New Analytical Solution of Stagnation Point Flow and Heat Transfer in a Porous Medium non-linear ordinary

A new analytical method called q-homotopy analysis method is applied for solving nonlinear differential equations. In this paper, the mathematical study of stagnation points  ow and heat transfer phenomena in a porous medium are discussed. The governing coupled nonlinear partial differential equations are converted into coupled nonlinear ordinary differential equations using similarity transformations and solved analytically for the values of Prandtl number Pr, Porous medium K, Casson  uid parameter β, Ratio parameter c using q-homotopy Analysis Method. The in uence of the skin-friction coef cients for different parameters is discussed and presented in tabular form. The obtained q-homotopy analysis method solutions are compared with numerical results and it gives a


Introduction
Stagnation point fl ow plays a vital rule in fl uid mechanics. It has greatest importance for the prediction of Skin friction and local Nusselt numbers. The solution for two dimensional fl ow of a fl uid near stagnation point was fi rst proposed by Hiemeanz [1].
Many researchers [2] have been investigated about porous and non-porous boundary layers fl ow near the stagnation point of a stretching /shrinking sheet. Recently, Nandeppanavar et. al [3] discussed the stagnation point fl ow and heat transfer of Casson fl uid over a stretching sheet. Nandeppanavar et. al has presented a numerical solution of Casson fl uid fl ow using Runge Kutta Fourth order method [4][5][6].
The aim of this present work is to fi nd the fl uid fl ow over a stagnation point fl ow with uniform heat transfer [7,8]. The governing partial differential equations are converted into the non-linear ordinary

OPEN ACCESS
Volume: 8  [9,10]. The transformed ordinary differential equations are solved analytically. The solution for velocity profi le and temperature distribution is obtained using the q-homotopy analysis method [11][12][13][14][15]. Estimation of Skin-friction Co-effi cient and local Nusselt number is also presented in analytically. The effects of the pertinent parameters on the velocity components, temperature distribution are also analyzed.

Mathematical Formulation
The boundary layer equations can be written as follows [4], ∂u/∂x+∂v/∂y=0 (2.1) where u and v are the velocity components of the fl uid in x and y directions respectively.
where ψ is the stream function.
The system of Eqs (2.6) and (2.9) is a coupled system of nonlinear differential equations

Results and Discussion
The velocity profi les and temperature distribution are discussed for various physical parameters β, K, Pr, c in the graphs. Eqs. (2.6) and (2.9) solved by q-homotopy analysis method and homotopy analysis method. The obtained analytical results are compared with the numerical result in Figs. (2) - (11) for different values of parameters. It gives good agreement with the numerical result.
The velocity of the fl ow for different parameters are presented in the graphs Figs.2 -5. From the Fig.2 exhibits that velocity profi le f'(η) decreases with β increases. In Fig.3 indicates that an increases K various values of the velocity profi le is also increases. Fig.4 visualizes the parameter Prandtl number Pr on the velocity fl ow f'(η). There is prominent decreases in the velocity and parameter Prandtl number Pr increases. Fig.5 illustrate that an increases in the ratio parameter c also increases the velocity profi le. The temperature distribution of the diffi cult parameters are plotting in the graphs Figs.6 -9. From Fig.6 depict to analyze the temperature for various values β. It is noticed that the θ(η) increases with an increases in the β parameter. In Fig.7 it is evident that the temperature with on the parameter K. It is clearly that the parameter K increases with temperature decreases. From Fig.8, when the parameter Pr increases the temperature decreases. Fig.9 shows the infl uence of the temperature increases with ratio parameter c is also increases.

Conclusion
Analytical solution of velocity fl ow and heat transfer are obtained for all values of parameter using q-homotopy analysis method. The effects of casson fl uid parameter β and porosity parameter K on velocity fl ow and temperature are opposite. When the casson parameter increases, the skin friction co-effi cients decreases. The q-homotopy analysis method, will be applicable for other strongly nonlinear problems.

Appendix A
The q-Homotopy analysis method is used to solve the Eqs.(2.6) and (2.9) with the suitable initial guess [10]. The auxiliary linear operator as L = ∂/∂T with L(C 1 ) = 0 where C 1 is arbitrary constant.