ANALYTICAL SOLUTION OF TWO-DIMENSIONAL VISCOUS FLOW BETWEEN SLOWLY EXPANDING OR CONTRACTING WALLS WITH WEAK PERMEABILITY

In this article the problem of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability is presented and Homotopy Perturbation Method are employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. Comparisons are made between the Numerical solution (NM) and the results of the He's Homotopy Perturbation Method (HPM). The results reveal that these methods are very effective and simple and can be applied for other nonlinear problems.


INTRODUCTION
The flow of Newtonian and non-Newtonian fluids in a porous surface channel has attracted the interest of many investigators in view of its applications in engineering practice, particularly in chemical industries.Examples of these are the cases of boundary layer control, transpiration cooling and gaseous diffusion.Theoretical research on steady flow of this type was initiated by Berman [1] who found a series solution for the two-dimensional laminar flow of a viscous incompressible fluid in a parallel-walled channel for the case of a very low cross-flow Reynolds number.After his work, this problem has been studied by many researchers considering various variations in the problem, e.g., Choi et al. [2] and references cited therein.For the case of a converging or diverging channel with a permeable wall, if the Reynolds number is large and if there is suction or injection at the walls whose magnitude is inversely proportional to the distance along the wall from the origin of the channel, a solution for laminar boundary layer equations can be obtained [3].Most scientific problems such as two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability and other fluid mechanic problems are inherently nonlinear.Except a limited number of these problems, most of them do not have analytical solution.Therefore, these nonlinear equations should be solved using other methods.In the analytical perturbation method, we should exert the small parameter in the equation.Therefore, finding the small parameter and exerting it into the equation are difficulties of this method.Since there are some limitations with the common perturbation method, and also because the basis of the common

MATHEMATICAL FORMULATION
Consider the laminar, isothermal, and incompressible flow in a rectangular domain bounded by two permeable surfaces that enable the fluid to enter or exit during successive expansions or contractions [10].One side of the cross section, representing the distance (2a) between the walls is taken to be smaller than the other two (W and L).Both walls are assumed to have equal permeability and to expand uniformly at a time dependent rate a .Furthermore, the origin 0 x = is assumed to be the center of the classic squeeze film problem.This enables us to assume flow symmetry about 0 x = .Under these assumptions, the equations for continuity and motion become 0, u v x y where , , p ρ υ and t are the dimensional pressure, density, kinematic viscosity, and time.Auxiliary conditions can be specified such as After some modification and special variable [10] and then we have

APPLICATION OF HPM TO TWO-DIMENSIONAL VISCOUS FLOW
In this section, we will apply the HPM to nonlinear ordinary differential equation ( 6).According to HPM, we can construct homotopy of equation ( 6) as follows: We consider F as follows: By substituting F from Eq. (26) into Eq.( 25) and after some simplifications and rearranging based on powers of p-terms, we have: (0) 0, (0) 0, (0) 0, (0) 0, Solving Eqs. ( 9)-(11) with boundary conditions, we have: where and to are determined from the boundary conditions.Solutions to were too long to be mentioned here; therefore, they are shown graphically.The solution of this equation, when , will be as follows:

RESULT AND DISCUSSION
In this section, we compare the results of analytical solution and the numerical solution.

CONCLUSIONS
Results clearly show that HPM, which was applied to the two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability problem, was capable of solving them with successive rapidly convergent approximations without any restrictive assumptions or transformations causing changes in the physical definition of the problem.The results show that this scheme provides excellent approximations to the solution of this nonlinear equation with high accuracy.Finally, it has been attempted to show the capabilities and wide-range applications of the Homotopy Perturbation Method (HPM) in comparison with the numerical solution of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability problems.

Figure ( 1 )Figure 1 :
Figure 1: The results of HPM and Numerical methods for and for and ( ) F y ( ) F y ′ Re 5 = 0.5 α = .

Figure 3 :
Figure 3: Effect of various Reynolds number on by HPM when (Re)

Figure 4 :
Figure 4: Effect of various α on by HPM when Re ( ) F y 1 = .