Lie group analysis of magnetohydrodynamic tangent hyperbolic fluid flow towards a stretching sheet with slip conditions

In this paper, we studied MHD two dimensional flow of an incompressible tangent hyperbolic fluid flow and heat transfer towards a stretching sheet with velocity and thermal slip. Lie group analysis is used to develop new similarity transformation, using these similarity transformation the governing nonlinear partial differential equation are reduced into a system of coupled nonlinear ordinary differential equation. The obtained system is solved numerically by applying shooting method. Effects of pertinent parameters on the velocity and temperature profiles, skin friction, local Nusselt number are graphically presented and discussed. Comparison between the present and previous results are shown in special cases.


Introduction
The study of the non-Newtonian fluids has gained a considerable interest in the last few decades due to its numerous applications in different fields of science and The flow due to stretching sheet has attracted a considerable attention of the researchers. Sakiadis [6] studied laminar bounded-layer behavior on a moving continuous flat surface and has obtained numerical solation for the boundarylayer equations. Crane [7] has extended the work for both liner and exponentially stretching sheet. The flow due to a stretching surface with partial slip has been study by Wang [8]. Nazarn et al. [9]  Sophius Lie [15] developed a classical method known as Lie group analysis to find point transformation that map a given differential equation to itself. This method unifies almost all known exact integration techniques for both ordinary and partial equation [16]. Lie group analysis used to find the similarity reduction of the nonlinear differential equations. In such analysis, one reduces the number of variable governing the partial differential equations. This reduction of variables changes the system of partial differential equations to self similar system of the ordinary

Model
We consider a steady, two dimensional flow of an incompressible tangent hyperbolic fluid. We assume that the fluid is coincident with the plane = 0, such that the flow is limited to the region > 0. We further assume that the generation of the fluid is due to the linear stretching. As describe in [25] the governing equation of the fluid is given bȳ Here ̄, 0 , ∞ , Γ and denote the extra stress tensor, the infinite shear rate viscosity, the zero shear rate viscosity, the time dependent material constant and the flow behavior index respectively. The term ̄i s given bȳ̇= with Π = 1 2 ( + ( )) 2 . As we are concerned with infinite shear rate viscosity problem, so in the following we restrict ourselves to the case ∞ = 0. The fluid under consideration that described shear thinning, so we must take Γ̄< 1.

Then Eq. (1) reduces to
The governing equations of the continuity, momentum and energy for the proposed model are Here, and are the velocity components in the -and -direction, respectively.
Here, is the stretching rate, is the velocity slip factor and 1 ( ) is the thermal slip factor. For = 0 = 1 ( ) the no slip condition are recovered.
Our first step is to transform the given system to a non-dimensionalized form. For this purpose, we introduce the following dimensionless quantities Insert the scaling (9) into the system described by Eqs (4)-(6) and dropping the bars, the continuity, the momentum and the energy equations become In the scenario of scaling given in (9), the boundary conditions (7) and (8) Again, the induction of the stream function converts the boundary conditions (13) and (14) to → 0, → 0 as → ∞.

Analysis
In this section, we perform Lie group analysis to find the new similarity transformations of Eqs (15) and (16). This will reduce the nonlinear partial differential equations to nonlinear ordinary differential equations. For this purpose, we consider the following scaling group of transformation Γ ∶ * = 1 , * = 2 , * = 3 , * = 4 , Γ * = Γ 5 .
Similarly, equating the first and the last term of (28) and integrating both sides, we obtain Here primes denoted differential with respectively . We

Method of solution
In order to solve the system of nonlinear ordinary differential equations given by (

Discussion
The numerical solutions are presented through graphs (see Figures 1-5) for the physical interpretation of the proposed study. These figures describes the influence of Hartmann number , Weissenberg number , power law index , Prandtl number , source/sink parameter , velocity slip parameter and thermal slip parameter on the dimensionless velocity, temperature, skin friction coefficient and local Nusselt number. Figure 1(a) shows the influence of Hartmann number on the velocity profile. Hartmann number is the ratio of electromagnetic force to the viscous force. From the graph it is clear that by increasing the values of , the velocity profile decrease and boundary layer thickness also deceases. This is happening due to increasing value of which tends to increase the Lorentz force, and produces more resistance to the transport phenomena. Figure 1(b) shows the influence of the power law index on the velocity profile of the flow. It is observed that by increasing the valves of both boundary layer thickness and the velocity profile decreases. Figure 1(b) depicts that the behavior of law index on the velocity profile is qualitatively same in Figure 1(a). Figure 1(c) shows the effects of Weissenberg number over the velocity function ′ ( ). It is clear that velocity profile decreases by increasing the values of , because after increasing Weissenberg number the relaxation time increase which offers more resistance to flow. Consequently, the boundary layer thickness decreases with an increase in . Figure 1(d) Table 1 in the absence of velocity slip parameter [1]. Figure 5 shows the influence of governing parameters on dimensionless heat transfer rates. Figure 5

Conclusions
We