Numerical Solution of Two Dimensional Stagnation Flows of Newtonian Fluid Towards a Shrinking Sheet

The two dimensional stagnation flows towards 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 to ordinary differential equations. The results have been calculated on three different grid sizes to check the accuracy of the results. The problem relates to the flows towards a shrinking sheet when 0 α < and if 0 α > the flows towards a stretching sheet. The numerical results for Newtonian fluids are found in good agreement with those obtained previously.


Introduction
The two dimensional fluid flow near a stagnation point is among the fundamental problems in fluid mechanics. The study of stagnation point flow has been extended in numerous ways including MHD flow, heat transfer, and porous medium and stretching surfaces. In recent years, the stagnation flow problem is being investigated for shrinking boundaries. Fang and Zhang [1] 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 [2] and Fang [3]. Nadeem et al [4] and Ara et al [5] 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 [6,7]. Sajid and Hayat [8] applied homotopy analysis method for MHD viscous flow due to a shrinking sheet. The problem of [8] is studied by Noor et al. [9] by using simple non-perturbative method. Wang [10] studied the stagnation flow towards a shrinking sheet.
In this paper, the numerical solutions of stagnation flow towards 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 [10]. The resulting equations are solved numerically by using SOR method and Simpson's (1/3) rule, for various values of the parameter α and the Prandtle number Pr. When 0 α > , the problem relates to the stagnation flow towards a stretching sheet. When 0 α = , it becomes Hiemenz [11] flow towards a solid plate. The problem relates to the flow towards a shrinking sheet when 0 α < .

Mathematical Analysis
The continuity equation and the Navier-Stokes equations for incompressible fluid in the absence of body forces are given by where ρ is the density, V is the velocity vector, π is the pressure and μ viscosity coefficient. The fluid temperature is T where as Ф, C p and K denote the dissipation function (in this case the dissipation function is neglected), specific heat and heat conductivity respectively.
The flow is in the frame of Cartesian coordinates. The velocity vector is represented by The two dimensional potential stagnation flow at infinity is given by Under these assumptions, the set of equations (1) to (3) become:

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Numerical Solution of Two Dimensional Stagnation Flows of Newtonian Fluid Towards a Shrinking Sheet The following similarity transformations of [10] are used to convert the governing partial differential equations into ordinary differential form: The equation of continuity (4) is readily satisfied. From equation (6), the pressure π is obtained as: The equations (5) and (7) by using (8) yield a set of non linear ordinary differential equations: 2 1 , , Pr 0. f θ θ ′′ ′ + = (12) The boundary conditions are 0, , 1, 1 at 0 1, 0, 0 as .

Computational Procedure
We now solve numerically the first order ordinary differential equation (14)

Discussion on Numerical Results
The  (0) g′ have been compared with the previous results by Wang [10]. This comparison of the results is shown in the Table 6.     Graphically, the results for the function f are shown in Figure 1 and Figure 2. It is noticed that the function f increases with the increasing values ofα positive. When α =0, f behaves like the Hiemenz [11] flow towards a solid plate. For negative values ofα , the function f is initially negative. The situation depicts the regions of reverse cellular flow. The effect of α on the universal function g is shown in Figure 3.The effect is smaller in case of stretching sheet but larger for shrinking sheet. The graph of (0) f ′′ and (0) g′ have been plotted respectively in Figure 4 and Figure   5. The non-alignment function g has no effect on heat transfer. For increasing values of the shrinking parameterα , the boundary layer thickness increases and hence the heat transfer rate decreases as depicted in Figure 6.

Conclusions
The effects of parameter α in the extended range 1.2475 10 α − ≤ ≤ are observed on the similarity, velocity, non-alignment and temperature profiles. Consequently, the function f increases with the increasing values ofα positive.
When α =0, f behaves like the Hiemenz flow towards a solid plate. The non-alignment function g has no effect on heat transfer. For increasing values of the shrinking parameter α , the boundary layer thickness increases and hence the heat transfer rate decreases. Thus the numerical results in this work are found in good agreement with those obtained previously.