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\title{On the Squeezing Flow of Nanofluid Through a Porous Medium With Slip Boundary and Magnetic Field: A Comparative Study of Three Approximate Analytical Methods}
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             \author[1]{M. G.  Sobamowo}

             \affil[1]{  University of Lagos}

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\date{\small \em Received: 7 December 2016 Accepted: 31 December 2016 Published: 15 January 2017}

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\begin{abstract}
        


This paper presents a comparative study of approximate analytical methods is carried out using differential transformation, homotopy perturbation and variation parameter methods for the analysis of a steady two-dimensional axisymmetric flow of nanofluid under the influence of a uniform transverse magnetic field with slip boundary condition. Also, parametric studies are carried out to investigate the effects of fluid properties, magnetic field and slip parameters on the squeezing flow. It is revealed from the results that the velocity of the fluid increases with increase in the magnetic parameter under the influence of slip condition while an opposite trend is recorded during no-slip condition. Also, the velocity of the fluid increases as the slip parameter increases but it decreases with increase in the magnetic field parameter and Reynold number under the no-slip condition. The approximate analytical solutions are verified by comparing the results of the approximate analytical methods with the numerical method using Runge-Kutta coupled with shooting method. Although, very good agreements are established between the results, the results of variation parameter method provide excellent agreement with the results of numerical method.

\end{abstract}


\keywords{nanofluid; squeezing flow; slip boundary; differential transformation method; homotopy perturbation method; variation parameter method.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
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 \hyperref[b0]{[1]}, there have been improved works on the flow phenomena. However, the earlier studies \hyperref[b0]{[1]}\hyperref[b1]{[2]}\hyperref[b2]{[3]} on squeezing flow were based on Reynolds equation. Jackson \hyperref[b3]{[4]} and Usha and Sridhar an \hyperref[b4]{[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 \hyperref[b25]{[27]}\hyperref[b26]{[28]}\hyperref[b27]{[29]}\hyperref[b28]{[30]}\hyperref[b29]{[31]}\hyperref[b30]{[32]}\hyperref[b31]{[33]}\hyperref[b32]{[34]}\hyperref[b33]{[35]}\hyperref[b34]{[36]}\hyperref[b35]{[37]}\hyperref[b36]{[38]}\hyperref[b37]{[39]}\hyperref[b38]{[40]}\hyperref[b39]{[41]}\hyperref[b40]{[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.\par
Therefore, Navier \hyperref[b41]{[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 \hyperref[b42]{[44]}. In a past study on slip effects on flow conditions of fluids, Ebaid \hyperref[b43]{[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. \hyperref[b44]{[46]}. Also, Hayat and Abelman \hyperref[b45]{[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. \hyperref[b46]{[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.\par
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.\par
II. 
\section[{Problem Formulation}]{Problem Formulation}\par
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: . 0v ? = ? (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 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? = ? + ?? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?\textbf{(2)}\par
(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 ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + × + ? + = ? × ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?\textbf{(4)}\par
Introducing the stream function ( ), r z ? , vorticity function ( ) , r z ?\par
and a generalized pressure for the cylindrical coordinate system as follows:( ) ( ) ( ) 2 2 2 1 1 1 1 , , , ,\textbf{2}f s u v r z p u v r z r r r ? ? ? ? ? ? ? ? ? ? + ? ? ? ? = = ? ? = ? = + ? ? ?\textbf{(5)}\par
Eliminating the pressure term from Eqs. ( \hyperref[formula_26]{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 , 1\textbf{1}f 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)\par
The boundary conditions are given as 0, 0 0, w v z v and z v z H v V and v z ? ? = = = ? ? = = ? = ? (8)\par
Applying a transformation ( ) ( ) 2 , r z r f z ? = , the compatibility Eq. ( \hyperref[formula_64]{6}) reduces to Eq. ( \hyperref[formula_9]{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 ? ? ? ? ? + ? ? ? ? ? ? ? + + = ? ? ? ?\textbf{(9)}\par
And the slip boundary conditions as( ) ( ) ( ) ( ) ( ) '' ' '' 0 0, 0 0, , 2 f f v f h f h f h ? = = = =\textbf{(10)}\par
Using the following dimensionless parameters in Eq. ( \hyperref[formula_11]{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 ? ? = = = + = + ? ? ? ? ? ? . (\textbf{11})\par
The dimensionless form of Eq. ( \hyperref[formula_9]{9}) is given as( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5 ''' 2 '' 1 1 0 iv s f F z R F z F z G F z ? ? ? ? ? ? ? + ? + ? ? = ? ? ? ? ? ?\textbf{(12)}\par
And the dimensionless boundary conditions in Eq. ( \hyperref[formula_10]{10}) as( ) ( ) ( ) ( ) ( ) 0 0, 0 0 1 1, 1\textbf{1}F F F F F ? ?? = = ? ?? = = (13)\par
where the asterisk, * has been omitted in Eqs. \hyperref[b11]{(12)} and Eq. (  {\ref 13}) for the sake of conveniences. 
\section[{III. Approximate Analytical Methods Of Solution: Differential Transform Method}]{III. Approximate Analytical Methods Of Solution: Differential Transform Method}\par
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:\par
If ( ) u x is analytic in the domain T, then it will be differentiated continuously with respect to space x.\par
( ) ( , )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 ? = ? ? = = ? ? ? ? (\textbf{18})\par
where p U is called the spectrum of ( ) u( ) ( ) ( ) ! p i p x x u x U p p ? ? ? ? = ? ? ? ? ? ? ?\textbf{(19)}\par
where Eq. ( \hyperref[formula_19]{19}) is called the inverse of ) (k U using the symbol 'D' denoting the differential transformation process and combining Eq. ( \hyperref[formula_16]{18}) and Eq. \hyperref[b18]{(19)}, it is obtained that ( )1 0 ( ) ( ) ( ) ! p i p x x u x U p D U p p ? ? = ? ? ? = = ? ? ? ? ? ? ?\textbf{(20)}\par
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 = , then\textbf{1 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 ? ? ? ? ? = + + + + + ? ? ? + ? + ? ? ? ? + ? ? ? + ? + ? ? ? ? ? ? ? ? + + + ? + = ?\textbf{(21)}\par
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\par
Using Eqs. ( \hyperref[formula_22]{21}) and (  {\ref 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\par
The differential transform of (  {\ref 15}) and (  {\ref 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 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? + + + ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ( ) ( ) ( ) ( ) ( ) ( )\textbf{3}1 1 1 1 5 s f s s f f a b R a bG R a b R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ?\par
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)\par
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 ? = ? ?\textbf{(24)}\par
with the boundary conditions, 0, u B u r ? ? ? ? = ? Î?" ? ? ? ? ? (\textbf{25})\par
where A is a general differential operator, B is a boundary operator, 
\section[{( )}]{( )}\par
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.( \hyperref[formula_30]{24}) can be therefore rewritten as follows( ) ( ) ( ) 0 L u N u f r + ? =\textbf{(26)}\par
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 ? = ? ? + ? = ? ? ? ? ? ? ? ? ?\textbf{(27)}\par
Or( ) ( ) ( ) ( ) ( ) ( ) , 0 H U p L U L U p LU p N U f r ? ? = ? + + ? = ? ? ? ?\textbf{(28)}\par
In the above Eqs. ( \hyperref[formula_34]{27}) and ( \hyperref[formula_35]{28}),[ ] 0,1 p ? is an embedding parameter, o\par
u is an initial approximation of equation of Eq.( \hyperref[formula_30]{24}), which satisfies the boundary conditions. Also, from Eqs. ( \hyperref[formula_34]{27}) and ( \hyperref[formula_35]{28}), we will have( ) ( ) ( ) , 0 0 o H U L U L U = ? = (29) ( ) ( ) ( ) , 0 0 H U A U f r = ? =\textbf{(30)}\par
The changing process of p from zero to unity is just that of ( ), U r p from ( ) o u r to ( )\par
u r . This is referred to homotopy in topology. Using the embedding parameter p as a small parameter, the solution of Eqs. ( \hyperref[formula_34]{27}) and ( \hyperref[formula_35]{28}) can be assumed to be written as a power series in p as given in Eq. ( \hyperref[formula_35]{28})2 1 2 ... o U U pU p U = + + +\textbf{(31)}\par
It should be pointed out that of all the values of p between 0 and 1, p=1 produces the best result.\par
Therefore, setting 1 p = , results in the approximation solution of Eq.( \hyperref[formula_30]{24})1 2 1 lim ... o p u U U U U ? = = + + +\textbf{(32)}\par
The basic idea expressed above is a combination of homotopy and perturbation method.  
\section[{A}]{A}\par
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.( \hyperref[formula_40]{32}) is convergent for most cases. 
\section[{b) Application of the homotopy perturbation method to the present problem}]{b) Application of the homotopy perturbation method to the present problem}\par
According to homotopy perturbation method (HPM), one can construct an homotopy for Eq. (  {\ref 16}) as( ) ( ) ( ) ( ) ( ) ( ) 2.5 2 , 1 1 1 "' " iv iv s f H z p p p R F F R FF F G ? ? ? ? ? ? ? ? ? = ? + + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?\textbf{(33)}\par
Using the embedding parameter p as a small parameter, the solution of Eqs. (  {\ref 16}) can be assumed to be written as a power series in p as given in Eq. ( \hyperref[formula_41]{33})2 3 1 2 3 ... o p p p F F F F F = + + + + ? ? ? ? ?\textbf{(34)}\par
On substituting Eqs. (34) and into Eq.( \hyperref[formula_41]{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 ? ? ? ? ? ? ? ? + ? + ? = ? ? ? ? ? ? ? ? ? ?\textbf{(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 =\textbf{(}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 ? = = = = ? ? ? ? ?\par
On solving the above Eqs. \hyperref[b33]{(35)}\hyperref[b34]{(36)}\hyperref[b35]{(37)}\hyperref[b36]{(38)}\hyperref[b37]{(39)}\hyperref[b38]{(40)}, we arrived at ( ) ( ) ( ) 3 0 3 2 1 2 3 1 z z F z ? ? ? + = ? ? (\textbf{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)\textbf{( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )}\par
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 ? ? ? + + = (\textbf{44})\par
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.\par
( ) 1 0 0 ( ) ( ) ( , ) ( ) ( ) ( ) n n n f f Rf Nf g d ? ? ? ? ? ? ? ? ? ? + = + ? ? ? ?\textbf{(45}f ? is given by 0 0 (0) ( ) ! i m i i k f f i ? = = ?\textbf{(46)}\par
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.\par
( )1 1 1 1 1 ( ) ( , )\textbf{( 1)!( )! ( 1)}! i i m m m i i m i m ? ? ? ? ? ? ? ? ? ? ? ? ? = = ? ? ? ?\textbf{(47)}\par
From the above, one can easily obtain the expressions of the multiplier forLf(?)= f n (?) ( ) ( ) ( ) ( ) ( ) 2 2 3 2 2 3 4 3 2 2 3 4 1, ,\textbf{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 ? ? ? ? ? ? ? ?? ? ? ? ? = = ? + ? + ? ? ?\par
In the same manner, the expressions for ( ) ( ) ( ) ( ) ( )2 3 4\textbf{5 6 , , , , ..}. z z z F F F F F z z ? ? ? ? ?\par
were obtained. However, they are too large expressions to be included in this paper. Setting 1 p = , results in the approximation solution of Eq. ( \hyperref[formula_30]{24})\par
The VPM provides the general iterative scheme for Eq. ( \hyperref[formula_51]{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 ? ? ? ? ? ? ? ? ? ? ? ? + = + + + ? ? ? ? ? ? ? + + + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?\textbf{(48)}\par
The above equation can also be written as  
\section[{Global}]{Global}( ) ( ) ( ) ( ) ( )\textbf{2 3 1 3 2 2 3 2.5 ''' 2 '' 0 ( ) (0) (0) (0) (0) 2 6 1}1 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 ? ? ? ? ? ? ? ? ? ? ? ? + ? ?? ??? = + + + ? ? ? ? ? ? ? + + + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?\textbf{(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 ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? = = ? + ? + ? + ? ? ? ? ?\par
Consequently, an exact solution can be obtained when n approaches infinity.\par
Using the standard procedure of VPM as stated above, one can write the solution of Eq. (  {\ref 15}) as\par
From the boundary conditions in Eq. (  {\ref 16})\par
(0) 0, (0) 0F F?? = =\par
Using the above statement and inserting the boundary conditions of Eq. (  {\ref 16}) into Eq. ( \hyperref[formula_61]{49}), we have( ) ( ) ( ) ( ) ( ) 3 2 1 1 3 2 2 3 2.5 ''' 2 '' 0 ( )\textbf{6}1 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)\par
From the iterative scheme, it can easily be shown that the series solution is given as3 2 0 1 ( ) 6 k z F z k z = +\textbf{(51)}\par
Here, k 1 , k 2 , k 3 , and k 4 are constants obtained by taking the highest order linear term of Eq. (  {\ref 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 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? = + + + ? ? ? ? ? ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? ( )\textbf{( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2}s f f s s f f s f R k z k G z R k k z R k G z R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? + ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ( ) ( ) ( ) ( )\textbf{3}s 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.\par
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.  {\ref 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. 
\section[{VI.}]{VI.} 
\section[{Conclusion}]{Conclusion}\par
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\begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
         \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.6982758620689655\textwidth}P{0.15172413793103448\textwidth}}
On the Squeezing Flow of Nanofluid Through Porous Medium With Slip Boundary and Magneti Field: A\tabcellsep \\
Comparative Study of Three Approximate Analytical Methods\tabcellsep \\
\tabcellsep Year 2017\\
\tabcellsep ( ) Volume XVII Issue VI Version I A\\
For 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 as\tabcellsep Global Journal of Researches in Engineering\\
i x x =\tabcellsep \\
If ( ) u x can be expressed by Taylor's series, the ( ) u x\tabcellsep \\
can be represented as\tabcellsep \end{longtable} \par
 
\caption{\label{tab_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.85\textwidth}}
( ) Volume XVII Issue VI Version I A\\
Researches in Engineering\end{longtable} \par
 
\caption{\label{tab_2}}\end{figure}
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\noindent\textbf{} \par 
\begin{longtable}{P{0.008095238095238096\textwidth}P{0.024285714285714285\textwidth}P{0.024285714285714285\textwidth}P{0.024285714285714285\textwidth}P{0.008095238095238096\textwidth}P{0.024285714285714285\textwidth}P{0.024285714285714285\textwidth}P{0.024285714285714285\textwidth}P{0.024285714285714285\textwidth}P{0.6638095238095237\textwidth}}
4\tabcellsep ( )\tabcellsep " 4\tabcellsep ( )\tabcellsep 4\tabcellsep ( )\tabcellsep ' 4\tabcellsep ( )\tabcellsep " 4\tabcellsep ( )\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep ( ) Volume XVII Issue VI Version I A\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Global Journal of Researches in Engineering\end{longtable} \par
 
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Journal of Researches in Engineering ( ) Volume XVII Issue VI Version A\\
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\tabcellsep \tabcellsep \tabcellsep ( )\tabcellsep 1260\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \multicolumn{2}{l}{22680}\tabcellsep A ( ) Volume XVII Issue VI Version I\\
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 			\footnote{T © 2017 Global Journals Inc. (US)} 			\footnote{\hyperref[b5]{(6)} where © 2017 Global Journals Inc. (US)} 			\footnote{© 2017 Global Journals Inc. (US)} 			\footnote{On the Squeezing Flow of Nanofluid Through Porous Medium With Slip Boundary and Magneti Field: A Comparative Study of Three Approximate Analytical Methods} 		 		\backmatter  			 \par
... 396249600\par
where the constants k 1 and k 2 are determined using the boundary conditions in Eq. (  {\ref 16}) i.e.\par
( )\par
The equations are solved for the corresponding values of k 1 and k 2 for the different values of ?.\par
V. 
\subsection[{Results and Discussion}]{Results and Discussion}\par
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.  {\ref 2-7}. 			 			  				\begin{bibitemlist}{1}
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\end{bibitemlist}
 			 		 	 
\end{document}
