Two-Dimensional Stagnation-Point Velocity-Slip Flow and Heat Transfer over Porous Stretching Sheet

Present paper investigates 2D (Two-Dimensional) stagnation-point velocity-slip flow over porous stretching sheet. The governing non-linear PDEs (Partial Differential Equations) are non-dimensionlized by using the similarity transformation technique that results into coupled non-linear ODEs (Ordinary Differential Equations). Such ODEs are then solved by using shooting technique with fourth-order Runge-Kutta method. Since the behavior of boundary layer stagnation-point flow depends on the rate of cooling and stretching. Therefore, the main objective of this paper is to analyze the effects of different working parameters on shear stress, heat transfer, velocity and temperature of fluid. The results revealed that the velocity-slip has significant effect on the fluid flow as well as on the heat transfer. The numerical results are also compared with existing work for no-slip condition and found to have good agreement with improved asymptotic behavior.

Few applications are electronic chips, drawing and stretching of plastics films, hot rolling. Such applications are of great interests because the quality of the product depends upon the rate of stretching and cooling. The study has its origin from the work of Sakiadis [1] and Crane [2]. Since then numerous researchers [3][4][5][6] have studied it for different kind of flows.
A stagnation-point flow is a flow around a point, stagnation-point, where the velocity of the considered object is zero [7]. Such kind of fluid flows have been of shrinking surface. Stagnation-point flow with heat generation/absorption and convective boundary conditions were studied by Alsaedi, et. a.l [11]. Attia [12] investigated porosity effects on the stagnation-point flow over stretching surface in the presence of heat generation and absorption. She used finite difference method to study such problem. Her work was improved by Kazem, et. al. [13] and produced converged results by using semi-analytic method called HAM (Homotopy Analysis Method) of Liao [14] and numerical relaxation method of Zwillinger [15]. However, the characteristics of the fluid flow remained same.
Literature depicts that much attention has been paid toward the study of boundary layer fluid flow with noslip boundary condition but there exists physical problems where such condition is not appropriate.
Considering the velocity-slip condition at the surface boundary certainly helps to understand the characteristics of fluid. In this regards, first-order velocity-slip effects on stagnation-point flow over both stretching and shrinking surface was studied by Aman, et. al. [16]. Turkyilmazoglu [17] investigated analytically the mangnetohydrodaynamic slip flow past a stretching sheet and reported multiple solutions. Rosca and Pop [18] analyzed the effects of second-order velocity-slip over both vertically stretching and shrinking sheet. Dual solutions were obtained by Singh and Chamkha [19] of flow and heat transfer of second-order velocityslip flow. The flow was considered over vertically permeable shrinking sheet. Aly and Vajravelu [20] studied magnetic and velocity-slip effects of the 2D axisymmetric flow over stretching surface and presented both numerical and exact solution. Hakeem, et. al. [21] took into consideration the second-order slip flow of a nanofluid over both shrinking and stretching sheet under the influence of magnetic and thermal effects. Dual solutions were obtained by Mishra and Singh [22] taken into consideration thermal and velocity-slip flow of a mixed convection flow over permeable shrinking cylinder.
Behavior of fluid flow may be better understood by considering velocity-slip at the boundary. Literature depicts that very little attention has been paid to this problem. Therefore, in the present work, partial-slip effects on the flow and heat transfer of a 2D stagnation-point flow through porous medium is studied. Review of the literature shows that no such study has been done before.

MATHEMATICAL FRAMEWORK
Where u and v, respectively represents the velocities of fluid in x and y direction,  is fluid density, K is Darcy permeability,  is kinematic viscosity, T is fluid temperature, T e is ambient fluid temperature, Q is rate of heat generation/absorption, k is fluid thermal conductivity and C p is specific heat capacity at a constant pressure.
Setting the boundary conditions as: where U slip is the first-order velocity-slip which was proposed by Maxwell [23] and T w is the boundary wall temperature. Suitable similarity transformation variables for the above set of Equations (1-4) are: Where f() and (), respectively, are the dimensional stream temperature function of independent similarity variable . Stream function, which identically satisfies Equation (1) and can be checked easily, is defined as u = y and v = -x. Using Equation (5) where prime represents the ordinary differentiation of the variable with respect to . Using Equations (5-6) in Equations (2-4), we get: Where M is porosity parameter, C is sheet stretching parameter, Pr is Prandtl number, B is heat generation/ absorption coefficient and S is velocity-slip parameter, which are defined as:

NUMERICAL PROCEDURE
In order to use Runge-Kutta fourth-order method to solve system of ordinary differential Equations (7-8) boundary conditions Equation (9), it is necessary to find the unknown initial conditions, that are, f"(0) and '(0) such that the given asymptotic boundary conditions remain satisfied. This way given boundary value problem will be transformed into initial value problem and Runge-Kutta fourth-order method can be easily applied. To find such magical values shooting technique [24] has been adopted.

RESULTS AND DISCUSSION
The numerical study has been done to study the velocity- simultaneously.

Two-Dimensional Stagnation-Point Velocity-Slip Flow and Heat Transfer over Porous Stretching Sheet
Mehran

. IMPROVED PROFILES OF FLUID VELOCITY f '() AT M=1
stretching parameter is less than 1. In the absence of velocity-slip, the data for variation of shear stress is compared with Kazem, et. al. [13]. It is found that present results are better than that of Kazem, et   surface stretching less than 1 and increasing for surface stretching greater than 1. In the absence of velocity-slip, the present results are compared with Kazem, et. al. [13].
It can be seen that the present results are also improved as they satisfy the asymptotic condition of boundary.
From     The temperature of fluid is also influenced by velocityslip. By decreasing the slip temperature is increases when the value of stretching parameter is greater than 1, and temperature decreases by decrease in slip when the value of stretching parameter is less than 1 as shown in Fig. 8. whereas it increases with the increase in heat generation parameter (Fig. 11).

CONCLUSIONS
The shear stress f"(0) is directly proportional to sheet stretching C, that is, by decreasing/ increasing C we notice decrease/increase in f"(0). For all values of porosity of medium M, the shear stress is negative when the surface stretching parameter value is less than 1 and positive when greater than 1.
(ii) Increasing the boundary slip S decreases the fluid velocity f'(h) and temperature q(h) when surface stretching parameter value is less than 1 and inverse profile is observed when it is greater than 1.
(iii) Boundary layer thickness decreases by increase in boundary slip parameter S, sheet stretching parameter C and Prandtl number Pr.
(iv) For all values of porosity parameter M, increasing sheet stretching C decreases heat transfer q(h).
(v) Temperature of the fluid q(h) decreases by increase in Prandtl number Prand sheet stretching parameter C.
(vi) For all values of heat generation/absorption parameter B, the fluid temperature q(h) decreases/increases by the decrease/increase in sheet stretching C.
(vii) Temperature for heat absorption is greater than heat generation.
More numerical investigations on stagnation-point fluid flow and heat transfer over stretching sheet under secondorder velocity-slip condition will be made in the future work by using shooting technique and other higheraccuracy numerical methods.