Combined Analytical-Numerical Solution for MHD Viscous Flow over a Stretching Sheet

We studied the two-dimensional flow of viscous and electrically conducting fluid over a stretching sheet under the influence of constant magnetic field. Approximate analytical solution of governing nonlinear boundary layer equation via optimal homotopy asymptotic method (OHAM) is obtained. For numerical comparison we used Runge-Kutta-Fehlberg fourth-fifth-ordermethod.The effect of different parameters on fluid flow is analyzed. It is found that the OHAM solution is very close to the numerical solution for different assigned values of parameters; this thus indicates the feasibility of the proposed method (OHAM).


Introduction
Magnetohydrodynamics (MHD) is the study of interaction of conducting fluids with electromagnetic phenomena. The flow of an electrically conducting fluid in the presence of magnetic field has many applications in engineering. Also the flow influence by a moving boundary is of crucial importance in the extrusion processes in chemical industries [1,2].
Sakiadis [3,4] is a pioneer in this area who has investigated the boundary layer flow with uniform speed over continuously stretching surface. Later on, Tsou et al. [5] experimentally verified the work of Sakiadis. Crane [6] studied the steady-state two-dimensional boundary layer flow caused by a stretching sheet whose velocity varies linearly with the distance from a fixed point on the sheet. Chiam [7] and Dandapat and Gupta [8] considered the motion of micropolar and power-law fluids, respectively. Most attention has so far been devoted to the analysis of flow of viscoelastic fluids [9][10][11][12][13] and the joint effect of viscoelasticity and magnetic field has been worked out by Ariel [14]. Beside this, Liao [15,16] has provided solution for impermeable and permeable stretching sheets, which shows that multiple solutions for the stretching surfaces are possible under definite conditions. The optimal homotopy asymptotic method is a powerful approximate analytical technique that is straightforward to use and does not require the existence of any small or large parameter. Optimal homotopy asymptotic method (OHAM) is employed to construct the series solution of the problem. This method is a consistent analytical tool and it has already been applied to a number of nonlinear differential equations arising in science and engineering [17][18][19]. So far, as we are aware, there have been no solutions for MHD viscous flow over stretching sheet via OHAM. This paper is organized as follows. First, in Section 2, we formulate the problem. In Section 3 we present basic principles of OHAM. The OHAM solution for MHD viscous flow is given in Section 4. In Section 5, we analyze the comparison of the solutions using OHAM with numerical method (NM). Section 6 is devoted to the concluding remarks.

Governing Equation
We consider the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a stretching sheet. A uniform magnetic field is applied and further assumes that the induced magnetic field is negligible due to small magnetic Reynolds number fluid. Under such 2 Journal of Computational Engineering assumption, the MHD boundary layer flow over a flat plate is governed by where and V are the velocity components in the and directions, respectively, ] is the kinematic viscosity, is fluid density, is the electrical conductivity, and is the magnetic field strength, where ( ) = 0 ( −1)/2 . The boundary conditions are given below: We introduce dimensionless variables for nondimensionalized form of momentum and energy equations [13]: Using (3), the governing equations can be reduced to nonlinear differential equation, where is a function of the similarity variable : subject to the boundary conditions where is the nondimensional parameter and is the magnetic parameter.

Basic Principles of OHAM
We review the basic principles of OHAM as developed in [17][18][19][20] in the following five steps.
(i) Let us consider the following differential equation: where Ω is problem domain, (V) = (V) + (V), where , are linear and nonlinear operator, V( ) is an unknown function, and ( ) is a known function.
(ii) Construct an optimal homotopy equation as where 0 ≤ ≤ 1 is an embedding parameter, ( ) = ∑ =1 is auxiliary function on which the convergence of the solution is greatly dependent. The auxiliary function ( ) also adjusts the convergence domain and controls the convergence region.
(v) Finally, substitute these constants in (11) and one can get the approximate solution.

Runge-Kutta-Fehlberg Method
The Runge-Kutta-Fehlberg method (RKF45) is a technique to resolve the below problem: It is called RKF45 because the fourth-order method with five stages is used together with a fifth-order method with six stages that uses all of the points of the first one. It has a procedure to determine if the proper step size ℎ is being used. At each step, two different approximations for the solution are made and compared. If the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step size is reduced. If the answers agree to more significant digits than required, the step size is increased. Each step requires the use of the following six values.
where the four function values 1 , 3 , 4 , and 5 are used.

Solution of the Problem via OHAM
According to the OHAM, applying (8) to (4), where primes denote differentiation with respect to . We consider and ( ) as the following: Using (16) in (15) and some simplifying and rearranging of the terms based on the powers of , the zeroth-, first-, and second-order problems are as follows.

Zeroth-Order Problem. We have
with boundary conditions Its solution is

First-Order Problem. We have
with boundary conditions Its solution is

Conclusion
In this study, we have successfully applied the optimal homotopy asymptotic method for MHD viscous flow over a stretching sheet. Both numerical and approximate analytical results are obtained for the problem. The results obtained via OHAM are presented in tabular and graphical forms. An excellent agreement is observed in OHAM and numerical solution. It was found that OHAM results are matched well with numerical solution. The approximate solution obtained using OHAM is also consistent with the numerical solution for various values of and . The objective of this work was to show the usefulness of OHAM. The proposed method is simple in applicability, as it does not require discretization like numerical methods. Furthermore, this method provides a convenient way to control the convergence by optimally determining the auxiliary constants. Additionally, this method converges rapidly at lower order of approximations. Therefore, OHAM shows its latent strength and potential for the solution of nonlinear physical problems in real-life applications.