Applied Mathematics and Computation 215 (2009) 681–685

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Symmetry groups and similarity solutions for the system of equationsfor a viscous compressible fluid

Manoj Pandey a, B.D. Pandey b, V.D. Sharma a,*

a Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, Indiab Department of Mathematics, Ohio State University, Marion, OH 43302, USA

a r t i c l e i n f o a b s t r a c t

Keywords:Similarity solutionsLie group analysisViscous compressible fluidsExact solutions

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.05.070

* Corresponding author.E-mail address: vsharma@math.iitb.ac.in (V.D. Sh

Here, using Lie group transformations, we consider the problem of finding similarity solu-tions to the system of partial differential equations (PDEs) governing one-dimensionalunsteady motion of a compressible fluid in the presence of viscosity and thermal conduc-tion, using the general form of the equation of state. The symmetry groups admitted by thegoverning system of PDEs are obtained, and the complete Lie algebra of infinitesimal sym-metries is established. Indeed, with the use of the entailed similarity solution the problemis transformed to a system of ordinary differential equations(ODEs), which in general isnonlinear; in some cases, it is possible to solve these ODEs to determine some special exactsolutions.

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1. Introduction

For nonlinear systems, we do not have the luxury of complete exact solutions, and for analytical work we have to rely onsome approximate analytical or numerical methods which may be useful to set the scene and provide useful information to-wards our understanding of the complex physical phenomenon involved. Special exact solutions of a system of nonlinear PDEsare of great interest; these solutions play a major role in designing, analysing and testing of numerical methods for solving spe-cial initial/ or boundary-value problems. One of the most powerful methods in order to determine particular solutions to PDEs isbased upon the study of their invariance with respect to one parameter Lie group of point transformations [1–4].

Besides similarity methods, another use of Lie point symmetries admitted by given PDEs consists in introducing someinvertible point transformations that map the original system to an equivalent one, admitting special solutions [5,6]. In[7] various classes of exact solutions to the equations that govern the unsteady motion of a prefect gas are explicitly deter-mined by introducing some transformations, referred to as substitution principle, that map the given equations to an equiv-alent autonomous form. In [8,9], group theoretic methods are used to obtain various classes of exact solutions inmagnetogasdynamics. Indeed, with the help of symmetry generators of the governing system of equations, one can constructsimilarity variables which lead to a reduced system of equations with one independent variable less at each step, and even-tually to a system of ordinary differential equations (ODEs). In some cases it is possible to solve these equations exactly;however, in most cases the ODEs must be solved numerically. Besides these similarity solutions, the symmetries admittedby the given PDEs enable us to look for appropriate canonical variables which transform the original system to an equivalentone whose simple solutions provide non-trivial solution of the original system.

Applications of Lie group analysis for unsteady one-dimensional problems may be found in [10–15]. In this paper, we ap-ply the symmetry method to the system of PDEs governing one-dimensional unsteady motion of a viscous compressible fluid

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682 M. Pandey et al. / Applied Mathematics and Computation 215 (2009) 681–685

using the general form of the equation of state. The symmetry groups, admitted by the governing system of PDEs, which areindeed continuous groups of transformations under which the system of PDEs remains invariant, are obtained. We then pro-ceed to the determine the similarity solutions, which in turn are used to transform the system of PDEs to a system of ODEs.Finally, we give some exact particular solutions to the system of PDEs.

2. Basic equations and Lie group analysis

The basic equations governing the unsteady, one-dimensional motion of a real gas in the presence of dissipative effectsare as follows [16]:

qt þ uqx þ qux ¼ 0;

ut þ uux þ1q

pqqx þ pT Tx �43guxx

� �¼ 0;

sT Tt þ uTxð Þ � sqqux �1qT

kTxx þ43gu2

x

� �¼ 0;

ð1Þ

where q is the density, u the particle velocity in the x-direction, p the pressure, s the entropy and T the temperature of thefluid. The variables x and t are the space and time coordinates, respectively. The constants g and k are respectively the coef-ficients of viscosity and thermal conductivity. We hereby consider q and T as independent state variables, then p and s arefunctions of these variables given by

p ¼ pðq; TÞ and s ¼ sðq; TÞ: ð2Þ

In order to determine a similarity solution and the similarity curves, we seek a one parameter Lie group of transformation[1–5]

t� ¼ t þ �sðx; t;q;u; TÞ þ Oð�2Þ; x� ¼ xþ � nðx; t;q;u; TÞ þ Oð�2Þ;q� ¼ qþ �/ðx; t;q;u; TÞ þ Oð�2Þ; u� ¼ uþ � wðx; t;q;u; TÞ þ Oð�2Þ;T� ¼ T þ �pðx; t;q; u; TÞ þ Oð�2Þ;

ð3Þ

where the generators s; n;/;w and p are to be determined in such a way that the PDEs (1) are invariant with respect to thetransformations (3); the entity � is so small that its square and higher powers may be neglected. The existence of such agroup allows the number of independent variables in the problem to be reduced by one, and thereby allowing the system(1) to be replaced by a system of ODEs. Following the usual procedure outlined in [10,12,13], the invariance of the system(1) under the infinitesimal group of transformations (3) yields the following forms of the infinitesimals

s ¼ a1xþ b1t þ c1; n ¼ a2t þ b2; / ¼ ða2 � 2a1Þq;w ¼ ða1 � a2Þuþ b1; p ¼ 2ða1 � a2ÞT;

ð4Þ

where a1; a2; b1; b2 and c1 are arbitrary constants. The invariance of (2) gives the following forms of s and p

s ¼ f ðlÞ þ b1 ln T; p ¼ T�a2=2ða1�a2ÞGðlÞ þ ð2ða1 � a2Þ=a2Þa1T�ða2=2ða1�a2ÞÞ; ð5Þ

where f and G are arbitrary functions of l, defined as l ¼ qT�ða2�2a1Þ=2ða1�a2Þ with a1 and b1 as arbitrary constants.

Definition. U ¼ HðxÞ is an invariant solution of the system of PDEs (1), admitted by the symmetry group with theinfinitesimals given by Eq. (4) if, and only if

(1) U ¼ HðxÞ is an invariant surface of (1),(2) U ¼ HðxÞ solves (1).

Thus, the invariant surface condition is given by the relations

s oqoxþ n

oqot¼ /; s ou

oxþ n

ouot¼ w; s oT

oxþ n

oTot¼ p: ð6Þ

The arbitrary constants, occurring in the expression for the generators of the local group of transformations, give rise to sev-eral cases of possible invariant solutions, which we discuss below.

Case I: When a1 ¼ 0; a2 – 0 and b1 ¼ 0, the invariant surface conditions (6) yield the following form of the flow variables

q ¼ t~qðfÞ; u ¼ ð1=tÞ~uðfÞ; T ¼ ð1=t2ÞeT ðfÞ; s ¼ f ðlÞ þ b1 ln T;

p ¼ fGðlÞ þ 2a1gT1=2; l ¼ qT1=2;ð7Þ

where f ¼ x is the similarity variable. Thus, substituting (7) in the governing PDEs (1), we obtain the following set of ODEs in~q; ~u and eT ,

M. Pandey et al. / Applied Mathematics and Computation 215 (2009) 681–685 683

~u~q0 þ ~q~u0 þ ~q ¼ 0;

~u~u0 þeT~q

G0ðlÞ~q0 � 1~q

a1 �12

GðlÞ� �eT�1

2eT 0 þ 12

G0ðlÞðlÞeT 0 þ ~u� 43~q

g~u00 ¼ 0;

b1eT þ12

f 0ðlÞeT 12

� �~ueT 0 � 2eTn o

f 0ðlÞ~qeT 12~u0 � 1

~qeT keT 00 þ 43g~u02

� �¼ 0; ð8Þ

where prime denotes differentiation with respect to f.Case II: When a1 – 0; a2 ¼ 0 and b1 ¼ 0, the invariant surface conditions (6) yield the following form of the flow variables

q ¼ ð1=x2Þ~qðfÞ; u ¼ x~uðfÞ; T ¼ x2eT ðfÞ;s ¼ f ðlÞ þ b1 ln T; p ¼ gðlÞ þ a1 ln T; l ¼ qT;

ð9Þ

where the similarity variable f is given by f ¼ t. Thus, substituting (9) in the governing PDEs (1), we obtain the following setof ODEs in ~q; ~u and eT ,

~q0 � ~q~u0 ¼ 0;

~u0 þ ~u2 þ 2a1

~q¼ 0;

~qf 0ðlÞ þ b1eT� �

ðeT 0 þ 2~ueT Þ � eT ~u~qf 0ðlÞ � 1eT ~qð2keT þ 4

3g~u2Þ ¼ 0:

ð10Þ

Case III: When a1 – a2 – 0 and b1 – 0, the invariant surface conditions yield

q ¼ ða2t þ b2Þða2�2a1Þ=a2 ~qðfÞ; u ¼ ða2t þ b2Þða1�a2Þ=a2 ~uðfÞ � ðb1=a1 � a2Þ; T ¼ ða2t þ b2Þ2ða1�a2Þa2 eT ðfÞ: ð11Þ

where s and p are given by Eq. (5). The functions ~q; ~u and eT depend only on the similarity variable f, which is given by therelation

f ¼ ða2t þ b2Þ�a1=a2 xþ b1t þ c1

a2

� �þ b1

a1 � a2

� �ða2t þ b2Þ�ða1�a2Þ=a2 : ð12Þ

Substituting (11) in the governing system of PDEs (1) and using (12), we obtain the following system of ODEs in ~q; ~u and eT ,

ð~u� a1fÞ~q0 þ ~q~u0 þ ða2 � 2a1Þ~q ¼ 0;

ð~u� a1fÞ~u0 þeT~q

G0ðlÞ~q0 � 1~q

a1 þa2

2ða1 � a2ÞGðlÞ

� �eT a2�2a12ða1�a2 ÞeT 0

� a2 � 2a1

2ða1 � a2ÞG0ðlÞeT 0 þ ða1 � a2Þ~u�

43~q

g~u00 ¼ 0;

b1eT �a2 � 2a1

2ða1 � a2Þf 0ðlÞeT� a2

2ða1�a2Þ

� �ð~u� a1fÞeT 0 þ 2ða1 � a2ÞeTn o

� f 0ðlÞ~qeT� a2�2a12ða1�a2 Þ~u0 � 1

~qeT keT 00 þ 43g~u02

� �¼ 0;

ð13Þ

where f and G are arbitrary functions of l.Case IV: When a1 – a2 – 0 and b1 ¼ 0, the invariant surface conditions (6) yield the following form of the similarity

solution

q ¼ tða2�2a1Þ=a2 ~qðfÞ; u ¼ tða1�a2Þ=a2 ~uðfÞ;T ¼ t2ða1�a2Þ=a2 eT ðfÞ; ð14Þ

where s and p have the same form as obtained in case I, and the similarity variable f is given by f ¼ xt�a1=a2 . Thus, substituting(14) in the governing PDEs (1), we obtain the following system of ODEs

ð~u� a1

a2fÞ~q0 þ ~q~u0 þ a2 � 2a1

a2

� �~q ¼ 0;

ð~u� a1

a2fÞ~u0 þ

eT~q

G0ðlÞ~q0 � 1~q

a1 þa2

2ða1 � a2ÞGðlÞ

� �eT a2�2a12ða1�a2ÞeT 0

� a2 � 2a1

2ða1 � a2ÞG0ðlÞeT 0 þ ða1 � a2Þ~u�

43~q

g~u00 ¼ 0;

b1eT �a2 � 2a1

2ða1 � a2Þf 0ðlÞeT� a2

2ða1�a2Þ

� �ð~u� a1

a2fÞeT 0 þ 2ða1 � a2Þ

a2

eT� �

� f 0ðlÞ~qeT� a2�2a12ða1�a2 Þ~u0 � 1

~qeT keT 00 þ 43g~u02

� �¼ 0:

ð15Þ

684 M. Pandey et al. / Applied Mathematics and Computation 215 (2009) 681–685

3. Special solutions

In this section, we look for some special exact solutions to the system of ODEs, obtained in the last section, which in turnyield exact solutions to the governing system (1).

Case (1): When a1 ¼ 0; a2 – 0 and b1 ¼ 0, the system of ODEs (9) on integration yields

~qðfÞ ¼ expf�fg; ~uðfÞ ¼ u0 expffg þ 1; eT ðfÞ ¼ ð1=4Þ expf�2fg; ð16Þ

where u0 is given by

u20 �

43gu0 þ 1 ¼ 0

and s and p have the following particular forms

s ¼ f ðlÞ þ b2; p ¼ GðlÞ � 2ð Þ eT 1=2 þ a2; ð17Þ

with GðlÞ ¼ 1=l;l ¼ ~qeT 1=2, and f satisfying the equation

f 0ðlÞ ¼ �2ð3kl2 þ gu20Þ=ð3l3ðu0 þ ð2lÞ1=2ÞÞ:

Thus, the similarity solution to the governing system (1) using (7) and (16) is given by

q ¼ t expf�xg; u ¼ ð1=tÞðexpfxg þ 1Þ; T ¼ ð1=4t2Þ expf�2xg; ð18Þ

where s and p are given by (17) with l ¼ qT1=2. It may be noticed that the state such as this, where the density distribution isexponential with regard to the special coordinate, was considered by Hayes [17] for motion of a rising strong shock wave.

Case (2): When a1 – 0; a2 ¼ 0 and b1 ¼ 0, the system of ODEs (10) on integration yields

~qðfÞ ¼ q0f; ~uðfÞ ¼ ð1=fÞ; eT ðfÞ ¼ T0fð4k=q0Þ � 4k

3ð4kþ 2q0Þ1f2 ; ð19Þ

where q0 and T0 are some reference values of density and temperature; the variables s and p have the following particularform

s ¼ f ðlÞ � ð1=2Þ ln T; p ¼ gðlÞ; ð20Þ

where f is given by f 0ðlÞ ¼ 1=l, and gðlÞ is an arbitrary function of l defined as l ¼ ~qeT . Thus, the similarity solution to thegoverning system (1) using (9) and (19) is given by

q ¼ q0ðt=x2Þ; u ¼ ðx=tÞ; T ¼ x2 T0tð4k=q0Þ � 4k3ð4kþ 2q0Þ

1t2

� �; ð21Þ

where s and p are given by (20) with l ¼ qT . It may be remarked that the state such as this, where the particle velocityexhibits linear dependence on the spatial coordinate, has been discussed by Pert [18], Sharma et al. [19] and Clarke [20]; Perthas shown that such a form of velocity distribution is useful in modelling the free expansion of polytropic fluids and is at-tained in the large time limit.

4. Conclusion

Lie group analysis is used to obtain special exact solutions to the system of PDEs governing one-dimensional unsteadymotion of a viscous compressible fluid having the general form of the equation of the state. The Lie symmetry groups, areused to transform the governing system of PDEs to a system of ODEs. Exact solutions to the system of ODEs are discovered,which in turn via the transformation of variable give special exact solutions to the governing system. Special exact solutionsof a system of nonlinear PDEs are of great interest; these solutions play a major role in designing, analysing and testing ofnumerical methods for solving special initial-and/or boundary-value problems. The form of special solutions obtained here,such as the one having the exponential density distribution with regard to the spatail coordinate and the one where the par-ticle velocity exhibits linear dependence on the spacial coordinate, has been discussed by some of the researchers in the past.

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