Computational Analysis of the Non – Linear Boundary Layer Flow over a Flat Plate Using Variational Iterative Method (VIM)

This paper applied the variational iterative method to solve the problem of the two-dimensional incompressible laminar boundary layer flow over a flat plat also called the Blasius problem. The problem is governed by the Navier-Stokes and continuity equations which were first transformed into an ordinary differential equation using similarity transforms and the resulting problem solved using variational iterative method. The results obtained for the similarity stream function and velocity were tabulated and were highly comparable in terms of accuracy with that obtained by Ganji et al. (2009) who studied the same problem using the homotopy perturbation technique and results obtained by Blasius. The results were found to be very accurate especially for η ≤ 4 when using the variational iterative method. The method is convenient as it greatly reduces the amount of computational work.


Introduction
Steady flow of viscous incompressible fluids has attracted considerable attention in recent years due to its crucial role in numerous science and engineering applications. In particular, the classical boundary layer flow past a static flat plate problem has attracted considerable interest of many researchers since introduced by Blasius (1908). Since then, the problem has been analyzed by various authors; Falkner & Skan (1931), Asaithambi (2005), Zhang & Chen (2009), Ganji et al (2009) and several others in order to obtain results of the Blasius equation. We are to solve the problem using He's Variational iterative method.
The Variational iterative method (VIM) which is based on the use of restricted variations and correction functional has found a wide application for the solution of linear and nonlinear ordinary and partial differential equations. VIM has been used for solving the 7 th order Swada-kotera equa- Our aim is to apply the VIM to find solution to the laminar boundary flow over a flat plate which is one of the very popular problems arising in fluid mechanics. We employ the use of a similarity transformation of the Navier -Stokes and continuity equations to obtain an ordinary differential equation with the necessary boundary conditions. This problem is solved using VIM in this paper and the results are then compared to those obtained by Ganji et al (2009) using homotopy perturbation method (HPM) and Blasius (1908).

He's Variational Iteration Method (VIM)
For the purpose of illustrating the methodology of He's variational iteration method, we begin by considering a differential equation in the formal form, where λ is a general Lagrangian parameter, which may be optimally identified in the variational theory. The subscript n denotes the n th order approximation, � is considered as a restricted variation i.e., � = 0.
We therefore determine λ as defined above after which the successive approximations U n+1 (x, t), n ≥ 0 of the solution U(x, t) will be readily obtained upon using the computed λ from a selective function U 0 . Consequently, the solution is given by The solution of the linear problems can be solved in a single iteration step due to the exact identification of the Lagrange multiplier. The method also provides an efficient method for handling nonlinear behaviour; a few iterations can be used to achieve a high degree of accuracy.

Materials and Method
Equations of the flow in the boundary layer over a flat plate are governed by the continuity equation; and the reduced Navier-Stokes equation where u and v are respectively the components of the velocity vector and represents the kinematic viscosity of the fluid.
Boundary conditions are In order to simplify the above problem, the Blasius similarity transform is applied in order to convert (2.5) into an ordinary differential equation.
Where a and b are constants and are chosen to make η and f(η) dimensionless, Therefore, η is called the dimensionless similarity variable and f(η) is called the dimensionless stream function. To solve the problem using VIM we reformulate by using the dependent variable; Hence equation (2.7) becomes, (2.11) and the boundary conditions hence, ′′ ( ) = 3 ′′ ( ) imposing the condition, (2.13) Solution to equation (2.11) with boundary conditions (2.12) using variational iterative method (VIM) is obtained as follows.
The correction functional is written as Making the correction functional stationary with respect to and noting that = 0 � is a restricted variation, therefore � = 0 18) The stationary conditions are;

Results
We compare results obtained through Variational Iterative method (VIM) with that of Ganji

Discussion
In this paper, we have used the method of iterative variational method to investigate the problem posed by the nonlinear boundary layer flow over a flat plat and the results obtained are highly comparable in term of accuracy with that obtained by Ganji et al who studied the same problem using the homotopy perturbation technique. The results were found to be very accurate especially when η ≤ 4 using the variational iterative method.