Approximate symmetries of creeping flow equations of a second grade fluid

International Journal of Non-Linear Mechanics 39 (2004) 1603–1619

Approximate symmetries of creeping $ow equations ofa second grade $uid

I.T. Dolap,c-a, M. Pakdemirlib;∗

aDepartment of Mathematics, Dumlup�nar University, K�utahya, TurkeybDepartment of Mechanical Engineering, Celal Bayar University, 45140 Muradiye, Manisa, Turkey

Received 26 May 2003; accepted 7 January 2004

Abstract

Creeping $ow equations of a second grade $uid are considered. Two current approximate symmetry methods and amodi7ed new one are applied to the equations of motion. Approximate symmetries obtained by di8erent methods andthe exact symmetries are contrasted. Approximate solutions corresponding to the approximate symmetries are derived foreach method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed indetail.? 2004 Elsevier Ltd. All rights reserved.

Keywords: Approximate symmetries; Creeping $ow; Second grade $uid

1. Introduction

Under the slow motion assumption, the Navier–Stokes equations can be linearized and closed form solutionsare available. When a similar assumption is made for the equations of motion of a second grade $uid, theresulting creeping $ow equations are not linear and it becomes a hard task to determine analytical solutionsfor the highly non-linear equations.The Newtonian and second grade creeping $ow solutions are contrasted before [1–4]. Tanner [1] stated that

the Newtonian creeping $ow solution is also a solution of the second grade $ow for given speci7c boundaryconditions. Huilgol [2] established the criteria under which Newtonian solutions are the only solutions availablefor a second grade $uid. A generalization of the theorem of Tanner was given by Fosdick and Rajagopal [3]who derived a drag formula for an immersed body in a second grade $uid.The problem of an additional boundary condition for a second grade $uid was discussed by Rajagopal

[4] who considered the creeping $ow over a porous $at plate. The extensions of Tanner’s theorem tothree-dimensional $ows were discussed as well. The problems arising in having additional boundary con-ditions in the case of di8erential type $uids were reviewed by Dunn and Rajagopal [5]. A similarity solutionfor non-parallel porous walls was developed by Bourgin and Tichy [6]. The slow motion of a body in asecond grade $uid was investigated by Galdi and Rajagopal [7]. Finally, YHurHusoy et al. [8] calculated the

∗ Corresponding author. Tel.: +90-236-241-2144; fax: +90-236-241-2143.E-mail address: [email protected] (M. Pakdemirli).

0020-7462/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2004.01.002

mailto:[email protected]

1604 I .T. Dolapc(�, M. Pakdemirli / International Journal of Non-Linear Mechanics 39 (2004) 1603–1619

exact symmetries of creeping $ow equations of a second grade $uid and developed analytical solutions usingtranslational and scaling symmetries.In this work, with the utilization of newly developed approximate symmetry theories, new analytical so-

lutions are searched for the special case of small second grade e8ects compared to viscous e8ects. A briefreview of the Lie Group theory and the associated di8erent approximate symmetry theories will be informativefor the readers.Especially for non-linear problems, Lie Group theory provides a systematic and uni7ed approach in search

of analytical solutions. Instead of solving directly the non-linear equations, using the theory, a system ofover-determined linear equations are treated. Once the so-called symmetries of the equations are calculated,similarity solutions can be produced or from a known solution another solution can be retrieved. By de7ningcanonical coordinates, it is also possible to transform the equations into a much simpler from.Another technique which is widely used especially for non-linear problems is perturbation analysis. By

expanding the dependent variables asymptotically in terms of a small parameter (may be a physical parameteror arti7cially introduced), approximate analytical solutions can be found.In the last century, these analytical techniques proved to be very useful in analyzing problems in non-linear

mechanics. To have the utmost gain from the techniques, combination of Lie Group theory and perturbationsare considered recently and two di8erent so-called “approximate symmetry” theories have been developed. Inthe 7rst method due to Baikov et al. [9,10], the dependent variable is not expanded in a perturbation seriesas should be done in an ordinary perturbation problem, rather, the in7nitesimal generator is expanded in aperturbation series. In this way, an approximate generator is found from which approximate solutions can beretrieved. In the second method due to Fushchich and Shtelen [11] and later followed by Euler et al. [12,13],Euler and Euler [14], the dependent variables are expanded in a perturbation series 7rst as also done in usualperturbation analysis. Terms are then separated at each order of approximation and a system of equations to besolved in a hierarchy is obtained. The system is assumed to be coupled and the approximate symmetry of theoriginal equation is de7ned to be the exact symmetry of the system of equations obtained from perturbations.In most of the problems, the unperturbed equations are linear and perturbed equations contain non-linear

terms. When expanded in a perturbation series, one thus obtains linear non-homogenous equations to be solvedin order. Actually, the system is not coupled and can be solved in hierarchy starting from the 7rst equation.The non-homogenous term is a known function but di8erent at each order of approximation. Requiring thisterm to be an arbitrary function, the approximate symmetry of the original equation is de7ned as the exactsymmetry of the non-homogenous linear equation [15]. This method (Method III) is actually a modi7cation ofMethod II and consistent with the perturbation theory. Since non-homogenous term is considered as an arbitraryfunction, equation(s) dictating the form of this function arise from the symmetry calculations. Alternatively,equivalence transformation method recently developed by Ibragimov et al. [16], Ibragimov and Torrisi [17] maybe employed in 7nding the form of the arbitrary function. For the application of equivalence transformations tothe exterior calculus approach, see [18]. Note that this last approximate symmetry method proposed eliminatesa great deal the algebra involved in Method II in which the system is assumed to be coupled. Anotherdisadvantage of Method II is that one usually has to solve coupled non-linear equations. However, in themodi7ed approach the equations are always linear.The outline of the paper is as follows: In Chapter 2, the creeping $ow equations and the exact symmetries are

given. In Chapter 3, the approximate symmetries are calculated using three di8erent approximate symmetrytheories. Symmetry results are contrasted then. In Chapter 4, approximate solutions corresponding to theapproximate symmetries are calculated for each method and comparisons are made. It is observed that MethodI failed to produce the solutions which can be obtained by Method II and III. The calculated approximatesymmetries and the analytical solutions developed are new. The advantages and disadvantages of the methodsare discussed in the concluding remarks section.Apart from the approximate symmetry techniques discussed in this paper, another interesting approximate

solution using Lie Group techniques has been presented by HOzkaya and Pakdemirli [19]. In that work, the

I .T. Dolapc(�, M. Pakdemirli / International Journal of Non-Linear Mechanics 39 (2004) 1603–1619 1605

exact solution obtained by Lie Group techniques does not satisfy the boundary conditions exactly. Therefore,the exact solution is expanded by using the perturbation parameter and then requiring the boundary conditionsto be satis7ed approximately, an approximate solution has been constructed for the vibrations of a beammoving with variable axial velocity.

2. Equations of motion and exact symmetries

The plane dimensionless creeping $ow equations of a second grade $uid in cartesian coordinates are [8]9u9x +

9v9y = 0; (1)

− 9p9x + �

(92u9x2 +

92u9y2

)+ �1

(59u9x92u9x2 +

9u9x92u9y2 + u

93u9x3 + v

93u9y3 + u

93u9y29x + 2

9v9x92v9x2

+9u9y

92u9x9y +

9u9y

92v9x2 + v

93u9x29y

)= 0; (2)

− 9p9y + �

(92v9x2 − 92u

9x9y

)+ �1

(59u9x

92u9y9x − 9u

9x92v9x2 − v

93u9x9y2 + u

93v9x3 − v

93u9x3 + 2

9u9y

92u9y2

− 9v9x92u9x2 +

9v9x92u9y2 − u

93u9x29y

)= 0; (3)

where u and v are the velocity components in the x and y directions, respectively, and p is the pressure.� = 1=Re (Re is the well-known Reynolds number in $uids) and �1 is the dimensionless non-NewtoniancoeScient. �1 is assumed to be small (�1��) and hence selected as the perturbation parameter.Exact symmetries of the equations were calculated previously [8]. If the in7nitesimal generator is

X = �199x + �2

99y + 1

99u + 2

99v + 3

99p (4)

then the symmetries are as follows:

�1 = ax + b; �2 = ay + c; 1 = au;

2 = av; 3 = d: (5)

Exact symmetries contain only scaling and translational symmetries. Four 7nite parameter Lie Group trans-formations exist in the exact case. Symmetries do not depend on the perturbation parameter and, as will bediscussed later, derivation of approximate symmetries of Method I using the exact symmetries is impossiblefor this speci7c example.

3. Approximate symmetries

In this section, approximate symmetries of the creeping $ow equations will be calculated using three di8erentmethods.

3.1. Approximate symmetries by Method I

This method is developed in Refs. [9,10]. Basically the dependent variables are not expanded in a per-turbation series, rather the approximate generator is developed in terms of the perturbation parameter. As a


7rst-order approximation, the approximate generator can be written as follows:

X = X0 + ∈X1; (6)

where X0 corresponds to the symmetries of the unperturbed equation and X1 is a correction to this symmetryfor the perturbed equation. Details of calculating (6) are given in Refs. [9,10]. For our creeping $ow equations,the result is

�1 = ax + b+ �1(Ax + B);

�2 = ay + c + �1(Ay + C);

1 = au+ �1(Du+ E(x; y));

2 = av+ �1(Dv+ F(x; y));

3 = g+ �1((D − A)p+ G(x; y)); (7)

where the functions satisfy the below equations:

9E9x +

9F9y = 0;

− 9G9x + �

(92E9x2 +

92E9y2

)= 0;

− 9G9y + �

(92F9x2 − 92E

9x9y

)= 0: (8)

For this speci7c problem, it turns out that when �1 = 0, the exact symmetries are retrieved. The 7nite LieGroup transformations increased from 4 to 8 together with the in7nite parameter transformations expressedby arbitrary functions. However, as will be discussed later, these additional symmetries are not Lie pointsymmetries of the equations and cannot yield group invariant solutions.

3.2. Approximate symmetries by Method II

In this method, which is developed and used in Refs. [11–14], the dependent variable is expanded in aperturbation series. The separated equations at each order are assumed to be coupled and the approximatesymmetry of the original equation is de7ned as the exact symmetry of these coupled equations. Although thelogic behind this approximate symmetry is simple, this analysis cannot be criticized from the perturbationpoint of view as well as from the Lie Group theory point of view. Expanding the dependent variables to the7rst order yields

u= u0 + �1u1 + · · ·v= v0 + �1v1 + · · · �1��; �1�1:

p= p0 + �1p1 + · · · (9)

Inserting the expansions into the original equations and separating at each order of perturbation parameter,one hasOrder 1

9u09x +

9v09y = 0; (10)


− 9p0

9x + �[92u09x2 +

92u09y2

]= 0; (11)

− 9p0

9y + �[92v09x2 − 92u0

9x9y

]= 0: (12)

Order �19u19x +

9v19y = 0; (13)

− 9p1

9x + �[92u19x2 +

92u19y2

]+[59u09x

92u09x2 +

9u09x

92u09y2 + u0

93u09x3 + v0

93u09y3

+ u093u09x9y2 + 2

9v09x

92v09x2 +

9u09y

92u09x9y +

9u09y

92v09x2 + v0

93u09x29y

]= 0; (14)

−@p1

@y+ �[@2v1@x2

− @2u1@x@y

]+[5@u0@x

@2u0@x@y

− @u0@x

@2v0@x2

− v0@3u0@x@y2 + u0

@3v0@x3

− v0@3u0@x3

+ 2@u0@y

@2u0@y2 − @v0

@x@2u0@x2

+@v0@x

@2u0@y2 − u0

@3u0@x2@y

]= 0: (15)

Approximate symmetries of the original equation correspond to the exact symmetries of system (10)–(15).De7ning the in7nitesimal generator

X= �199x + �2

99y + 1

99u0

+ 299v0

+ 399p0

+ 499u1

+ 599v1

+ 699p1

(16)

and performing the standard Lie Group algebra, one 7nally has

�1 = ax + b;

�2 = ay + c;

1 = du0;

2 = dv0;

3 = (d − a)p0 − e;

4 = fu0 + (2d − a)u1 + E(x; y);

5 = fv0 + (2d − a)v1 + F(x; y);

6 = fp0 + 2(d − a)p1 + G(x; y); (17)

where the functions satisfy the below equations

9E9x +

9F9y = 0; −9G9y + �

{92F9x2 +

92F9y2

}= 0;

9G9x + �

{92E9y2 − 92F

9x9y

}= 0: (18)


Since there are six dependent variables and the highest order derivatives are of third-order, the algebra increasesenormously and the problem can only be handled by a symbolic computer package. The algorithm Biglieof Mumath is used in calculations. Comparing with the exact symmetries, the 7nite Lie Point symmetriesincreased from 4 to 6 in addition to the three in7nite Lie Point transformations represented by arbitraryfunctions.Since the dependent variables are increased by three, the algebra involved in 7nding the symmetries in-

creases substantially. If higher order perturbations are considered, the algebra would increase further. The basicassumption in Method II is that system (10)–(15) is a coupled system. However, this is not the case actuallyand the 7rst-order terms in the expansions are uncoupled and can be solved. In the next section, utilizingMethod III, one removes the “coupled-equations” assumption and hence decreases the algebra substantially.

3.3. Approximate symmetries by Method III

The “coupled-equations” assumption in Method II is removed in this method. It will be better to discussthe method in a more general form 7rst. Consider a non-linear equation with a small parameter ∈ as follows:

$(u) + ∈N (u) = 0; (19)

where $ is an arbitrary linear and N is an arbitrary non-linear di8erential operator. xi are the independentvariables and u is the dependent variable such that u= u(x1; x2; : : : ; xn). Expanding u in a perturbation series

u= u0 + ∈ u1 + ∈2 u2 + · · · (20)

and substituting into the original equation, one has

$(u0) = 0 = h0(x1; x2; : : : ; xn);

$(u1) = −N (u0) = h1(x1; x2; : : : ; xn);

$(u2) = −N (u0; u1) = h2(x1; x2; : : : ; xn): (21)

Note that the left-hand side of all the equations are the same and the right-hand sides of the equations can beconsidered as arbitrary functions of the independent variables which are to be determined sequentially startingfrom the 7rst equation. The following de7nition for approximate symmetry is stated.

De�nition. The approximate symmetry of the non-linear equation (19) is the exact symmetry of the followinglinear non-homogenous equation

$(u) = h(x1; x2; : : : ; xn) (22)

with h considered as an arbitrary function. The approximate symmetries at each level are determined aftersubstitution of the speci7c h values obtained from the preceding levels of approximation.Returning now to the creeping $ow equations, the dependent variables are again expanded in a perturbation

series similar to Method II. In search of symmetries, the below equations should be considered

9u9x +

9v9y = 0; (23)

− 9p9x + �

[92u9x2 +

92u9y2

]= h1(x; y); (24)

− 9p9y + �

[92v9x2 − 92u

9x9y

]= h2(x; y): (25)


At order 1, h1(x; y) = h2(x; y) = 0 and at order �1 they are

h1(x; y) =−[59u09x

92u09x2 +

9u09x

92u09y2 + u0

93u09x3 + v0

93u09y3 + u0

93u09x9y2

+ 29v09x

92v09x2 +

9u09y

92u09x9y +

9u09y

92v09x2 + v0

93u09x29y

]; (26)

h2(x; y) =−[59u09x

92u09x9y − 9u0

9x92v09x2 − v0

93u09x9y2 + u0

93v09x3 − v0

93u09x3

+ 29u09y

92u09y2 − 9v0

9x92u09x2 +

9v09x

92u09y2 − u0

93u09x29y

]; (27)

where h1(x; y) and h2(x; y) are known functions determined by order 1 solutions. For the in7nitesimal generator

X= �199x + �2

99y + 1

99u + 2

99v + 3

99p; (28)

the symmetries of system (23)–(25) are found to be

�1 = ax + b;

�2 = ay + c;

1 = du+ E(x; y);

2 = dv+ F(x; y);

3 = (d − a)p+ G(x; y); (29)

where above functions satisfy the below equations

9E9x +

9F9y = 0;

− 9G9x + �

{92E9x2 +

92E9y2

}= (2a − d)h1 + (ax + b)

9h19x + (ay + c)

9h19y

− 9G9y + �

{92F9x2 − 92E

9x9y

}= (2a − d)h2 + (ax + b)

9h29x + (ay + c)

9h29y : (30)

Note that h1 = h2 = 0 corresponds to the symmetries of order 1 equation and at order �1 the speci7c formof the functions should be inserted into Eq. (30). The advantage of this method is that the symmetries arecalculated once for each order of approximation and there is no need for a repeat of the lengthy calculations.An alternative to the standard Lie Group techniques in calculating symmetries of Eqs. (23)–(25) is the

use of equivalence transformations [16,17]. The equivalence operator would include h1 and h2 functions inaddition to the other variables.

4. Approximate solutions

In this section, some approximate solutions will be retrieved using the approximate symmetries. The solutionswill be contrasted with each other.


4.1. Scaling symmetry with parameters d and a

Using the three di8erent symmetries obtained by di8erent methods, similarity solutions will be produced.Both scaling parameters “d” and “a” will be considered by selecting d=ma where m is an arbitrary parameter.The analysis will be carried for an arbitrary “m” parameter and special choices will be selected then forsimplicity.

4.1.1. Method IIn Eqs. (7) and (8), selecting D=mA and all other parameters zero yields 7nally, the transformed equations

mf − �f′ + g′ = 0; (31)

− (m − 1)r + �r′ + �{m(m − 1)f − 2(m − 1)�f′ + (1 + �2)f′′}+ xm−1�1{(mf − �f′)f′′ + f′((m − 1)f′ − �f′′) + 5(mf − �f′)((m − 1)(mf − 2�f′)

+ �2f′′) + f′((m − 1)(mg − 2�g′) + �2g′′) + 2(mg − �g′)((m − 1)(mg − 2�g′)

+ �2g′′) + gf′′′ + f((m − 2)f′′ − �f′′′) + f((m − 2)(m − 1)mf

+3(m − 2)�(−(m − 1)f′ + �f′′) − �3f′′′) + g((m − 2)(m − 1)f′

+ �(−2(m − 2)f′′ + �f′′′))} = 0; (32)

− r′ + �{m(m − 1)g − 2(m − 1)�g′ + �2g′′ − (m − 1)f′ + �f′′} + xm−1�1{5�(1 − m)(f′)2

+ 4�2(1 − m)f′g′ + (2 + 5�2)f′f′′ + (�3 − �)g′f′′ + �3f′g′′ + g(6(m − 1)2�f′

+(2 + (6 − 4m)�2)f′′ + �(1 + �2)f′′′) − f(2(m − 1)m2g+ (2 + 2m − 4m2)f′

+�(−6(m − 1)g′ + (4 + 3m)f′′ + �(−2(m − 3)g′′ + f′′′ + �g′′′)))} = 0: (33)

The above equations can be ordinary if and only if m = 1. However, this case corresponds to one of theexact symmetries of the original equation. Therefore, approximate solutions cannot be produced other thanthis choice of “m” using Method I. A similarity solution corresponding to m= 1 is presented previously [8].

4.1.2. Method IISelecting d = ma in the given symmetries of Method II (i.e. Eqs. (17) and (18)), equations determining

the similarity variables are as follows:dxax

=dyay

=du0mau0

=dv0mav0

=dp0

(m − 1)ap0

=du1

(2m − 1)au1=

dv1(2m − 1)av1

=dp1

(2m − 2)ap1: (34)

Solving the above equations, one de7nes the similarity variable and functions

�=yx; u0 = xmf0(�); v0 = xmg0(�);

p0 = xm−1r0(�); u1 = x2m−1f1(�);

v1 = x2m−1g1(�); p1 = x2m−2r1(�): (35)


Substituting these variables into Eqs. (10)–(15) yields

mf0 − �f′0 + g′

0 = 0; (36)

− (m − 1)r0 + �r′0 + �{m(m − 1)f0 − 2(m − 1)�f′

0 + (1 + �2)f′′0 } = 0; (37)

− r′0 + �{m(m − 1)g0 − 2(m − 1)�g′

0 + �2g′′0 − (m − 1)f′

0 + �f′′0 } = 0; (38)

(2m − 1)f1 − �f′1 + g′

1 = 0; (39)

− 2(m − 1)r1 + �r′1 + �{2(m − 1)(2m − 1)f1 − 4(m − 1)�f′

1 + (1 + �2)f′′1 } + {g0f′′′

0

+f′0((m − 1)f′

0 − �f′′0 ) + 5(mf0 − �f′

0) ((m − 1)(mf0 − 2�f′0) + �2f′′

0 )

+ (mf0 − �f′0)f

′′0 + f′

0((m − 1)(mg0 − 2�g′0) + �2g′′

0 )

+ 2(mg0 − �g′0)((m − 1)(mg0 − 2�g′

0) + �2g′′0 ) + f0((m − 2)f′′

0 − �f′′′0 )

+f0((m − 2)(m − 1)mf0 + 3(m − 2)�(�f′′0 − (m − 1)f′

0) − �3f′′′0 ) + g0((m − 2)(m − 1)f′

0

+ �(−2(m − 2)f′′0 + �f′′′

0 ))} = 0; (40)

− r′1 + �{2(m − 1)(2m − 1)g1 − 4(m − 1)�g′

1 + �2g′′1 − 2(m − 1)f′

1 + �f′′1 } + {�3f′

0g′′0

+ 4�2(1 − m)f′0g

′0 + (2 + 5�2)f′

0f′′0 + (�3 − �)g′

0f′′0 + 5�(1 − m)f′2

0

+ g0(6(m − 1)2�f′0 + (2 + (6 − 4m)�2)f′′

0 + �(1 + �2)f′′′0 ) − f0(2(m − 1)m2g0

+ (2 + 2m − 4m2)f′0 + �(−6(m − 1)g′

0 + (4 + 3m)f′′0 + �(−2(m − 3)g′′

0

+f′′′0 + �g′′′

0 )))} = 0: (41)

In contrast to method I, the partial di8erential system has been reduced to an ordinary di8erential system forarbitrary m values by this method. Eliminating r0 between Eqs. (37) and (38) yields

�{(m − 2)[(m − 1)(2(1 + 3�2)f′0 − mg0 − 3m�f0) − 4�(1 + �2)f′′

0 ]

+ (1 + �2)2f′′′0 } = 0: (42)

This equation can be solved with Eq. (36). For simplicity, two special choices will be considered belowm = 0 case: This choice corresponds to a scaling in parameter “a” only since d = 0. For this choice,

Eq. (42) yields

f0(�) =12�

{c1 Arc tan �+

c1� − c21 + �2

+ c3

}(43)

and then inserting this solution and m= 0 into (36)–(41) 7nally yields

g0(�) =12�

{c2 Arc tan � − c1 + c2�

1 + �2+ c4

}; (44)


r0(�) =c1� − c21 + �2

; (45)

f1(�) =c22 − c2c3 − c1c4 + 2�3�(c6 − c5�2)

2�3(1 + �2)2; (46)

g1(�) =(c22 − c2c3 − c1c4 + 2�3�(c6 − �2c5))�

2�3(1 + �2)2+ c5; (47)

r1(�) =(c1(c2 − c3) + c2c4 + 2(c5 + c6)�3)�+ (c2 − c1�)(c1 + c2�)Arc tan �

�2(1 + �2)2: (48)

Substituting these functions into Eq. (35) and then into the perturbation expansions (9), returning back to theoriginal independent variables, one 7nally has

u(x; y) =12�

{c1 Arc tan(y=x) +

c1(y=x) − c21 + (y=x)2

+ c3

}+

�1x

×{c22 − c2c3 − c1c4 + 2�3(y=x)(c6 − c5(y=x)2)

2�3(1 + (y=x)2)2

}; (49)

v(x; y) =12�

{c2 Arc tan(y=x) − c1 + c2(y=x)

1 + (y=x)2+ c4

}+

�1x

×{(c22 − c2c3 − c1c4 + 2�3(y=x)(c6 − (y=x)2c5))(y=x)

2�3(1 + (y=x)2)2+ c5

}; (50)

p(x; y) =1x

{c1(y=x) − c21 + (y=x)2

}+

�1x2

×{(c1(c2 − c3) + c2c4 + 2(c5 + c6)�3)(y=x) + (c2 − c1(y=x))(c1 + c2(y=x))Arc tan(y=x)

�2(1 + (y=x)2)2

}:

(51)

This non-trivial approximate solution cannot be retrieved by Method I.m= 2 case: For this case, Eq. (42) takes the simple form

�(1 + �2)2f′′′0 = 0: (52)

This choice 7nally yields

f0(�) = c1 + c2�+ c3�2; (53)

g0(�) = c4 − 2c1� − 12 c2�

2; (54)

r0(�) = �{2(c1 + c3) + (2c4 − c2)�}; (55)

f1(�) = a1 + a2�+ a3�2 + a4�3; (56)

g1(�) =2a2 + 3a4

3− 3a1� − a2�2 − 1

3a3�3; (57)


r1(�) =c222

+ c2c4 + 4c24 + 2c21(5 + �2) + 2c1(c3 + 5c2� − 4c4� − c3�2)

+ �(3a1 + a3) + 12 �(5c

22�+ 8c3(c4 + c3�)

+ 2c2(4c3 − c4�) + 4�(3a4 + a2) − 2��(3a1 + a3)): (58)

Back substitution into the perturbation expansion with the original variables yield

u(x; y) = x2{c1 + c2(y=x) + c3(y=x)2} + x3�1{a1 + a2(y=x) + a3(y=x)2 + a4(y=x)3}; (59)

v(x; y) = x2{c4 − 2c1(y=x) − 1

2 c2(y=x)2}

+ x3�1

{2a2 + 3a4

3− 3a1(y=x) − a2(y=x)2 − 1

3a3(y=x)3

}; (60)

p(x; y) = x�{2(c1 + c3) + (2c4 − c2)(y=x)} + x2�1

{c222

+ c2c4 + 4c24 + 2c21(5 + (y=x)2)

+ 2c1(c3 + 5c2(y=x) − 4c4(y=x) − c3(y=x)2) + �(3a1 + a3) + 12 (y=x)[5c

22(y=x)

+ 8c3(c4 + c3(y=x)) + 2c2(4c3 − c4(y=x)) + 4�(3a4 + a2) − 2�(y=x)(3a1 + a3)]}: (61)

Once again, this solution cannot be obtained by Method I.

4.1.3. Method IIIFor the 7rst-order symmetries, h1 = h2 = 0 should be selected in Eqs. (30). Selecting d=ma in symmetries

(29) with all other parameters zero yieldsdxax

=dyay

=du0mau0

=dv0mav0

=dp0

(m − 1)ap0; (62)

from which the similarity variables and functions are calculated to be

�=yx; u0 = xmf0(�);

v0 = xmg0(�); p0 = xm−1r0(�): (63)

Inserting these variables into order 1 equations yields

g′0 − �f′

0 + mf0 = 0; (64)

− (m − 1)r0 + �r′0 + �{m(m − 1)f0 − 2(m − 1)�f′

0 + (1 + �2)f′′0 } = 0; (65)

− r′0 + �{m(m − 1)g0 − 2(m − 1)�g′

0 + �2g′′0 − (m − 1)f′

0 + �f′′0 } = 0: (66)

Similar to Method II, r0 can be eliminated between the equations

�{(m − 2)[(m − 1)(2(1 + 3�2)f′0 − mg0 − 3m�f0) − 4�(1 + �2)f′′

0 ] + (1 + �2)2f′′′0 } = 0; (67)

which is exactly the same equation obtained by Method II. Speci7c m values will be selected belowm = 0 case: For this speci7c choice, f0 is calculated from (67) and then other functions from (64)–(66).

The results are

f0(�) =12�

{c1 Arc tan �+

c1� − c21 + �2

+ c3

}; (68)


g0(�) =12�

{c2 Arc tan � − c1 + c2�

1 + �2+ c4

}; (69)

r0(�) =c1� − c21 + �2

: (70)

Order 1 solutions are found. One needs to calculate right-hand sides of order � equations, namely Eqs. (26)and (27). Substituting (68)–(70) into (63) and all into (26) and (27), one has

h1 =1

�2x3(1 + �2)3{−(c22 − c2c3 − c1c4) + (4c1c2 − 3c1c3 + 3c2c4)�

− (c21 + c2(3c3 − 4c2) + 3c1c4)�2 − (2c1c2 − c1c3 + c2c4)�3

+ [2c1c2 − 3(c21 − c22)� − 6c1c2�2 + (c21 − c22)�3] Arc tan �}; (71)

h2 =1

�2x3(1 + �2)3{−(2c1c2 − c1c3 + c2c4) + (c21 + c2(3c3 − 4c2) + 3c1c4)�

+(4c1c2 − 3c1c3 + 3c2c4)�2 + (c22 − c2c3 − c1c4)�3 + [(c21 − c22)

+ 6c1c2� − 3(c21 − c22)�2 − 2c1c2�3] Arc tan �}: (72)

Examining the symmetries given in Eq. (30), when d = −a and b = c = 0 are selected, the right-hand sideterms can be annulled and this choice leads to a simple selection such that E = F = G = 0. The similarityvariables are

�=yx; u1 =

f1(�)x

;

v1 =g1(�)x

; p1 =r1(�)x2

: (73)

Substituting into Order � equations together with the h1 and h2 de7nitions and solving yields

f1(�) =c22 − c2c3 − c1c4 + 2�3�(c6 − c5�2)

2�3(1 + �2)2; (74)

g1(�) =(c22 − c2c3 − c1c4 + 2�3�(c6 − �2c5))�

2�3(1 + �2)2+ c5; (75)

r1(�) =(c1(c2 − c3) + c2c4 + 2(c5 + c6)�3)�+ (c2 − c1�)(c1 + c2�)Arc tan�

�2(1 + �2)2: (76)

Back substitution and returning to original variables yields exactly the same solutions given in Eqs. (49)–(51)obtained by Method II.m = 2 case: For this choice, Eq. (67) is solved for f0 and then Eqs. (64) and (66) for the remaining

variables

f0(�) = c1 + c2�+ c3�2; (77)

g0(�) = c4 − 2c1� − 12 c2�

2; (78)

r0(�) = �{2(c1 + c3) + (2c4 − c2)�}: (79)


The right-hand sides of O(�) equations are calculated as follows:

h1 = −x{c22 + 4c1(5c1 + c3) + 2c2c4 + 8c24 + 2(c1(5c2 − 4c4) + 2c3(c2 + c4))�}; (80)

h2 = −x{4c21�+ 5c22�+ 4c3(c4 + 2c3�) − c1(8c4 − 10c2 + 4c3�) − c2(2c4� − 4c3)}: (81)

Examining the symmetries given in Eq. (30), when d = 3a and b = c = 0 are selected, the right-hand sideterms can be annulled and this choice leads to a simple selection such that E = F = G = 0. The similarityvariables are

�=yx; u1 = x3f1(�);

v1 = x3g1(�); p1 = x2r1(�) (82)

and the solutions at this order are 7nally:

f1(�) = a1 + a2�+ a3�2 + a4�3; (83)

g1(�) =2a2 + 3a4

3− 3a1� − a2�2 − 1

3a3�3; (84)

r1(�) =c222

+ c2c4 + 4c24 + 2c21(5 + �2) + 2c1(c3 + 5c2� − 4c4� − c3�2)

+ �(3a1 + a3) + 12 �(5c

22�+ 8c3(c4 + c3�) + 2c2(4c3 − c4�)

+ 4�(3a4 + a2) − 2��(3a1 + a3)): (85)

Back substitutions yield exactly the same solutions (59)–(61) given by Method II.

4.2. Translational symmetry with parameters b and c

In this section, approximate solutions will be calculated by three approximate symmetry methods usingtranslational symmetries.

4.2.1. Method ISince the translational symmetries of the independent variables are in fact exact symmetries of the original

equation, it is obvious that Method I will yield a non-trivial solution. However, this solution would be the exactsolution of the original equation. Since this solution was already presented in Ref. [8] (i.e. Eqs. (16)–(18) inthat reference), the details are not given for this method.

4.2.2. Method IISelecting only b and c parameters in the symmetries (17) of Method II and all others zero yields

dxb

=dyc

=du00

=dv00

=dp0

0=

du10

=dv10

=dp1

0: (86)

Solving the system, one has

�= y − mx; u0 = f0(�); v0 = g0(�);

p0 = r0(�); u1 = f1(�);

v1 = g1(�); p1 = r1(�); (87)


where m= c=b. Substituting these variables into Eqs. (10)–(15) yields

− mf′0 + g′

0 = 0; (88)

mr′0 + �(m2 + 1)f′′

0 = 0; (89)

− r′0 + �m(mg′′

0 + f′′0 ) = 0; (90)

− mf′1 + g′

1 = 0; (91)

mr′1 + �(1 + m2)f′′

1 = m{(2 + 5m2)f′0f

′′0 − m(f′

0 − 2mg′0)g

′′0 } + (1 + m2)(mf0 − g0)f′′′

0 ; (92)

− r′1 + �m(mg′′

1 + f′′1 ) = −f′

0{(2 + 5m2)f′′0 + m3g′′

0 } − m{(m2 − 1)g′0f

′′0 + (1 + m2)

× g0f′′′0 − mf0(f′′′

0 + mg′′′0 )}: (93)

Eliminating r0 between (89) and (90) gives the simple equation

(1 + m2)2f′′0 = 0: (94)

For m �= ∓i, f0(�) is

f0(�) = c1�+ c2: (95)

The remaining functions are then solved

g0(�) = mc1�+ c3; (96)

r0(�) = c4; (97)

f1(�) = k1�+ k2; (98)

g1(�) = mk1�+ k3; (99)

r1(�) = k4: (100)

Back substitution into original variables yield 7nally

u(x; y) = c1(y − mx) + c2 + �1(k1(y − mx) + k2); (101)

v(x; y) = mc1(y − mx) + c3 + �1(mk1(y − mx) + k3); (102)

p(x; y) = c4 + �1k4: (103)

The above solutions can be retrieved from the exact solutions given in [8] by considering vanishing perturbationparameter �1 and neglecting higher order and exponentially small terms. It can be said that this solution is avalid approximate solution of the original equation.


4.2.3. Method IIIFor the 7rst-order symmetries, h1 =h2 =0 should be selected in Eqs. (30). Selecting b and c in symmetries

(29) with all other parameters zero yields

dxb

=dyc

=du00

=dv00

=dp0

0: (104)

Similarity variables and functions are

�= y − mx; u0 = f0(�);

v0 = g0(�); p0 = r0(�); (105)

where m= c=b. The 7rst-order equations transform into the following ordinary di8erential system:

− mf′0 + g′

0 = 0; (106)

mr′0 + �(m2 + 1)f′′

0 = 0; (107)

− r′0 + �m{mg′′

0 + f′′0 } = 0: (108)

Following similar steps with Method II, the functions are found to be

f0(�) = c1�+ c2; (109)

g0(�) = mc1�+ c3; (110)

r0(�) = c4: (111)

The right-hand sides of O(�1) equations are calculated next

h1 = 0; (112)

h2 = 0: (113)

Since h1 and h2 vanishes, this results in a similar symmetry with the O(1) case, yielding similar variables

�= y − mx; u1 = f1(�);

v1 = g1(�); p1 = r1(�) (114)

and hence similar solutions with O(1) case

f1(�) = k1�+ k2; (115)

g1(�) = mk1�+ k3; (116)

r1(�) = k4: (117)

Back substitution to original variables yields exactly the same solutions given in Eqs. (101)–(103) obtainedby Method II.


5. Concluding remarks

New approximate solutions of the creeping $ow equations of second grade $uids are found by employingthree di8erent approximate symmetry methods. Approximate symmetries as well as approximate solutionsobtained by di8erent methods are contrasted. From the comparisons, the following remarks are given:

(1) Starting from the exact generator, it is not possible to obtain the approximate generator of Method I forthis speci7c example.

(2) Since the dependent variable is not expanded in a perturbation series, approximate solutions obtainedby Method I using a 7rst-order approximate generator may contain higher order terms also. This occurswhen exact symmetries and approximate symmetries of Method I coincide.

(3) When the approximate symmetries of Method I do not coincide with the exact symmetries, groupinvariant (similarity) solutions cannot be produced by Method I.

(4) Method I failed to produce some approximate group-invariant solutions which can be obtained by Meth-ods II and III. Three such examples are given for creeping $ow equations.

(5) Method II and Method III are consistent with the perturbation theory and yield correct terms for theapproximate solutions. They are better in producing approximate solutions also.

(6) Method II requires more algebra than Method III. If higher order perturbations are considered, sincethe dependent variable increases and the equations become highly coupled, the algebra involved incalculating symmetries increases enormously in Method II.

(7) Method III is more e8ective and the algebra reduces much if the equations at each order are same withthe exceptions of non-homogenous terms.

(8) Method I and II employ non-linear systems. Some approximate solutions produced by superpositionprinciple (a feature of linear systems) cannot be retrieved due to this non-linearity. This problem doesnot arise in Method III. This issue was discussed in detail in a previous paper [15].

(9) When the equations appearing at each order of approximation are simple and can be solved in a straight-forward manner, usage of approximate symmetry methods is unpractical and unnecessary. Those methodsbecome eScient when solutions of equations appearing at di8erent orders are involved and cannot beretrieved in a straightforward calculation. Such equations were already treated by Method II previously[11–14]. The creeping $ow equations considered here is a good example for involved problems.

(10) Authors recommend Method III as the approximate symmetry method.

References

[1] R.I. Tanner, Plane creeping $ow of incompressible second order $uids, Phys. Fluids 9 (1966) 1246.[2] R.R. Huilgol, On uniqueness and non-uniqueness in the plane creeping $ows of second order $uids, Soc. Ind. Appl. Math. 24 (1973)

226.[3] R.L. Fosdick, K.R. Rajagopal, Uniqueness and drag for $uids of second grade in steady motion, Int. J. Non-linear Mech. 13 (1978)

131.[4] K.R. Rajagopal, On the creeping $ow of the second order $uids, J. Non-Newtonian Fluid Mech. 15 (1984) 239.[5] J.E. Dunn, K.R. Rajagopal, Fluids of di8erential type: critical review and thermodynamic analysis, Int. J. Eng. Sci. 33 (1995) 689.[6] P. Bourgin, J.A. Tichy, The e8ect of an additional boundary condition on the plane creeping $ow of a second-order $uid, Int. J.

Non-linear Mech. 24 (1989) 561.[7] G.P. Galdi, K.R. Rajagopal, Slow motion of a body in a $uid of second grade, Int. J. Eng. Sci. 35 (1997) 33.[8] M. YHurHusoy, M. Pakdemirli, HO.F. Noyan, Lie Group analysis of creeping $ow of a second grade $uid, Int. J. Non-linear Mech. 36

(2001) 955.[9] V.A. Baikov, R.K. Gazizov, N.H. Ibragimov, Approximate symmetries of equations with a small parameter, Mat. Sb. 136 (1988)

435 (English Translation in Math. USSR Sb. 64 (1989) 427).[10] V.A. Baikov, R.K. Gazizov, N.H. Ibragimov, in: N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Di8erential

Equations, Vol. 3, CRC Press, Boca Raton, FL, 1996.


[11] W.I. Fushchich, W.H. Shtelen, On approximate symmetry and approximate solutions of the non-linear wave equation with a smallparameter, J. Phys. A: Math. Gen. 22 (1989) 887.

[12] N. Euler, M.W. Shulga, W.H. Steeb, Approximate symmetries and approximate solutions for a multi-dimensional Landau–Ginzburgequation, J. Phys. A: Math. Gen. 25 (1992) 1095.

[13] M. Euler, N. Euler, A. KHohler, On the construction of approximate solutions for a multi-dimensional nonlinear heat equation,J. Phys. A: Math. Gen. 27 (1994) 2083.

[14] N. Euler, M. Euler, Symmetry properties of the approximations of multidimensional generalized Van der Pol equations, NonlinearMath. Phys. 1 (1994) 41.

[15] M. Pakdemirli, M. YHurHusoy, Approximate symmetries of a nonlinear wave equation using di8erent methods, Math. Comput. Appl.5 (2000) 179.

[16] N.H. Ibragimov, M. Torrisi, A. Valenti, Preliminary group classi7cation of equation vtt = f(x; vx)vxx + g(x; vx), J. Math. Phys. 32(1991) 2988.

[17] N.H. Ibragimov, M. Torrisi, A simple method for group analysis and its application to a model of detonation, J. Math. Phys. 33(1992) 3931.

[18] M. Pakdemirli, M. YHurHusoy, Equivalence transformations applied to exterior calculus approach for 7nding symmetries: an exampleof non-Newtonian $uid $ow, Int. J. Eng. Sci. 37 (1999) 25.

[19] E. HOzkaya, M. Pakdemirli, Lie group theory and analytical solutions for the axially accelerating string problem, J. Sound Vibration230 (2000) 729.

Approximate symmetries of creeping flow equations of a second grade fluid

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