Numerical Solution of Boundary Layer Flow of Viscous Fluid Via Successive Linearization Method

In the recent years, a great deal of interest has been gained to fluids applications. Some fluids not easy to expressed by particular constitutive relationship between shear rates and stress and which is totally different than the viscous fluids [1,2]. These fluids including many home items namely, toiletries, paints, cosmetics certain oils, shampoo, jams, soups etc. have different features and are denoted by non-Newtonian fluids. In general, the categorization of non-Newtonian fluid models is given under three class which are named the integral, differential, and rate types [3-6]. In the present study, the main interest is to discuss the heat transfer flow of hydrodynamic viscous fluid over a flat plate in a uniform stream of fluid with dissipation effect. The most phenomena in the field of engineering and science that occur is nonlinear. With this nonlinearity the equations become more difficult to handle and solve. Some of these nonlinear equations can be solved by using approximate analytical methods such as Homotopy analysis method (HAM) proposed by liao S [7,8], Homotopy Perturbation method (HPM) it was found by Ji-Huan [9] and Adomain decomposition method (ADM) Q Esmaili et al. [10], Makinde OD et al. [11] and Makinde OD [12].


Introduction
In the recent years, a great deal of interest has been gained to fluids applications. Some fluids not easy to expressed by particular constitutive relationship between shear rates and stress and which is totally different than the viscous fluids [1,2]. These fluids including many home items namely, toiletries, paints, cosmetics certain oils, shampoo, jams, soups etc. have different features and are denoted by non-Newtonian fluids. In general, the categorization of non-Newtonian fluid models is given under three class which are named the integral, differential, and rate types [3][4][5][6]. In the present study, the main interest is to discuss the heat transfer flow of hydrodynamic viscous fluid over a flat plate in a uniform stream of fluid with dissipation effect. The most phenomena in the field of engineering and science that occur is nonlinear. With this nonlinearity the equations become more difficult to handle and solve. Some of these nonlinear equations can be solved by using approximate analytical methods such as Homotopy analysis method (HAM) proposed by liao S [7,8], Homotopy Perturbation method (HPM) it was found by Ji-Huan [9] and Adomain decomposition method (ADM) Q Esmaili et al. [10], Makinde OD et al. [11] and Makinde OD [12].
However, some of these equations are solved via traditional numerical techniques such as finite difference method,shooting method and Keller box method, Runge-Kutta. Recently some studies have presented a new method called Successive Linearization Method (SLM). This method has been applied successfully in many nonlinear problems in sciences and engineering, such as the MHD flows of non-Newtonian fluids and heat transfer over a stretching sheet [13], viscoelastic squeezing flow between two parallel plates [14], two dimensional laminar flow between two moving porous walls [15] and convective heat transfer for boundary layer with pressure gradient [16,17]. Therefore, the effectiveness, validity, accuracy and flexibility of the SLM are verified among of all these successful applications. Presently a new investigation on the heat transfer flow of hydrodynamic viscous fluid over a flat plate in a uniform stream of fluid with dissipation effect is discussed. The numerical solution to the resulting nonlinear problem is computed by using the SLM approach. The embedded flow parameters are discussed and illustrated graphically.

Mathematical formulation of the problem
The governing equations are Where , w T T ∞ are constants. Introducing the following dimensionless variables Utilizing equation (6), equation (1) is satisfied automatically and equations (2) and (3) characterize to the following problems The related boundary conditions

Solution of the problem
Here successive linearization method (SLM) [14][15][16] is implemented to obtain the numerical solutions for nonlinear system (8) and (10) corresponding to the boundary condition Eq. (11) -(13) ( Table 1). The convergence for numerical values of "(0) f and '(0) θ − for different order of approximation when Ec = 0.01, Pr =1 and n =1.00 (Table 1). The numerical values of ( ) f η and '( ) f η when, n =1, Pr =1 for Ec= 0.01. (Table 2). The numerical values of ( ) θ η and '( ) θ η − when, n =1, Pr =1 for Ec =0.01 (Table 3). Comparison of numerical values of ( ) f η with Ref: [16] when, n= Ec = 0, Pr =1 (Table 4).  noticing that by increasing the parameter η reveals that buoyancy because of augments of gravity which boosts on the velocity. Figure   3 is sketched for the variation of Prandtl number Pr on ( ) θ η . It is noted that for lager Pr ,the thermal field is lower and then this reduce the temperature. In fact law Prandtl number Pr assist fluid with higher thermal conductivity and this create thicker thermal boundary layer than that for lager Pr. Finally, Figure 4 shows the effect of Ec on velocity and temperature profiles over the plate, and we note that by increasing in Ec parameter is seen that the effect is very big for the temperature.

Conclusion
In this research, the problem of heat transfer of an incompressible viscous fluid over flat pate is solved numerically. The numerical solutions are well established by SLM. The influence of various parameters is shown through different graphs. The present results have been tested and compared with the available published results in [16], in a limiting situation shown in tables v and an excellent agreement is found [17].