011;16(2):690–7. [15] Liu Y, Sun W. Riemann problem and wave interactions in magnetogasdynamics. J Math Anal Appl 2013;397(2):454–66. [16] Sharma VD. Quasilinear hyperbolic systems, compressible flows, and waves. Boca Raton, FL: CRC Press; 2010.

Acknowledgments Research support from, Ministry of Minority Affairs through UGC, Government of India (Ref. F1-17:1/2010/MANF-CHRORI-1839/(SA-III/Website) is gratefully acknowledged by the first authors.

Bibekananda Bira Research Scholar Department of Mathematics National Institute of Technology Rourkela, Rourkela 769008.

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T. Raja Sekhar Assistant Professor, Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur.