Numerical Solutions of MHD Viscous Flow of Newtonian Fluids due to a Shrinking Sheet by SOR Iterative Procedure

The problem of Magneto Hydrodynamic viscous flow due to a shrinking sheet of Newtonian fluids has been solved numerically by using SOR Iterative Procedure. The similarity transformations have been used to reduce the highly nonlinear partial differential equations of motion to ordinary differential equations. The results have been calculated on three different grid sizes to check the accuracy of the results. The numerical results for Newtonian fluids are found in good agreement with those obtained by the previous results.


Introduction
The flow problems of stretching surfaces have relevance to several technological processes. Chiam [1] investigated steady two dimensional stagnation point flow of an incompressible fluid towards a stretching surface. Mahapatra and Gupta [2,3] combined both the stagnation point flow and stretching surface. Shafique and Rashid [4] examined the three dimensional fluid motion caused by the stretching of a flat surface. In recent years, the flow problem is being investigated for shrinking boundaries. Wang [5] studied the stagnation flow towards a shrinking sheet. He considered the two dimensional and axisymmetric studies of this problem. His results represent a rare class of exact similarity solutions with reverse flow. Fang and Zhang [6] considered MHD flow over a shrinking sheet and obtained closed form exact solution for the problem. The MHD boundary layer flow of fluid over a shrinking sheet has been studied by Hayat et al [7] and Fang [8]. Nadeem et al [9] and Ara et al [10] have been investigated MHD boundary layer flow of fluid over an exponentially permeable shrinking sheet. The steady boundary layer flow and steady two-dimensional flow of a nanofluid past a nonlinearly permeable stretching/ shrinking sheet is numerically studied by Zaimi et al [11,12]. Sajid and Hayat [13] applied homotopy analysis method for MHD viscous flow due to a shrinking sheet. The problem of [13] is studied by Noor et al. [14] by using simple non-perturbative method.
In this research, the numerical solutions of MHD viscous flow due to a shrinking sheet for Newtonian fluid have been discussed. In order to find the numerical solution of the problem, the Navier Stokes equations are reduced to ordinary differential equations by using similarity transformations [13]. This system is solved numerically by using SOR Iterative Procedure with Simpson (1/3) Rule. The calculations have been carried out using three different grid sizes to check the accuracy of the results. The present numerical results have also been compared with the previous results in a particular case and found in good agreement. The numerical results have been discussed in both tabular as well as graphically.

Mathematical Analysis
The continuity equation and the Navier-Stokes equations for incompressible fluid in the presence of body forces are given by Where ρ and V are respectively, the density and the Under the above assumptions, the equations (1) and (2) become 0, Where p is the pressure and Here 0 a > is the shrinking constant and W is the suction velocity. When m=1, the sheet shrinks in the x-direction and when m=2, the sheet shrinks axisymmetrically.
In order to solve equations (3) to (6), the similarity transformations of [13] are given: The resulting partial differential equations by using (8) we obtain: with the boundary conditions: , 1 at 0, 0 as . Where

Finite Difference Equations
For numerical purpose, we rewrite the equations (9) and (10) by putting The equation (9) as follows: and the boundary conditions (10) takes the form: Now if we approximate equation (12) n n n n n n n h M h P P P P mhf P P Where h denotes a grid size and the symbols used denote as

Computational Procedure
We now integrate numerically equation (11) and solve the system of finite difference equation (14) at each required grid point of the interval [0,∞). The equation (11) is integrated by using the Simpson's (1/3) rule [15] as given below at the gird point Whereas the system of finite-difference equations (14) are solved by using S.O.R. iterative procedure [15] { } ( ) n n n n n n n n Subject to the appropriate boundary conditions (13), the computation has been checked for different of the relaxation parameter ω between 1 and 2. The optimum value of the relaxation parameter for the problem under consideration is 1.5. The SOR iterative procedure is terminated when the following criterion is satisfied:

Numerical Results and Discussions
The numerical computation has been performed to study the effect of the flow parameters namely S and M. The accuracy of the results is checked by comparing them on different grid sizes. The results for the non-dimensional velocity components f ′′ (the skin friction) with the previous results by [13,14] is given in table 4.               Numerical Solutions of MHD Viscous Flow of Newtonian Fluids due to a Shrinking Sheet by SOR Iterative Procedure