1 Introduction

Non-Newtonian fluids are useful in many areas such as chemical engineering,electromagnetic propulsion,and nuclear reactors ^{[1, 2, 3, 4]} . The magnetohydrodynamic (MHD) boundary layer behavior of non-Newtonian fluids on stretching surface is important in many engineering and industrial applications,e.g.,petroleum drilling,paper production,glass blowing,plastic sheet formation,and extrusion of polymeric fluids and melts. To study the non-Newtonian fluid flow, boundary-layer assumptions are usually used in the power law model.

Recently,many researchers have presented works on MHD flow and heat transfer of an electrically conducting power law fluid past a continuously stretching sheet ^{[5, 6, 7, 8]} . Pavlov ^[9] studied the MHD boundary layer flow of an incompressible fluid due to a stretching plane surface. Wu ^[10] investigated an MHD boundary layer flow with uniform suction/injection. Andersson et al. ^[11] studied the MHD flow of a non-Newtonian power-law fluid over a stretching surface with a linear velocity in the ambient fluid. Liao ^[12] obtained the analytic solution of the MHD flows of non-Newtonian fluids over a stretching surface by using the homotopy analysis method (HAM). Chen et al. ^[13] extended the works by considering the problem of the MHD boundary layer flow over a moving surface with power-law velocity. The analytical approximate solutions were obtained in terms of a rapid convergent power series by introducing Crocco’s transformation and the Adomian decomposition method,and the values of the skin friction were established. In studying the motion of non-Newtonian fluids,numerical and approximate methods were also explored due to the nonlinearity of the basic equation and the complexity of the boundary conditions ^{[14, 15, 16, 17]} .

Motivated by the studies mentioned above,this paper investigates the MHD boundary layer flow of the non-Newtonian fluid over a moving surface with power-law velocity and special injection/blowing. In Section 2,the MHD boundary layer flow equations and the constitutive relations are given,and the partial differential equations are transformed into the ordinary differential equations by the Lie-group transformation. The analytical approximate solutions for the integer power-law index and special injection/blowing are developed in Section 3. The results and discussion are presented in Section 4. In Section 5,the conclusions of this paper are summarized. 2 Mathematical description

The MHD boundary layer flow of an electrically conducting power-law fluid over a moving surface with special injection/blowing is considered ^[18] . The basic equations for the boundary layer flow are

where u and v are the velocity components parallel and normal to the plate,respectively. ρ is the fluid density,σ is the electrical conductivity,B(x) is the magnetic field,and τ _xy is the shear stress tensor defined by The boundary conditions are where A and B are constants.

We now introduce the following relations for u and v:

where ψ is the stream function.

Substituting the relations (5) into the boundary layer equation (2),we can obtain the following equation:

Introduce the Lie-group transformation ^[19] as follows: It may be considered as a point-transformation,transforming the coordinates (x,y,ψ,u,v) to the coordinates (x ^* ,y ^* ,ψ ^* ,u ^* ,v ^* ). Substituting E_q. (7) in E_q. (6) yields The system remains invariant under the group transformation Γ. Therefore,we can get the following relations among the transformation parameters:

In terms of the boundary conditions,we have Thus,we obtain

We now calculate f (η) and η. Let us assume that

Since

we have

We assume that B(x) has the special form as follows:

Then,E_qs. (1)-(4) can be transformed into with the boundary conditions where β is the constant related to the power law indices N and P ,M is the magnetic parameter, and ξ is the suction/blowing parameter. They are defined by

where positive ξ corresponds to suction ( v| _y=0 < 0) if B < 0,while negative ξ corresponds to blowing ( v| _y=0 > 0) if B > 0. The skin friction coefficient Cf at the wall is given by Here,ρ(Ax ^p ) ^2−N x ^N /K is the local Reynolds number based on the surface velocity Ax ^p . C_f is directly related to the dimensionless velocity gradient f ^′′ (0) at the wall. 3 Mathematical analysis for HAM solution

The basic idea of the HAM is introduced in Ref. ^[20]. In this paper,the HAM is used to solve E_q. (10) where N is a positive integer. For the solution obtained by the HAM ^{[20, 21, 22, 23]} ,we select

as the initial approximation of f (η),and choose as the auxiliary linear operator. We construct the deformation equation where B_n (n = 0,1,2,· · · ,+∞) are decomposition polynomials representing the nonlinear operator

When N = 1, when N = 2, and when N 3, The homotopy decomposition method employs the recursive relation subject to the boundary conditions

Substituting E_q. (20) in E_q. (19) yields the following solution:

where a _n,k are coefficients. Due to the initial approximation of f (η),we have the first two coefficients as follows:

Starting from these two coefficients,the coefficients a _n,k can be calculated by means of the above recursive formulas. Therefore,we can obtain the explicit analytic solution as follows: 3.1 Solutions when N =1

Substituting E_q. (22) into E_qs. (19) and (20),we have

where 3.2 Solutions when N=2

When N = 2,we have

for 2 k 2n + 1,and where A_n−1,k−1 and B_n−1,k−1 are defined by E_qs. (25) and (26),respectively,and 3.3 Solutions when N 3

When N 3,we have the recursive formulae as follows:

for 2 k Nn + 1,and where for N k − 1 Nn, for N − 1 j Ni + N − 1, for 1 k − 1 Nn − N + 2,and for 2 k − 1 Nn − N + 2. 4 Results and discussion

The systems of E_q. (11) subject to the boundary condition (12) are solved by the homotopy decomposition method ^[20] . The asymptotic and numerical results are listed in Table 1 ,and the main results are shown in Figs. 1-6.

As pointed out by Liao ^[12] ,the value of h determines the convergence region and the rate of approximations for the HAM solution. The h-curves are plotted in Fig. 1 for the 30th-order of approximations where

β = −1.5,ξ = 0,M = 5,N = 1.

Figure 1 clearly indicates that when −1.5 < h −0.5,the values are admissible.

To assess the accuracy of the present numerical results,the values of F ^′′ (0) obtained in this paper are compared with those obtained in Ref. ^[24] (see Table 1). From this table,it is obvious that our results show excellent agreement with the results in Ref. ^[24].

Figure 2 shows the variations of f ^′′ (0) for the 30th-order of approximations versus the magnetic parameter M . For β = −1.5 and a small range of M near zero,there is a decrease in the skin friction as M increases. However,when β = 1.0 and β = 5.0,the skin friction increases with the increase in the magnetic field strength. The present work is consistent with the previous numerical results in Ref. ^[24].

Figure 3 shows the variations of −f ^′′ (0) for the 30th-order of approximations versus the suction/blowing parameter ξ. It is seen that there is an increase in the skin friction as ξ increases.

For the power-law index N = 2,the shear force is plotted for varying the suction/blowing parameter. In Fig. 4 (there is a suction from the wall to fluid),the shear force decreases with the increases in η. The largest shear force occurs at η = 0 and the smallest shear force occurs at the edge of the boundary layer. In Fig. 5 (there is a fluid blowing from the wall),the shear force first increases with the increase in η and arrives at the maximum at a certain point,and then decreases,which means that when there is a fluid blowing from the wall,the biggest shear force occurs at the interior of the boundary layer.

For the power-law index N = 4,the distribution of the dimensionless velocity f ^′ (η) is displayed versus the magnetic field parameter M . In Fig. 6,it is observed that the velocity component decreases with the increase in the parameter M .

5 Conclusions

In this study,the MHD flow on a moving surface with the power-law velocity and special injection/blowing is analyzed. The HAM is employed to find the analytical approximate solution of the boundary layer equations.

The convergence of the results is discussed explicitly. The results for the velocity,the skin friction,and the shear force are presented graphically. The mechanism of momentum transfer can be affected by varying ξ,M ,and β. Some of the published results ^{[13, 24]} are special cases of the results of the present work.