It is noticed that MHD axisymmetric

It is noticed that MHD axisymmetric flow of third-grade fluid over a stretching cylinder has not been conducted so far. The present analysis is an attempt in this direction. This paper is organized as follows. After formulation the arising nonlinear problem is analytically solved using homotopy analysis method (HAM) [18–32] which is a novel technique. The behavior of dimensionless parameter convergence is checked. Velocity components and skin friction daidzin coefficients noted are displayed. Whole analysis is summarized.

Formulation of the problem
Let us consider a steady axisymmetric incompressible electrically conducting third-grade fluid flow along a stretching cylinder. The flow is confined at . The -axis is taken along the stretching cylinder and the -axis is normal to it. The governing continuity and momentum equations under the boundary layer approximation are as follows [18]:
The appropriate boundary conditions for the considered flow problem are asfollows [17]:
It is worth mentioning that represents the velocity with which cylinder is stretched.

Solution procedure
Homotopy analysis method was derived from the fundamental concept of topology known as homotopy. Two functions are said to be homotopic if one function can be continuously deformed into the other function. If and are two continuous functions which maps from a topological space X into topological space Y then is homotopic to if there exists a continuous map Fsuch that for each
Then map F is called homotopic between and . Homotopy analysis method is proposed by Liao [27] in 1992 which is used to solve the highly nonlinear equations. This method is independent of small or large physical parameters. Homotopy is a continuous deformation or variation of a function or equation. It has several advantages over the other methods i.e., (i) it is independent of small or large parameters, (ii) it ensures the convergence of series solution and (iii) it provides great freedom to select the daidzin function and linear operator. Such flexibility and freedom help us in solving the highly nonlinear problems. It is also noted that linear part of the differential equation is selected as the linear operator for the homotopy analysis method. However in semi infinite domain it is preferred in such a way that the solution appears in the form of exponential functions for rapid convergence analysis. Homotopy analysis method requires initial guess and linear operator in the forms [18]: which satisfies the following property:where are the constants.

Convergence of the developed solution
Homotopy series solution (23) contains auxiliary parameter . This parameter has pivotal role in adjusting and controlling the convergence of the series solution (23). To obtain a suitable range of the h-curve is plotted at 13th order of approximation in Fig. 2. From this Fig. 1, it is noted that suitable range for is . Furthermore, convergence of series solution is checked and shown in Table 1. This Table shows that the series solution converges up to 12th order of approximation.

Results and discussion
In this section, we have discussed the effects of some interesting physical parameters on velocity field through graphs. Numerical values of skin friction coefficient are tabulated and behavior of dimensionless parameters on skin friction coefficient is analyzed.
Fig. 3 is plotted to see the variation of magnetic parameter M on the non-dimensional velocity . It is observed that with the increase in M the velocity profile decreases. The momentum boundary layer thickness also reduces. Physically the Lorentz force arising out of the imposed magnetic field acts as a retarding force, and causes the reduction in the velocity as well as in the momentum boundary layer. The variation of the shear thickening/thinning parameter is given in Fig. 4. It is depicted that with an increase in the velocity and momentum boundary layer thickness decrease. Thus almost the same behavior appears for both M and . Fig. 5 displays the variation of with for several values of . This figure reveals that both velocity profile and the momentum boundary layer thickness increase with the increase of . The similar effect is observed for in Fig. 6. Fig. 7 is drawn to see the influence of curvature parameter on the velocity profile. From this figure it is noted that the velocity and the momentum boundary layer thickness are increasing functions of curvature parameter . Physically, it is due to the fact that when we increase the curvature parameter the radius of the cylinder reduces and the area of the cylinder in contact with fluid also decreases which causes less resistance to the fluid motion. Hence the velocity profile increases. Fig. 8 elucidates the influence of the Reynolds number on the velocity profile versus . Increasing the Reynolds number leads to a decrease in the velocity profile. Also the momentum boundary layer thickness reduces.