# ANALYTICAL SOLUTION OF ORTHOGONAL FLOW IMPINGING ON A WALL WITH SUCTION OR BLOWING.

Abstract

In this paper, we have studied the effect of suction and injection on flow and shear stress in case of orthogonal flow. The emerging non-linear differential equation doesn't contain any small or large parameter called the perturbation quantity. Therefore, to solve the non-linear equation we have used the homotopy analysis method (HAM) to obtain the explicit analytic solution of the emerging non-linear equation. The comparison of numerical results with HAM results is also presented. It is noted that the behavior of the HAM solution for velocity profile and skin friction parameter is in good agreement with the numerical solution. It is also noted that the region of convergence can be increased by a better choice of the auxiliary linear operator.

Keywords: Homotopy Analysis Method, Orthogonal flow, Suction/Blowing.

INTRODUCTION

Analytic solutions are important than numerical solutions because these are valid on the whole domain of definition whereas the numerical solutions are only valid at chosen points in the domain of definition. Liao (1992, 2009), proposed an analytic method for nonlinear problems namely the homotopy analysis method (HAM). The HAM is independent upon small or large parameter and has been applied successfully to solve nonlinear problems such as viscous flow, heat transfer, nonlinear oscillations and Thomas Fermi atom model (Liao, 1999, 2002, 2003, 2006), Liao and Campo (2002), Ayub et al. (2003, 2010), Hayat et al. (2004, 2005, 2006, 2007, 2010), Abbasbandy (2006, 2007), Ariel et al. (2006), Sajid et al. (2006), Song et al. (2007), Zaman et al. (2010).

Further the HAM has certain other advantages over the perturbation expansion method, the delta expansion method and the Lypanov's expansion method that HAM allows us great freedom and flexibility: (i) To control the region of convergence. (ii) To choose the initial guess. (iii) To choose the auxiliary linear operator.

The process of suction and injection has its importance in many engineering activities such as in the design of thrust bearing and radial diffusers, and thermal oil recovery. Suction is applied to chemical processes to remove reactants. Blowing is used to add reactants, cool the surfaces, prevent corrosion or scaling and reduce the drag. Wang (2003), Mahapatra et al. (2004) and Ayub et al. (2008), studied stagnation point flow. Smart (1959), Tamada (1979) and Dorrepaal (1986) have discussed stagnation point flow impinges on the wall orthogonally.

Recently, Labropulu and Chandna (1996) have presented the numerical solution for stagnation-point flow of a viscous fluid impinging on a flat wall with suction or blowing. The aim of the present communication is to revisit the problem considered by Labropulu and Chandna (1996) for the HAM solution. The emerging non- linear differential equation doesn't contain any small or large parameter called the perturbation quantity. Therefore perturbation technique fails to convert completely the non-linear problem into linear sub-problems. However, the homotopy analysis method (HAM) solves the problem and provides an explicit, totally analytic solution of the problem. HAM solution for velocity profile is developed and comparison between the numerical and analytic solutions is provided. Graph for velocity profile is plotted and discussed.

HAM solution for dimensionless stream function

The boundary layer equation for the orthogonal flow

For comparison of tables, for the values of the shear stress on the wall, the absolute difference is calculated for numerical solution and HAM solution. These results are obtained for different values of h laying in the interval of convergence.

Table. Comparison of the values of the shear stress on the

wall f (0)

S###f Numerical###f HAM###by Absolute difference###

###0###1.232588###1.232618###0.000030###

0.1###1.290975###1.290929###0.000046###

1.0###1.889314###1.889357###0.000043###

2.0###2.670056###2.670132###0.000076###

3.0###3.526640###3.524437###0.002203###

-0.1###1.176041###1.176053###0.000012###

-1.0###0.756575###0.756864###0.000289###

-2.0###0.475811###0.475753###0.000057###

-3.0###0.329453###0.329432###0.000021

It is observed from the table that with the increase in parameter s the shear stress on the wall increases. It is also observed from the table that HAM results are in very good agreement with the numerical resluts of Labropulu and Chandna (1996) for all values of the suction parameter s.

CONCLUSION

The effect of suction and injection on flow and shear stress near a porous wall have been investigated. The homotopy analysis method has been used to solve the governing non linear differential equations, the governing non linear equations does not contain any small or large parameter which is necessary for the application of a perturbation technique. The HAM solves the equation and provides us an analytical solution. It is found that with an increase in suction/injection parameter s the velocity increases near the porous wall and shear stress on the wall also increases. It is also observed that a better choice of auxiliary linear operator increases the region of convergence.

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Zaman, H. and Ayub, M. 2010. Reply to the comments on: Series solution of hydromagnetic flow and heat transfer with Hall effect in a second grade fluid over a stretching sheet. Central European Journal of Physics, Cent. Eur. J. Phys. DOI: 10.2478/s11534-009-0147-0 (Online).