# Introduction

he flow of nanofluid in a channel, between two contracting or expanding plates and also, over a stretching sheet have aroused research interests in recent times. Among the recent studies, the analysis of squeezing flow of nanofluid or viscous fluid between two parallel plates have increased tremendously due to its various industrial and biological applications. After the pioneer work on squeezing flow by Stefan [1], there have been improved works on the flow phenomena. However, the earlier studies [1][2][3] on squeezing flow were based on Reynolds equation. Jackson [4] and Usha and Sridhar an [5] pointed out the insufficiencies of the Reynolds equation for some cases of flow situations. Consequently, there have been several attempts and renewed research interests by different researchers to properly analyze and understand the squeezing flows using different analytical and numerical methods . Also, effects of magnetic field, flow characteristics and fluid properties on the squeezing flow have been widely investigated under no slip conditions [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. However, in many cases of fluid and flow problems such as polymeric liquids, thin film problems, nanofluids, rarefied fluid problems, fluids containing concentrated suspensions, and flow on multiple interfaces, slip condition prevails at the boundary of the flow process.

Therefore, Navier [43] proposed the general boundary condition which demonstrates the fluid slip at the surface. Such consideration of slip condition in the flow analysis of fluids is of great importance especially when fluids with elastic character are under consideration [44]. In a past study on slip effects on flow conditions of fluids, Ebaid [45] investigated the effects of magnetic field and wall slip conditions on the peristaltic transport in an asymmetric channel. The influence of slip on the peristaltic motion of third-order fluid in asymmetric channel was analyzed by Hayat et al. [46]. Also, Hayat and Abelman [47] presented a study on the effects of slip condition on the rotating flow of a third grade fluid in a nonporous medium. Abelman et al. [48] extended their work to a porous medium and obtained the numerical solutions for the steady magnetodrodynamics flow of a third grade fluid in a rotating frame.

The past efforts in analyzing the squeezing flow problems have been largely based on the applications of various numerical and approximate analytical methods such as differential transformation method (DTM), Adomian Decomposition Method (ADM), homotopy analysis method (HAM), optimal homotopy asymptotic method (OHPM), variational iteration method (VIM). Moreover, most of the studies are based on viscous fluids. To the best of the authors' knowledge, a Approximate Analytical Methods study on squeezing flow of nanofluid under the influences of magnetic field and slip boundary conditions using variation parameter method (VPM) has not been carried out in literature. Also, a comparative study of the three approximate analytical methods (differential transformation, homotopy perturbation and variation parameter methods) has presented in this paper has not been analyzed in past work. Therefore, in the paper, a comparative study of approximate analytical methods is carried out using differential transformation, homotopy perturbation and variation parameter method for the analysis of a steady twodimensional axisymmetric flow of nanofluid under the influence of a uniform transverse magnetic field with slip boundary condition. The analytical solutions are used to investigate the effects of fluid properties, magnetic field and slip parameters on the squeezing flow.

II.


# Problem Formulation

Consider a squeezing flow of nanofluid squeezed between two parallel plates which are at distance 2h apart and they approach each other with slowly with a constant velocity under in the presence of a magnetic field as shown in Fig. 1. Assuming that the fluid is incompressible, the flow is laminar and isothermal, the governing equations of motion for the quasi steady flow of the nanofluid are given as: . 0
v ? = ? (1) ( ) ( ) ( ) ( ) 2 2 0 2.5 1 1 . 1 . B 1 f f s f s v v v f p v t k µ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? = ? + ?? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(2)
(3)
( ) ( ) ( ) ( ) 2 2 0 2.5 1 1 1 B 2 1 f s f f s v w v p w v k ? ? ? ? µ ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + × + ? + = ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(4)
Introducing the stream function ( )
, r z ? , vorticity function ( ) , r z ?
and a generalized pressure for the cylindrical coordinate system as follows:
( ) ( ) ( ) 2 2 2 1 1 1 1 , , , ,2f s u v r z p u v r z r r r ? ? ? ? ? ? ? ? ? ? + ? ? ? ? = = ? ? = ? = + ? ? ?(5)
Eliminating the pressure term from Eqs. ( 3) and (4), we have Neglecting the body force, the continuity and Navier-Stokes' equation for the problem is given as
( ) ( ) ( ) ( ) ( ) 2 2 2 4 2 0 2.5 2.5 2 , / 1 1 1 B , 11f f f s r r z r r z k ? ? µ µ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + = ? + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?. 0 v ? = ? 2 2 2 2 2 1 r r r z ? ? ? = ? + ? ? ? ? (7)
The boundary conditions are given as 0, 0 0
, w v z v and z v z H v V and v z ? ? = = = ? ? = = ? = ? (8)
Applying a transformation ( ) ( ) 2 , r z r f z ? = , the compatibility Eq. ( 6) reduces to Eq. ( 9) as
( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 2.5 2 0 '' ''' 2 1 1 B 1 1 0 f s iv f f f z f z f z f z k ? ? ? ? ? ? ? µ µ ? ? ? ? ? + ? ? ? ? ? ? ? + + = ? ? ? ?(9)
And the slip boundary conditions as
( ) ( ) ( ) ( ) ( ) '' ' '' 0 0, 0 0, , 2 f f v f h f h f h ? = = = =(10)
Using the following dimensionless parameters in Eq. ( 11)
( ) 2 * * 2 0 B 1 , , , = / 2 f f nf Hv f z F z R G h Da m v h k ? ? µ µ ? ? = = = + = + ? ? ? ? ? ? . (11)
The dimensionless form of Eq. ( 9) is given as
( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 ''' 2 '' 1 1 0 iv s f F z R F z F z G F z ? ? ? ? ? ? ? + ? + ? ? = ? ? ? ? ? ?(12)
And the dimensionless boundary conditions in Eq. ( 10) as
( ) ( ) ( ) ( ) ( ) 0 0, 0 0 1 1, 11F F F F F ? ?? = = ? ?? = = (13)
where the asterisk, * has been omitted in Eqs. (12) and Eq. ( 13) for the sake of conveniences.


# III. Approximate Analytical Methods Of Solution: Differential Transform Method

The differential transform method has widely been used to solve both singular and non-singular perturbed boundary values problems. It gives analytical solution to differential or integral solutions in the form of a polynomial by transforming each term in the differential equation or integral into a recursive form or relation of the equation which follows an iterative procedure for obtaining analytical series solutions of differential equation. The basic definitions of the method is as follows:

If ( ) u x is analytic in the domain T, then it will be differentiated continuously with respect to space x.

( ) ( , )
p p d u x x p dx ? = for all x T ? (17) ( ) ( ) ( , ) i p i p x x d u x U p x p dt ? = ? ? = = ? ? ? ? (18)
where p U is called the spectrum of ( ) 
u( ) ( ) ( ) ! p i p x x u x U p p ? ? ? ? = ? ? ? ? ? ? ?(19)
where Eq. ( 19) is called the inverse of ) (k U using the symbol 'D' denoting the differential transformation process and combining Eq. ( 18) and Eq. (19), it is obtained that ( )
1 0 ( ) ( ) ( ) ! p i p x x u x U p D U p p ? ? = ? ? ? = = ? ? ? ? ? ? ?(20)
a) Operational properties of differential transformation method If ( ) ( ) u x and v x are two independent functions with space (x) where ( ) U p and ( ) V p are the transformed function corresponding to ( ) u x and ( ) v x , then it can be shown from the fundamental mathematics operations performed by differential transformation that.
i. If ( ) ( ) ( ), z x u x v x = ± then ( ) ( ) ( ) p U p V p ? = ± ii. If ( ) ( ), z x u x ? = then ( ) ( ) Z p U p ? = iii. If ( ) ( ) , n n d u x z x dx = then ( ) ( 1) ( 2)( 3)...( ) ( ) p p p p p n U p n ? = + + + + + iv. If ( ) ( ) ( ), z x u x v x = then 0 ( ) ( ) ( ) p r p V r U p r = ? = ? ? v. If ( ) ( ) m z x u x = , then1 0 ( ) ( ) ( )p m r p U r U p r ? = ? = ? ? vi. If ( ) ( ) ( ), z x u x v x = then 0 ( ) ( 1) ( 1) ( ) p r p r V r U p r = ? = + + ? ? vii. If ( ) ( ) , n m n d u x z x x dx = then ( )( )( )( ) ( ) ( ) 0 ( ) 1 1 2 3 ... p l p l m p l p l p l p l n U p l n ? = ? = ? ? ? + ? + ? + ? + ? + ? viii. If 3 3 ( ) ( )) ( ) , d u x d u x z x dx dx = then ( )( )( )( ) ( ) 0 ( ) 1 2 3 3 p l Z p U p l l l l U l = = ? + + + + ? ix. If 2 2 ( ) ( )) ( ) , d u x d u x z x dx dx = then ( ) ( )( )( ) ( ) 0 ( ) 1 1 1 2 2 p l Z p p l U p l l l U l = = ? + ? + + + + ? x. If 2 ( ) ( ) du x z x dx ? ? = ? ? ? ? then ( ) ( )( ) ( ) 0 ( ) 1 1 1 1 p l Z p p l U p l l U l = = ? + ? + + + ? xi. If ( ) ( ) , du x z x u dx = then ( )( ) ( ) 0 ( ) 1 1 p l Z p U p l l U l = = ? + + ? If 2 2 2 ( )) ( ) , d u x z x dx ? ? = ? ? ? ? then ( )( ) ( )( )( ) ( ) 0 ( ) 1 2 2 1 2 2 p l Z p p l p l U p l l l U l = = ? + ? + ? + + + + IV. ( )( )( )( ) [ ] ( ) ( ) ( )( )( ) [ ] [ ] ( )( ) [ ] ( ) 0 2.5 1 2 3 4 4 3 2 1 3 2 2 0 1 1 1 k l s f k k k k F k R k l k l k l F l F k l G k k F k ? ? ? ? ? = + + + + + ? ? ? + ? + ? ? ? ? + ? ? ? + ? + ? ? ? ? ? ? ? ? + + + ? + = ?(21)
With differential transformed boundary conditions  Where a and b are unknowns to be determined later using the boundary conditions of Eq. (16b).
[ ] [ ] [ ] [ ] 0 0, 1 , 2 0, 3 , F F a F F b = = = = ? ? ? ? (22) ( 1) [ 1] ( 1)( 2) [ 2] k F k k k F k ? + + = + + + ? ? [ ] 0 4 F = ? [ ] ( ) ( ) 2.5 2 1 5 20 1 1 s f F bG abR ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? [ ] 0 6 F = ? [ ] ( ) ( ) ( ) ( ) ( ) ( ) 2.5 4 2.5 2. 2 2 5 2 2 6 4 1 7 1 1 1 1 1 84 3 1 0 s s f f s f bG b R abG R F a b R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? + ? ? ? ? ? ? ? ? [ ] 0 8 F = ? [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 2.5 2.s s f f s s f f s f bG b R abG R F ab R a bG R a G bR ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? [ ] 0 10 F = ? Year 2017 Global Journal of
Using Eqs. ( 21) and ( 22), the value of ( ), 1, 2,3, 4,5,... 19, 20.   i i F = ? are Application of the Differential Transform Method to the Present Problem

The differential transform of ( 15) and ( 16) is given by  
[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )s s f f s s f f s f bG b R abG R b R ab R F G bG R G a ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? + + + ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ( ) ( ) ( ) ( ) ( ) ( )31 1 1 1 5 s f s s f f a b R a bG R a b R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ?
And so on According to the definition of DTM, the solution is 
( ) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]F z F zF z F z F z F z F z F z F z F z F z F z F = + + + + + + + + + + + + ? ? ? ? ? ? ? ? ? ? ? ? (23)
a) The basic idea of homotopy perturbation method In order to establish the basic idea behind homotopy perturbation method, consider a system of nonlinear differential equations given as
( ) ( ) 0, A U f r r ? = ? ?(24)
with the boundary conditions
, 0, u B u r ? ? ? ? = ? Î?" ? ? ? ? ? (25)
where A is a general differential operator, B is a boundary operator,


# ( )

f r a known analytical function and Î?" is the boundary of the domain ? The operator A can be divided into two parts, which are L and N, where L is a linear operator, N is a non-linear operator. Eq.( 24) can be therefore rewritten as follows
( ) ( ) ( ) 0 L u N u f r + ? =(26)
By the homotopy technique, a homotopy
( ) [ ] , : 0,1 U r p R ? × ? can be constructed, which satisfies ( ) ( ) ( ) ( ) ( ) ( ) [ ] , 1 0, 0,1 H U p p L U L U p A U f r p ? = ? ? + ? = ? ? ? ? ? ? ? ? ?(27)
Or
( ) ( ) ( ) ( ) ( ) ( ) , 0 H U p L U L U p LU p N U f r ? ? = ? + + ? = ? ? ? ?(28)
In the above Eqs. ( 27) and ( 28),
[ ] 0,1 p ? is an embedding parameter, o
u is an initial approximation of equation of Eq.( 24), which satisfies the boundary conditions. Also, from Eqs. ( 27) and ( 28), we will have
( ) ( ) ( ) , 0 0 o H U L U L U = ? = (29) ( ) ( ) ( ) , 0 0 H U A U f r = ? =(30)
The changing process of p from zero to unity is just that of ( )
, U r p from ( ) o u r to ( )
u r . This is referred to homotopy in topology. Using the embedding parameter p as a small parameter, the solution of Eqs. ( 27) and ( 28) can be assumed to be written as a power series in p as given in Eq. ( 28)
2 1 2 ... o U U pU p U = + + +(31)
It should be pointed out that of all the values of p between 0 and 1, p=1 produces the best result.

Therefore, setting 1 p = , results in the approximation solution of Eq.( 24)
1 2 1 lim ... o p u U U U U ? = = + + +(32)
The basic idea expressed above is a combination of homotopy and perturbation method. 


# A

Hence, the method is called homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantages of the traditional perturbation techniques. The series Eq.( 32) is convergent for most cases.


# b) Application of the homotopy perturbation method to the present problem

According to homotopy perturbation method (HPM), one can construct an homotopy for Eq. ( 16) as
( ) ( ) ( ) ( ) ( ) ( ) 2.5 2 , 1 1 1 "' " iv iv s f H z p p p R F F R FF F G ? ? ? ? ? ? ? ? ? = ? + + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(33)
Using the embedding parameter p as a small parameter, the solution of Eqs. ( 16) can be assumed to be written as a power series in p as given in Eq. ( 33)
2 3 1 2 3 ... o p p p F F F F F = + + + + ? ? ? ? ?(34)
On substituting Eqs. (34) and into Eq.( 33) and expanding the equation and collecting all terms with the same order of p together, the resulting equation appears in form of polynomial in p . On equating each coefficient of the resulting polynomial in p to zero, we arrived at a set of differential equations and the corresponding boundary conditions as 0 0, 0 0, 1 0,
( ) 0 0 : 0, iv p F = ? ( ) ( ) ( ) ( ) ( ) " ' " 0 0 0 0 0 0 0, 0 0, 1 1, 1 1 F F F F F ? = = = = ? ? ? ? ? (35) ( ) ( ) ( ) 2.5 1 2 " ''' 1 0 0 0 : 1 1 0, iv s f F F G F p F R ? ? ? ? ? ? ? ? + ? + ? = ? ? ? ? ? ? ? ? ? ?(36) ( ) ( ) ( ) ( ) ( )" ' " 1 1 1 1 1 0 0, 0 0, 1 0, 1 1 F F F F F ? = = = = ? ? ? ? ? ( ) ( ) ( ) ( ) ( ) 2.5 2.5 2 2 " ''' ''' 2 1 1 0 0 1 : 1 1 1 1 0 , iv s s f f F F F F G F R F p R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? + ? + ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (37) ( ) ( ) ( ) ( ) ( ) " ' " 2 2 2 2 2 0 0, 0 0, 1 0, 1 1 F F F F F ? = = = = ? ? ? ? ? ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 2.5 3 2 " ''' ''' 3 2 2 0 1 1 2.5 ''' 0 2 : 1 1 1 1 1 1 0, iv s s f f s f F F F F F R R F R F p F G ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? (38) ( ) ( ) ( ) ( ) ( ) " ' " 3 3 3 3 3 0 0, 0 0, 1 0, 1 1 F F F F F ? = = = = ? ? ? ? ? ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 2.5 4 2 " ''' ''' 4 3 3 0 3 0 2.5 2.5 ''' ''' 2 1 1 2 2.5 '' 0 3 : 1 1 1 1 1 1 1 1 1 1 iv s s f f s s f f s f F F F F F F F F p G R R R R F R F F F ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ' 0 =(1 1 F F F F F ? = = = = ? ? ? ? ? ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 2.5 5 2 " ''' ''' 5 4 3 0 4 0 2.5 2.5 ''' ''' 3 1 2 2 2.5 1 3 : 1 1 1 1 1 1 1 1 1 1 iv s s f f s s f f s f p G R R F F F F F F F F F F F F R R R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( ) ( ) 2.5 ''' ''' 0 4 1 1 0 s f F R F ? ? ? ? ? ? ? + ? + ? = ? ? ? ? ? ? ? ? (40) ( ) ( ) ( ) ( ) ( ) " ' " 5 5 5 5 5 0 0, 0 0, 1 0, 1 1 F F F F F ? = = = = ? ? ? ? ?
On solving the above Eqs. (35)(36)(37)(38)(39)(40), we arrived at 
( ) ( ) ( ) 3 0 3 2 1 2 3 1 z z F z ? ? ? + = ? ? (41)( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 2.5 2 5 7 1 2 2 2.5 2.5 2 2 2 9 1 1 2 1 3 1 1 3 3 1 2 3 1 2 3 1 90 1 1 2 1 63 1 1 60 3 1 3 1 3 1 1 3 2 1 s s f f s s f f R R G z z z R R F G ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? ? ? + ? ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2.5 2.5 2 2 2 2.5 2 2 3 6 1 1 2 1 9 1 1 12 3 1 3 1 3 1 90 1 1 2 1 63 1 60 3 1 3 1 1 3 2 1 s s f f s s f f z R R G R R G ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? + ? ? ? ? ? ? ? ? ? ? + + ? ? + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 2 2.5 2.5 2 2 2 1 3 1 3 6 1 1 2 1 9 1 1 12 3 1 3 1 2 3 1 s s f f z R R G ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( ) ( ) ( ) ( ) 2.5 2 9 1 1 2 1 3 3 1 3 1 s f R G ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? + ? ? ( ) ( ) ( ) 2.5 2 2 3 1 1 2 3 1 s f R z ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? (42) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 3 4 lim ... o p F z F z F z F z F z F F z z ? = = + + + + + ? ? ? ? ? ? (43)( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
c) The Procedure of Variation Parameter Method The basic concept of VPM for solving differential equations is as follows: The general nonlinear equation is in the operator form
( ) ( ) ( ) Lf Rf Nf g ? ? ? + + = (44)
The linear terms are decomposed into L + R, with L taken as the highest order derivative which is easily invertible and R as the remainder of the linear operator of order less than L. where g is the system input or the source term and u is the system output, Nu represents the nonlinear terms.

( ) 
1 0 0 ( ) ( ) ( , ) ( ) ( ) ( ) n n n f f Rf Nf g d ? ? ? ? ? ? ? ? ? ? + = + ? ? ? ?(45f ? is given by 0 0 (0) ( ) ! i m i i k f f i ? = = ?(46)
m is the order of the given differential equation, k i s are the unknown constants that can be determined by initial/boundary conditions and ( , ) ? ? ? is the multiplier that reduces the order of the integration and can be determined with the help of Wronskian technique.

( )
1 1 1 1 1 ( ) ( , )( 1)!( )! ( 1)! i i m m m i i m i m ? ? ? ? ? ? ? ? ? ? ? ? ? = = ? ? ? ?(47)
From the above, one can easily obtain the expressions of the multiplier for
Lf(?)= f n (?) ( ) ( ) ( ) ( ) ( ) 2 2 3 2 2 3 4 3 2 2 3 4 1, ,1 2, , 3, , 2! 2! 4, , 3! 2! 2! 3! 5, , 4! 3! 2 2! 3! 4! n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? = = = = ? = = ? + = = ? + ? = = ? + ? + ? ( ) 5 4 3 2 2 3 4 5 6, , 5! 4! 2 3! 2 3! 4! 5! n ? ? ? ? ? ? ? ?? ? ? ? ? = = ? + ? + ? ? ?
In the same manner, the expressions for ( ) ( ) ( ) ( ) ( )
2 3 45 6 , , , , ... z z z F F F F F z z ? ? ? ? ?
were obtained. However, they are too large expressions to be included in this paper. Setting 1 p = , results in the approximation solution of Eq. ( 24)

The VPM provides the general iterative scheme for Eq. ( 45) as: ( ) 
? + ? + ? ? ? ? ? ( ) ( ) ( ) ( ) ( ) 2 3 1 1 2 3 4 3 2 2 3 2.5 ''' 2 '' 0 ( ) 2 6 1 1 3! 2! 2! 3! n z s n n n f z z F z k k z k k z z z R F F G F d ? ? ? ? ? ? ? ? ? ? ? ? + = + + + ? ? ? ? ? ? ? + + + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(48)
The above equation can also be written as 


# Global
( ) ( ) ( ) ( ) ( )2 3 1 3 2 2 3 2.5 ''' 2 '' 0 ( ) (0) (0) (0) (0) 2 6 11 3! 2! 2! 3! n z s n n n f z z F z F F z F F z z z R F F G F d ? ? ? ? ? ? ? ? ? ? ? ? + ? ?? ??? = + + + ? ? ? ? ? ? ? + + + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(49) ( ) 7 6 5 2 4 3 3 4 2 5 6 7 8, , 7! 6! 2 5! 6 4! 6 4! 2 5! 6! 7! n ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? = = ? + ? + ? + ? ? ? ? ?
Consequently, an exact solution can be obtained when n approaches infinity.

Using the standard procedure of VPM as stated above, one can write the solution of Eq. ( 15) as

From the boundary conditions in Eq. ( 16)

(0) 0, (0) 0
F F?? = =
Using the above statement and inserting the boundary conditions of Eq. ( 16) into Eq. ( 49), we have
( ) ( ) ( ) ( ) ( ) 3 2 1 1 3 2 2 3 2.5 ''' 2 '' 0 ( )61 1 3! 2! 2! 3! n s n n n f k z F z k z z z z R F F G F d ? ? ? ? ? ? ? ? ? ? ? ? ? + = + ? ? ? ? ? ? ? + + + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (50)
From the iterative scheme, it can easily be shown that the series solution is given as
3 2 0 1 ( ) 6 k z F z k z = +(51)
Here, k 1 , k 2 , k 3 , and k 4 are constants obtained by taking the highest order linear term of Eq. ( 15) and integrating it four times to get the final form of the scheme.     
s f s f k k R z k z G k z F z k z R k z ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = + ? + ? ? ? + ? ? ? ? ? ? ? ? (52)1 s s f f s s f f s R k k z R k z k z k k Rz F k z R k k G z R k k z R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? = + + + ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2s f f s s f f s f R k z k G z R k k z R k G z R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? + ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ( ) ( ) ( ) ( )3s f s f k k z R k G z R k z ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? (53) Therefore, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )s s f f s s f f R k k z R k z k z k k Rz F z F k z R k k G z R k k z ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? = = + + + ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? + Similarly, the other iterations F z F z F z F F z F z are obtained.
Although, analytically, the VPM and DTM are somehow easier and straight-forward as compared to HPM, there is no search for Wronskian multiplier (as carried out in VPM) or the rigour of developing recursive relations or differential transforms coupled with the search for included unknown parameter that will satisfy second the boundary condition lead to additional computational cost in the generation of the solution to the problem using DTM. This drawback is not only peculiar to VPM and DTM, other approximate analytical methods such as HAM, ADM, VIM, DJM, TAM also required additional computational cost and time for the determination of included unknown parameter that will satisfy second the boundary condition. Also, the VPM and DTM have their own operational restrictions that severely narrow there functioning domains as they are limited to small domain. Using VPM or DTM for large or infinite domain is accompanied with either the application of before-treatment techniques such as domain transformation techniques, domain truncation techniques and conversion of the boundary value problems to initial value problems or the use of aftertreatment techniques such as Pade-approximants, basis functions, cosine after-treatment technique, sine aftertreatment technique and domain decomposition technique. This is because VPM and DTM were initially established for initial value problems. Amending the methods to boundary value problems especially for large or infinite domains boundary value problems leads to the inclusion of unknown parameters (that will satisfy second the boundary condition) in the solution. This drawback in the other approximation analytical methods is not experienced in HPM as such tasks of before-and after-treatment techniques might not necessarily be required in HPM. This is because HPM is easily applied to the boundary value problems without any included unknown parameter in the solution as found in VPM and DTM.     In order to get an insight into the problem, the effects of pertinent flow, magnetic field and slip parameters on the velocity profile of the fluid are investigated. Fig. 2 and 4 shows the effects of magnetic field and porous parameter on the velocity of the fluid under the influence of slip condition, while Fig. 3 and 5 depicts the influence of the porous and magnetic on the velocity of the fluid under no-slip condition. It could be inferred from the figures that the velocity of the fluid increases with increase in the porous-magnetic parameter under slip condition while an opposite trend was recorded during no-slip condition as the velocity of the fluid decreases with increase in the porous-magnetic parameter under the no slip condition. Fig. 6 shows the influence of the slip parameter ?? on the fluid velocity. By increasing ??, it is observed that the velocity of the fluid increases. Fig. 7 presents the effects of Reynold's number on the velocity of the fluid. It is observed from the figure that by increasing the value R, the velocity of the fluid decreases.


# VI.


# Conclusion

In this work, a comparative study of three approximate analytical methods have been carried out for the analysis of two-dimensional axisymmetric flow of an incompressible viscous fluid through porous medium under the influence of a uniform transverse magnetic field with slip boundary condition. From the analysis, it is established that VPM give higher accurate results than DTM and HPM with faster rate of convergence. Also, from the parametric study, it was established from the results that, the velocity of the fluid increases with increase in the porous-magnetic parameter under slip condition while the velocity of the fluid decreases with increase in the porous-magnetic parameter under no slip condition. By increasing the slip parameter, the velocity of the fluid increases, and the fluid velocity decreases as the Reynolds number increases. The approximate analytical solutions have been verified by comparing the results of the approximate analytical method with the numerical method using Runge-Kutta coupled with shooting method
![Year 2017 Global Journal of Researches in Engineering ( ) Volume XVII Issue VI Version I A On the Squeezing Flow of Nanofluid Through a Porous Medium With Slip Boundary and Magnetic Field: A Comparative Study of Three](image-2.png "")








On the Squeezing Flow of Nanofluid Through Porous Medium With Slip Boundary and Magneti Field: AComparative Study of Three Approximate Analytical MethodsYear 2017( ) Volume XVII Issue VI Version I AFor belongs to the set of non-negative integers, denoted as i x x = , then ( , ) ( , ) i x p x p ? ? = , where p the p-domain. Therefore Eq. (17) can be rewritten asGlobal Journal of Researches in Engineeringi x x =If ( ) u x can be expressed by Taylor's series, the ( ) u xcan be represented as
( ) Volume XVII Issue VI Version I AResearches in Engineering
4( )" 4( )4( )' 4( )" 4( )( ) Volume XVII Issue VI Version I AGlobal Journal of Researches in Engineering
Year 201768IJournal of Researches in Engineering ( ) Volume XVII Issue VI Version AGlobal
35( ) 1( ) 12.52 1 27( ) 1( ) 12.52 7 22(z)12 61 2 12016805040( ) 1( ) 12.51 22 7( ) 1( ) 12.52 2 19( )126022680A ( ) Volume XVII Issue VI Version IGlobal Journal of Researches in Engineering
			T © 2017 Global Journals Inc. (US)
			(6) where © 2017 Global Journals Inc. (US)
			© 2017 Global Journals Inc. (US)
			On the Squeezing Flow of Nanofluid Through Porous Medium With Slip Boundary and Magneti Field: A Comparative Study of Three Approximate Analytical Methods
		
		
			

... 396249600

where the constants k 1 and k 2 are determined using the boundary conditions in Eq. ( 16) i.e.

( )

The equations are solved for the corresponding values of k 1 and k 2 for the different values of ?.

V.


## Results and Discussion

The above analyses show the applications of three approximate analytical methods of differential transformation, homotopy perturbation and variation of parameters methods for the analysis of a steady twodimensional axisymmetric flow of an incompressible viscous fluid under the influence of a uniform transverse magnetic field with slip boundary condition. Using VPM and DTM, closed form series solutions are obtained as they provide excellent approximations to the solution of the non-linear equation with higher accuracy than HPM. Also, the VPM and DTM shows to more convenient for engineering calculations compared to HPM as they appear more appealing than the HPM. However, higher accuracy and high rate of convergence was recorded in VPM than DTM as shown the table, the solution of VPM is used to carry out the parametric study shown in Figs. 2-7. 
			
			

				


* 
	
		Versuch Uber die scheinbare adhesion'', Sitzungsberichte der Akademie der Wissenschaften in Wien
		
			MJStefan
		
	
	
		Mathematik-Naturwissen
		
			69
			
			1874
		
	




* 
	
		On the theory of lubrication and its application to Mr Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil
		
			OReynolds
		
	
	
		Philos. Trans. Royal Soc. London
		
			177
			
			1886
		
	




* 
	
		Load capacity and time relations for squeeze films
		
			FRArchibald
		
		
			FR
		
	
	
		J. Lubr. Technol
		
			78
			
			1956
		
	




* 
	
		A study of squeezing flow
		
			JDJackson
		
	
	
		Appl. Sci. Res. A
		
			11
			
			1962
		
	




* 
	
		Arbitrary squeezing of a viscous fluid between elliptic plates
		
			RUsha
		
		
			RSridharan
		
		
			R
		
	
	
		Fluid Dyn. Res
		
			18
			
			1996
		
	




* 
	
		Unsteady laminar boundary layers in an incom-pressible stagnation flow
		
			WAWolfe
		
		
			KTYang
		
	
	
		J. Appl. Math. Trans. ASM E
		
			14
			
			1965. 1958
		
	
	Appl. Sci. Res.




* 
	
		Fluid inertia effects in squeeze films
		
			DCKuzma
		
	
	
		Appl. Sci. Res
		
			18
			
			1968
		
	




* 
	
		Inertial considerations in parallel circular squeeze film bearings
		
			JATichy
		
		
			WOWiner
		
	
	
		J. Lubr. Technol
		
			92
			
			1970
		
	




* 
	
		Squeezing flows of Newtonian liquid films: an analysis include the fluid inertia
		
			RJGrimm
		
	
	
		Appl. Sci. Res
		
			32
			2
			
			1976
		
	




* 
	
		
			GBirkh
		
		Hydrodynamics, a Study in Logic, Fact and Similitude
				
			Princeton University Press
			1960
			137
		
	
	Revised ed




* 
	
		The squeezing of fluid between two plates
		
			CYWang
		
	
	
		J. Appl. Mech
		
			43
			4
			
			1976
		
	




* 
	
		Squeezing of a viscous fluid between elliptic plates
		
			CYWang
		
		
			LTWatson
		
	
	
		Appl. Sci. Res
		
			35
			
			1979
		
	




* 
	
		Analysis of the squeezing flow of dusty fluids
		
			MHHamdan
		
		
			RMBaron
		
	
	
		Appl. Sci. Res
		
			49
			
			1992
		
	




* 
	
		Squeeze flow of a viscoelastic solid
		
			PTNhan
		
	
	
		J. Non-Newtonian Fluid Mech
		
			95
			
			2000
		
	




* 
	
		Unsteady Squeezing flow of Casson fluid between parallel plates
		
			UKhan
		
		
			NAhmed
		
		
			SI UKhan
		
		
			BSaima
		
		
			ST
		
	
	
		World J. Model. Simul
		
			10
			4
			
			2014
		
	
	Mohyud-din




* 
	
		Analytic approximate solutions for unsteady two dimensional and axisymmetric squeezing flows between parallel plates
		
			MMRashidi
		
		
			HShahmohamadi
		
		
			SDinarv
		
	
	
		Mathematical Problems in Engineering
		
			
			2008. 2008
		
	




* 
	
		On heat transfer effects of a viscous fluid squeezed and extruded between parallel plates
		
			HMDuwairi
		
		
			BTashtoush
		
		
			RADomesh
		
	
	
		Heat Mass Transfer
		
			14
			
			2004
		
	




* 
	
		Squeezing flow of non-Newtonian second grade fluids and micro polar models
		
			AQayyum
		
		
			MAwais
		
		
			AAlsaedi
		
		
			THayat
		
	
	
		Chinese Physics Letters
		
			29
			34701
			2012
		
	




* 
	
		Analysis of squeezing flow of dusty fluids
		
			MHamdam
		
		
			RMBaron
		
	
	
		Applied Science Research
		
			49
			
			1992
		
	




* 
	
		Squeezed flow and heat transfer over a porous surface for viscous fluid
		
			MMahmood
		
		
			SAssghar
		
		
			MAHossain
		
	
	
		Heat and mass Transfer
		
			44
			
		
	




* 
	
		Differential Transformation Method for Newtonian and non-Newtonian nanofluids flow analysis: Compared to numerical solution
		
			MHatami
		
		
			DJing
		
	
	
		Alexandria Engineering Journal
		
			55
			
		
	




* 
	
		On heat and mass transfer analysis for the flow of a nanofluid between rotating parallel plates
		
			STMohyud-Din
		
		
			ZAZaidi
		
		
			UKhan
		
		
			NAhmed
		
	
	
		Aerospace Science and Technology
		
			46
			
			2014
		
	




* 
	
		Nonlinear radiation effects on squeezing flow of a Cass on fluid between parallel disks
		
			STMohyud-Din
		
		
			SIKhan
		
	
	
		Aerospace Science & Technology
		
			48
			
			2016
			Elsevier
		
	




* 
	
		Modeling and Analysis of Unsteady Axisymmetric Squeezing Fluid Flow through Porous Medium Channel with Slip Boundary
		
			MQayyum
		
		
			HKhan
		
		
			MTRahim
		
		
			IUllah
		
		
			; HKhan
		
	
	
		Behavioral Study of Unsteady Squeezing Flow through Porous Medium
				
			2015 25. 2016
			10
			
		
	
	PLoS ONE




* 
	
		On heat and mass transfer in the unsteady squeezing flow between parallel plates
		
			MMustafa
		
		
			SHayat
		
		
			Obaidat
		
	
	
		Mechanica
		
			47
			
			2012
		
	




* 
	
		Unsteady squeezing flow of viscous MHD fluid between parallel plates
		
			AMSiddiqui
		
		
			SIrum
		
		
			ARAnsari
		
	
	
		Mathematical Modeling Analysis
		
			
			2008. 2008
		
	




* 
	
		Approximate analysis of MHD squeeze flow between two parallel disk with suction or injection by homotopy perturbation method
		
			GDomairry
		
		
			AAziz
		
	
	
		Mathematical Problem in Engineering
		
			
			2009. 2009
		
	




* 
	
		The squeezing flow of Cu-water and Cu-kerosene nanofluid between two parallel plates
		
			NAcharya
		
		
			KDas
		
		
			PKKundu
		
	
	
		Alexandria Engineering Journal
		
			55
			
		
	




* 
	
		Mohyud-Din. Magneto hydrodynamic (MHD) squeezing flow of a Casson fluid between parallel disks
		
			NAhmed
		
		
			UKhan
		
		
			XJYang
		
		
			SI UKhan
		
		
			ZAZaidi
		
		
			ST
		
	
	
		Int. J. Phys. Sci
		
			8
			36
			
			2013
		
	




* 
	
		MHD Flow of a Dusty Incompressible Fluid between Dilating and Squeezing Porous Walls
		
			NAhmed
		
		
			UKhan
		
		
			ZAZaidi
		
		
			SUJan
		
		
			AWaheed
		
		
			STMohyud-Din
		
	
	
		Journal of Porous Media
		
			17
			10
			
			2014
		
	




* 
	
		Mohyud-Din. On unsteady twodimensional and axisymmetric squeezing flow between parallel plates
		
			UKhan
		
		
			NAhmed
		
		
			SI UKhan
		
		
			ZAZaidi
		
		
			XJYang
		
		
			ST
		
	
	
		Alexandria Eng. J
		
			53
			
			2014 a
		
	




* 
	
		Mohyud-Din. MHD squeezing flow between two infinite plates
		
			UKhan
		
		
			NAhmed
		
		
			ZAZaidi
		
		
			MAsadullah
		
		
			ST
		
	
	
		Ain Shams Eng. J
		
			5
			
			2014b
		
	




* 
	
		MHD squeezing flow of second grade fluid between parallel disks
		
			THayat
		
		
			AYousaf
		
		
			MMustafa
		
		
			SObadiat
		
	
	
		International Journal of Numerical Methods
		
			69
			
			2011
		
	




* 
	
		Unsteady Squeezing Flow of Casson Fluid with Magneto hydrodynamic Effect and Passing through Porous Medium
		
			HKhan
		
		
			MQayyum
		
		
			OKhan
		
		
			MAli
		
		ID 4293721
	
	
		Mathematical Problems in Engineering
		
			2016
			14
			2016
		
	




* 
	
		Analytical Analysis of Squeezing Flow in Porous Medium with MHD Effect
		
			AUllah
		
		
			MTRahim
		
		
			HKhan
		
		
			MQayyum
		
	
	
		U.P.B. Sci. Bull., Series A
		
			78
			2
			2016
		
	




* 
	
		Squeezing flows of Newtonian liquid films an analysis including fluid inertia
		
			RJGrimm
		
	
	
		Applied Scientific Research
		
			32
			2
			
			1976
		
	




* 
	
		Magneto hydrodynamic lubrication flow between parallel rotating disks
		
			WFHughes
		
		
			RAElco
		
	
	
		Journal of Fluid Mechanics
		
			13
			
			1962
		
	




* 
	
		Inertia Effects in MHD hydrostatic thrust bearing
		
			SKamiyama
		
	
	
		Transacti ons ASME
				
			1969
			91
			
		
	




* 
	
		Magneto hydrodynamic squeeze film
		
			EAHamza
		
	
	
		Journal of Tribol ogy
		
			110
			2
			
			1988
		
	




* 
	
		Unsteady MHD squeezing flow between two parallel rotating discs
		
			SBhattacharyya
		
		
			A
		
	
	
		Mechanics Research Communications
		
			24
			6
			
			1997
		
	




* 
	
		Anaxisymmetric squeezing fluid flow between the two infinite parallel plates in a porous medium channel
		
			SIslam
		
		
			HKhan
		
		
			IAShah
		
		
			GZaman
		
		ID 349803
	
	
		Mathematical Problems in Engineer-ing
				
			2011
			2011
			10
		
	




* 
	
		Sur les lois de l' equilibre et du movement des corps solides elastiques
		
			C.-L.-M.-HNavier
		
		
			1823
			
		
	
	Bulletin des Sciences par la Societe Philomatique de Paris




* 
	
		Existence and uniqueness of the flow of second grade fluids with slip boundary conditions
		
			CLe Roux
		
	
	
		Archive for Rational Mechanics and Analysis
				
			1999
			148
			
		
	




* 
	
		Effects of magnetic field and wall slip conditions on the peristaltic transport of a Newtonian fluid in an asymmetric channel
		
			AEbaid
		
	
	
		Physics Letters A
		
			372
			24
			
			2008
		
	




* 
	
		Abrasive flow machining of cylinder heads and its positive effects on performance and cost characteristics
		
			LJRhooades
		
		
			RResnic
		
		
			TO' Bradovich
		
		
			SStegman
		
	
	
		Tech. Rep
		
			1996
		
	




* 
	
		The influence of slip on the peristaltic motion of third order fluid in an asymmetric channel
		
			THayat
		
		
			MUQureshi
		
		
			NAli
		
	
	
		Physics Letters A
		
			372
			
			2008
		
	




* 
	
		A numerical study of the influence of slip boundary condition on rotating flow
		
			THayat
		
		
			SAbelman
		
	
	
		International Journal of Computational Fluid Dynamics
		
			21
			1
			
			2007
		
	




* 
	
		Steady MHD flow of a third grade fluid in a rotating frame and porous space
		
			SAbelman
		
		
			EMomoniat
		
		
			THayat
		
	
	
		Nonlinear Analysis: Real World Applications
		
			10
			6
			
			2009