Boundary Layer Stagnation-Point Flow of Micropolar Fluid over an Exponentially Stretching Sheet

In this paper, the steady boundary layer stagnation point flow and heat transfer of a micropolar fluid flowing over an exponentially stretching sheet is investigated. The solution of the problem is obtained numerically using the Keller-box method and the series solutions are obtained with the help of homotopy analysis method (HAM). Comparisons of both the solutions are presented. At the end the effects of important physical parameters are presented through graphs and the salient features are discussed.


Introduction
The study of stretching sheet was initiated by Crane [1], has attained lot of attention due to its wide range of engineering applications. Literature is rich of the material concerning the boundary layer flow of steady/unsteady flow of Newtonian/non-Newtonian fluids. In a recent paper Merkin and Kumaran [2] has studied the unsteady boundary layer flow on a shrinking surface in an electrically conducting fluid. Moreover, the suction effects for the magneto hydrodynamic viscous flow over a shrinking sheet have been analyzed by Akyildiz and Siginer [3]. According to their analysis the velocity field exhibited a decreasing behavior with respect to the suction parameter. The problem of stagnation point flow of a viscous fluid towards a stretching sheet was discussed analytically by Nadeem et al. [4]. Further, the steady boundary layer stagnation point flow of a micropolar fluid towards a horizontal linearly stretching/shrinking sheet has been studied by Yacob et al. [5]. They solved the problem numerically using the Runge-Kutte-Fehhlberg method with shooting technique. Nadeem and Awais [6] have examined the effects of variable viscosity and variable thermo capillarity on the unsteady flow in a thin film on a horizontal porous shrinking sheet through a porous medium. Moreover, the stagnation point flow towards a shrinking sheet has been analyzed by Wang [7], the obtained results were reflecting that a region of reverse flow occurs near the surface and that for larger shrinking rates, the solution/similarity can't be obtained. Recently, the suction/blowing and thermal radiation effects on steady boundary layer stagnation point flow and heat transfer over a porous shrinking sheet has been investigated by Bhallacharrya and Layek [8].
The study of stretching/shrinking sheet phenomena for the boundary layer stagnation point flow has been considered by many researchers for linear/polynomial stretching but not a lot of work is available on these concepts with exponential stretching and stagnation point flow. Recently, Abdul Rehman et al. [9] have presented the solutions for the problem of boundary layer flow and heat transfer of a third grade fluid flowing over an exponentially stretching sheet. In another attempt, Abdul Rehman et al. [10] have also discussed the nanoparticles effect over the boundary layer flow of a Casson fluid flowing over an exponentially stretching surface. The purpose of the present work is to provide a solution of the boundary layer stagnation point flow of a micropolar fluid for exponentially stretching/shrinking sheets. The solutions are obtained through a second order difference scheme known as the Keller-box technique. Also the analytical solutions are obtained by using the homotopy analysis method (HAM). Comparisons of both the solutions are presented for Over an Exponentially Stretching Sheet compatibility. Details about the homotopy analysis method can be found in Refs. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].

Formulation
Let us consider a stagnation point flow of an incompressible micropolar fluid over an exponentially stretching sheet. The Cartesian coordinates (x, y) are used such that x is along the surface of the sheet, while y is taken as normal to it. The related boundary layer equations of motion in the presence of microrotation and heat transfer are where (u, v) are the velocity components along the (x, y) axes, ρ is the fluid density, µ is the coefficient of viscosity, k is the vertex viscosity, ν is the kinematic viscosity, j is the microrotation density, γ is the micropolar constant, N is the angular microrotation momentum, T is temperature, α is the thermal diffusivity, p is pressure and is the free stream velocity. The corresponding boundary conditions for the problem are where " is the stretching velocity and $ " is the surface temperature. For exponential stretching, the expression for , " and $ " are defined as = (. /0 , " = 1. /0 , $ " = $ + 2. /0 (7) in which a and b are constant velocities, c is constant temperature and L is the reference length.
To convert Equations (1) to (4) into a nondimensional form we introduce the following similarity transformations in which K=k/µ is the micropolar parameter, K = /L is the micropolar coefficient, M. = N /2 is the non-similar Reynolds number, OP = /! is the Prandtl number and Q = 0 R . The boundary conditions in nondimensional form can be written as 3%0' = 0, >%0' = 1, where S = 1/(. The skin friction coefficient and the local Nusselt numbers are obtained in dimensionless form as where M. 9 = & 8 /2 N is local Reynolds number.

Numerical Solution of the Problem
To solve system of Equations (10) to (12) with the help of Keller-box scheme we first reduce the system into a first order one by taking the relations with the help of Equation (17), Equations (10) to (12) can be written as The corresponding boundary conditions take the form , → 1, = → 0, > → 0, (, 5 → ∞ The above system of equations is first approximated by central differences and then those equations are linearized using Newton's method. The solution then can be obtained by applying block-elimination method over these linearized equations. The details of the procedure can be found in [29][30][31].

Results and Discussion
The convergence of HAM solution for velocity, microrotation and temperature have been discussed by plotting ℏ -curves for nondimensional f´´, M´´ and θ ′′ . It is found that the admissible values of ℏ -curves are  (1)-(3))). Figures.(4) and (5) are plotted for comparison of the results obtained from numeric and HAM solution for nondimensional velocity f´ for various values of microrotation parameter K and the stretching ratio ε. From these graphs it is clear that both the solutions are in good agreement. From Figure. (4) it is found that with the increase in the micropolar parameter K, the velocity profile f´ decreases and also the boundary layer thickness reduces. Figure. Figure. (6) and Figure. (7). The observed settlement of the two solutions is acceptable.
From Figure. (6) it is observed that as the micropolar parameter Λ increases, the microrotation velocity profile M is forced to decrease. Whereas, with increase in the stretching ratio ε, the microrotation profile M also increases. Figure. (8) is sketched to check the compatibility of the Keller-box and homotophy solutions obtained for the nondimensional heat transfer profile > for different values of the Prandtl number Pr, from Figure. (8) it is observed that both the solutions are in agreement and that due to increase in the Prandtl number Pr the temperature profile > decreases. Figure. (9) is designed to show the behavior of f´ for different combinations of the micropolar parameter K and the stretching ratio ε. It is observed that for the stretching ratio ε<0 (shrinking sheet) an increase in the micropolar parameter K demands a decrease in the velocity profile f´, while for the stretching ratio ε>0 ( ) stretching sheet velocity profile f´ has a dual behavior that is for the stretching ratio ε<1, an increase in the micropolar parameter K, the velocity profile f´ decreases, whereas for the stretching ratio ε>1, with an increase in the micropolar parameter K, the velocity profile f´ also increases. Figure.(10) is schemed to detect the manner revealed by the microrotation profile M when plotted for different combinations of the micropolar parameter Λ and the stretching ratio ε. It is observed that the behavior of micropolar parameter Λ for different values of the stretching ratio ε is similar to the behavior of the micropolar parameter K for different values of the stretching ratio ε over the microrotation profile M´ (Figure. (9)). The microrotation parameter M for various values of Q against different Λ is plotted in Figure. (11). From Figure. (11) it is observed that with an increase in Q , M increases. It is also observed from Figure. (11) that the rate of convergence for small Λ is much faster to that observed for a larger Λ. Figure. (12) is graphed to predict the influence of Re for different values of ε . It is observed that an increase in Re with ε<1 tends to decrease M, whereas for ε>1 the behavior is opposite. Figure. (13) is depicting over the effects of Pr over θ for different choices of ε. It is observed that temperature profile decreases with an increase in Pr the thermal boundary layer thickness also decreases and that the rate of convergence observed is minimizing with increase in ε. The coefficient of skin friction c f for different values of K and ε against different Pr are shown in Figure. (14). It is observed that c f increases with the increase in K and ε. The Nusselt number Nu for different values of Pr are shown in Figure. (15). Table. (1) is prepared to observe the behavior of skin friction coefficient for different combinations of the parameters K, ε and Λ It is observed that skin friction decreases for all the parameters. Table. (2) is displaying the behavior of local Nusselt numbers for different combinations of the parameters Pr, ε and K. It seems that local Nusselt numbers increases with an increase in Pr and ε, whereas with an increase in K local Nusselt numbers decreases. From Table. (1) and Table. (2) it is obvious that the numerical solutions and the analytical solutions both are in excellent resemblance.