Homotopy Perturbation Analysis in an energy transit problem closed to a Stretching Surface

Article History:Received:11 november 2020; Accepted: 27 December 2020; Published online: 05 April 2021 ABSTRACT : Present study explored the influence of stretching constraint in addition with inclusion of energy. An analytical solution for the system of non-linear equations of motion is worked out by adopting Homotopy Perturbation method (HPM). Physical and graphical reflections various parameters are demonstrated in the present problem. Utility of this model has been perceived in diverse industrial and chemical processes. 2010 AMS subject classification: 76W05


Introduction
Investigation of heat transfer in boundary layer has got tremendous application in Engineering, Plasma physics etc. and become an important subject for many researchers at the present time. In 1911, Hiemenz [9] considered the 2-D fluid flow near the point of stagnation and used similarity transformation. Thereafter, mass transfer, heat transfer, stagnation point flow, different physical effects, and stagnation point, calculation of skinfriction was exercised numerous ways.
The analytic form of 2-D steady flow of incompressible fluid because of the stretch sheet was estimated by Crane [5] with varying velocity. Also, Crane and Carragher [2] discussed the transfer of heat in above given flow model. Discussion of two-dimensional flow of stagnation point on the stretch surface of an incompressible viscous fluid was initiated by Chiam [4] and Mahapatra et al. [12] carried out the investigation by introducing temperature distribution. The pioneer work regarding the study of heat generation on a stretching surface was introduced in references [2], [7], [6], [13] and carried out their research by following the different phases of the problem.
The present research includes the experiment in laminar flow in a porous media when a viscous and incompressible fluid was used on a stretch surface with generation of heat considering the stream and wall temperatures to be constant. In this proposed study, the results of Attia [1], S. Kazem et al. [11] were modified by employing Homotopy Perturbation Method (HPM) He J. H.
[8] to obtain almost exact solution of the energy and momentum governing equations. HPM is a constructive tool to covenant with nonlinear differential equation, has been established by many authors like A. K. Jhankal [10]. Though a huge calculation is involved in this analytic method (HPM), but always provides a convenient solution to discuss the behavior of the parameters. We sketched and discussed the obtained results for discussing the heat and flow characteristics. In addition to this, the outcomes were compared with those of Attia [1] and S. Kazem et al. [11] to verify the accuracy of our result. Moreover, we have shown and discussed one comparison graph related to temperature with the graph obtained by Attia [1], to establish the accuracy of the method adopted here.

Mathematical formulation of the problem:
A 2-dimensional steady stagnation flow of point for an incompressible and viscous, electrically conducting fluid near O the stagnation point, on a surface which is considered along the plane y = 0 (x-axis). The substantial circumstances describing the model with initial velocity w u , w T be the temperature whereas    (1) uv xy Neglecting the dissipation, the governing energy equation with the temperature vales that depends on the absorption and generation of heat as per White M. F.
Where, the symbols have their usual meaning The boundary limitations describing the flow are given by: Here c is any positive constant, and is the temperature at the wall. Using, υ= μ / ρ as kinematic viscosity, using similarity transformations: Here the prime denotes the derivative value as per the value of η. For normalizing this model of flow, the non-dimensional temperature given below is introduced: The following nonlinear coupled differential equations are produced by applying (5), (6) in (2), (3) and (4):

Solution of the problem
The nonlinear coupled differential equations (7) and (8) can be rewritten as: Constants are mentioned in the appendix Applying HPM, the equations (10), (11) can take the following form: We consider "f" and "θ" are considered as under: Using (14) in (12) and (13) and equating the terms free from 'p', the following ordinary differential equations are obtained:  (14) in (12) and (13), the following ordinary differential equations are obtained:   Nu The present analysis reveals the expressions of boundary layer equations with viscous drag and Nusselt number are piled up to get a variety of graphs with their substantial interprtations by taking some standard values of different parameters implicated in the problem.  It was noted that temperature profileis continuously reducedfor varying the stretching parameter ( C = 0.5, 1, 1.5 ) i.e. the temperature distribution is continuously moves down on account of stretching parameter. Figure 5, shows the temperature profile for differentparameter values for heat generation and stretching parameter. It was observed that fluid temperature of the present flow problem is enlarged for boosting the heat generation which clearly satisfies the physical reality.

Comparison of results
For comparing the results of the presenet research, the experiment of H. A. Attia [1] was considered. When figure 6 and 7 were compared ( Figure 5