On Three-Dimensional Flow and Heat Transfer over a Non-Linearly Stretching Sheet: Analytical and Numerical Solutions

This article studies the viscous flow and heat transfer over a plane horizontal surface stretched non-linearly in two lateral directions. Appropriate wall conditions characterizing the non-linear variation in the velocity and temperature of the sheet are employed for the first time. A new set of similarity variables is introduced to reduce the boundary layer equations into self-similar forms. The velocity and temperature distributions are determined by two methods, namely (i) optimal homotopy analysis method (OHAM) and (ii) fourth-fifth-order Runge-Kutta integration based shooting technique. The analytic and numerical solutions are compared and these are found in excellent agreement. Influences of embedded parameters on momentum and thermal boundary layers are sketched and discussed.


Introduction
The fundamental problem of two-dimensional flow due to stretching plane surface, initially discussed by Crane [1], is involved in various industrial processes such as metal and polymer extrusion, drawing of plastic films, paper production etc. Owing to such applications, the researchers have discussed this problem under various aspects including suction or injection, variable surface temperature, convective boundary condition, mass transfer, mixed convection etc. The three-dimensional flow due to plane bi-directional linearly stretching sheet was first discussed by Wang [2]. He found an exact similarity solution of the classical Navier-Stokes equations. Later, Lakshmisha et al. [3] numerically examined the unsteady three-dimensional flow with heat and mass transfer over an unsteady stretching sheet. In contrast to this problem, Takhar et al. [4] investigated the three-dimensional flow of an electrically conducting fluid due to an impulsive motion of the stretching sheet. Ariel [4] derived approximate analytic and numeric solutions for steady three-dimensional flow over a stretching sheet. Xu et al. [5] provided uniformly valid series solutions for three-dimensional unsteady flow caused by the impulsively stretching sheet. Liu and Andersson [6] considered the heat transfer in three-dimensional flow due to non-isothermal stretching sheet. The unsteady three-dimensional flow of elasticoviscous fluid and mass transfer due to unsteady stretching sheet with constant wall concentration was studied by Hayat et al. [7]. In another paper, Hayat et al. [8] described the three-dimensional flow of Jeffrey fluid due to stretching sheet. Liu et al. [9] firstly discussed the three-dimensional flow due to exponentially stretching sheet numerically. Steady flow of nanofluid past a linearly bi-directional stretching sheet through Buongiorno's model was examined by Junaid et al. [10]. Sheikholeslami and Ganji [11] discussed the flow and heat transfer of nanofluid between parallel sheets in the presence of Brownian motion and thermophoresis effects.
The literature cited above deals only with the case of either linearly or exponentially driven velocity of the sheet. Vajravelu [12] numerically discussed the viscous flow due to stretching sheet when the velocity of the sheet was assumed to obey the power-law distribution, i.e.u w~c x n . He computed numerical solutions for various values of power-law index n: Cortell [13] extended this problem by considering viscous dissipation effects and variable surface temperature. The steady boundary layer flow of micropolar fluid over non-linearly stretching sheet was discussed by Bhargava et al. [14]. Radiation and viscous dissipation effects on the boundary layer flow above a non-linearly stretching sheet were explored by Cortell [15]. Homotopy analytic solutions for mixed convection flow of micropolar fluid past a non-linearly stretching vertical sheet were obtained by Hayat et al. [16]. Kechil and Hashim [17] derived analytic solutions for MHD flow over a nonlinearly stretching sheet by Adomian decomposition method. Hayat et al. [18] used modified decomposition method for the series solutions of MHD flow of an electrically conducting fluid over a non-linearly stretching surface. The impact of chemical reaction on the flow over a non-linearly stretching sheet embedded in a porous medium was investigated by Ziabakhsh et al. [19]. Rana and Bhargava [20] computed numerical solutions for twodimensional flow of nanofluid due to non-linearly stretching sheet by finite element method. Shahzad et al. [21] obtained closed form exact solutions for axisymmetric flow and heat transfer when the velocity of the stretching sheet was proportional to r 3 . Partial slip effects on the boundary layer flow past a non-linearly permeable stretching surface have been addressed by Mukhopadhyay [22]. In another paper, Mukhopadhyay [23] analyzed the flow and heat transfer of Casson fluid due to non-linearly stretching sheet. Rashidi et al. [24] derived homotopy based analytic solutions for flow over a non-isothermal stretching plate with transpiration.
To our knowledge, the three-dimensional flow due to nonlinearly stretching sheet has not been yet reported. It is obvious that three-dimensional flows can be appropriate in giving more clear physical insights of the real world problem when compared with the two-dimensional flows. The present work is therefore undertaken to fill such a void. The study also assumes that the temperature across the sheet is non-linearly distributed. Introducing a new set of similarity variables the boundary layer equations are first reduced into self-similar forms and then solved both analytically and numerically. It is pertinent to mention that computation of either approximate analytic or numerical solutions of the boundary layer equations governing the flow and heat transfer is often challenging (see [25][26][27][28][29][30][31][32][33] for details). Attention is focused on the physical interpretation of parameters including the power-law index n:

Mathematical Modeling
Let us consider the three-dimensionalincompressible flow over a plane elastic sheet located at z~0 as shown in the Fig. 1. The flow is induced due to stretching of the sheet in two lateral directions. Let U w x,y ð Þ~a(xzy) n and V w x,y ð Þ~b(xzy) n be the velocities of the sheet along the x{ and y{ directions respectively with a,b,nw0 are constants (see Table 1). T w x,y ð Þ~T ? zA(xzy) n is the variable surface temperature where Aw0 is constant and T ? is the ambient fluid temperature. Under the usual boundary layer assumptions, the equations governing the three-dimensional flow and heat transfer in the absence of viscous dissipation and internal heat generation/absorption can be expressed as (see Liu et al. [9]) where u,v and w are the velocity components along the x,y and z{ directions respectively, n is the kinematic viscosity, T is the fluid temperature and a is the thermal diffusivity (see Table 1).
The boundary conditions are imposed as under: We introduce the new similarity transformations as follows: We have modified the similarity transformations used by Liu et al. [9] for the current problem. Using (6), Eq.(1) is identically satisfied and Eqs. (2)-(5) become where Pr~n=a is the Prandtl number and l~b=a is the ratio of stretching rate along the y{ direction to the stretching rate along the x{ direction (see Table 1). The above equations reduce to the case of two-dimensional flow when l~0. Moreover, when l~1, the equations governing the axisymmetric flow due to non-linearly stretching sheet are recovered. When Pr~1 the solution of f 0 is also a solution of h. The quantities of practical interest are the skin friction coefficients and the local Nusselt number which are defined as below: where t zx and t zy are the wall shear stresses and q w is the wall heat flux. These are given by using Eqs. (6) and (12) in Eq. (11), one obtains where Re x~Uw (xzy)=n and Re y~Vw (xzy)=n are the local Reynolds numbers along the x{ and y{ directions respectively (see Table 1). The vertical component of velocity at the far field boundary can be expressed as Optimal homotopy analytic solutions The non-linear differential equations (7)-(9) with the boundary conditions (10) have been solved by optimal homotopy analysis Three-Dimensional Flow over a Non-Linearly Stretching Sheet method (OHAM) [34,35]. For this purpose, we first select the initial guesses f 0 , g 0 and h 0 of f (g), g(g) and h(g) as under: and the auxiliary linear operators are selected as below If q[½0, 1 is the embedding parameter and B the non-zero auxiliary parameter, then the generalized homotopic equations corresponding to (7)-(10) can be written as follows (1{q)L g ½G( g; q){g 0 (g)~qBN g F (g; q),G(g; q) ½ , ð18Þ LF ( g; q) Lg g??~0 , LG( g; q) Lg g??~0 , H(g; q) g??~0 , where the non-linear operators N f , N g and N h are N g F ( g; q), G( g; q) ½ N h F (g; q), G(g; q),H(g; q) ½ 1 Pr By Taylor's series expansion one obtains G(g; q)~g 0 (g)z X ?
Lf m (g) Lg g?z?~0 , , h m (?)~0, x m~0 , mƒ1, 1, mw1: In order to determine the optimal values of B we define the squared residuals of the governing Eqs. Such kind of error has been considered in other works [36][37][38][39][40][41]. The smaller f M 's, the more accurate the mth order approximation of the solution. The optimal values of B can be obtained by minimizing the f M 's, through the command Minimize of the software MATHEMATICA (see Liao [36] for details). Alternatively MATHEMATICA package bvph 2.0 can also be used to calculate such values (see [41] for details).

Results and Discussion
This section contains the physical interpretations of the behavior of the interesting parameters entering into the problem. We compare the 15 th -order OHAM solutions for temperature h with the numerical ones for different values of n: Fig. 2 shows that data retrieved from both solution methods are identical, demonstrating the validation of our findings.
Figs. 3 and 4 show the variations in horizontal and vertical components of velocity with an increase in stretching rates ratio l. It is clear that increase in l corresponds to an increase in the stretching rate along the y{direction. Due to this reason the vertical component of velocity increases with an enhancement in l while the velocity in the x{direction decreases correspondingly. The wall velocity gradients f 00 (0), g 00 (0) and entrainment velocity f (?)zg(?) as functions of stretching rates ratio l have been plotted in Fig. 5. Due to the bi-directional stretching sheet, there will be downward flow in the vertical direction. The vertical component at far field boundary is therefore expected to be negative in this situation. We notice that shear stresses at the wall increase when l is increased. Further, the larger values of l enhances the velocity of the cold fluid at the ambient. As a consequence, the entrainment velocity is an increasing function of l. Fig. 6 indicates that temperature h decreases with an increase in stretching rates ratio l for unity Prandtl number. Physically, an increase in l enhances the intensity of colder fluid at the ambient (as noticed in Fig. 6) towards the hot sheet which eventually corresponds to decrease the local fluid temperature. Fig. 7 perceives the behavior of Prandtl number Pr on the temperature. A bigger Prandtl number fluid has less thermal diffusivity and hence it allows less thermal effect to penetrate deeper into the fluid. As a result, temperature decreases and the thermal boundary layer becomes thinner when Pr is increased. This decrease in thickness of the thermal boundary layer is compensated with a larger wall slope of temperature function. Fig. 8 plots the wall temperature gradient against Pr with the variation in stretching rates ratio l. The wall heat transfer rate approaches the zero value for vanishing Prandtl number Pr ?0, a fact that is clear from the energy equation (9). Moreover, this Fig.  compliments the results of Fig. 4. In bigger Prandtl number fluids the convection is effective in transferring energy from the stretching sheet compared to pure conduction. Due to this reason the wall heat transfer rate is an increasing function of Pr. The reduction in thermal boundary layer thickness with an increase in l meets with the bigger magnitude of local Nusselt number. In other words the enhanced intensity of cold fluid at the ambient towards the hot fluid closer to the sheet results in larger heat transfer rate at the sheet. Tables 2 and 3 provide the numerical values of skin friction coefficients and local Nusselt number for different values of parameters by employing shooting method. The results are compared with the MATLAB built in function bvp5c and found in excellent agreement. We notice that wall shear stresses increase with an increase in l more rapidly at n~3 when compared with n~1. The thinner thermal boundary layer accounted for larger n accompanies with larger temperature gradient along the sheet. The magnitude of increase in wall temperature gradient h 0 (0) with an increase in Pr increases when n is increased.

Conclusions
For the first time, the flow and heat transfer over a plane surface stretched non-linearly in two lateral directions have been investigated. The simulation in this study assumes that the temperature across the sheet is non-linearly distributed. Both analytic and numerical solutions are obtained and found in excellent agreement. Following are the major results of this study.
I. It is seen that shear stress at the wall increase when the stretching rates ratio is increased. The entrainment velocity is negative, representing a downward flow in the vertical direction, which is a consequence of the bi-directional stretching sheet. II. The increased intensity of the cold fluid at the ambient towards the stretching sheet with an increase in stretching rates ratio l decreases the fluid temperature. III. The temperature decreases and thermal boundary layer thins when the power-law index n is increased. IV. The results for the case of two-dimensional and axisymmetric flows can be obtained as special cases of present study when l~0 and l~1 respectively.