Numerical and Series Solutions for Stagnation-Point Flow of Nanofluid over an Exponentially Stretching Sheet

This investigation is concerned with the stagnation-point flow of nanofluid past an exponentially stretching sheet. The presence of Brownian motion and thermophoretic effects yields a coupled nonlinear boundary-value problem (BVP). Similarity transformations are invoked to reduce the partial differential equations into ordinary ones. Local similarity solutions are obtained by homotopy analysis method (HAM), which enables us to investigate the effects of parameters at a fixed location above the sheet. The numerical solutions are also derived using the built-in solver bvp4c of the software MATLAB. The results indicate that temperature and the thermal boundary layer thickness appreciably increase when the Brownian motion and thermophoresis effects are strengthened. Moreover the nanoparticles volume fraction is found to increase when the thermophoretic effect intensifies.


Introduction
There has been great interest of researchers in the flow and heat transfer characteristics due to the impulsive motion of stretching sheet. A variety of technical processes involve the production of sheeting material which includes both metal and polymer sheets. The rate of heat transfer at the sheet is largely dependent on the quality of final product.
Flow past a flat plate with a uniform free stream was reported by Blasius [1]. In contrast to the Blasius problem, the boundary layer flow over a continuously moving plate in a quiescent ambient fluid was explored by Sakiadis [2]. Crane [3] extended this concept for a sheet which is stretched with the velocity linearly proportional to the distance from the origin. Since this pioneering work of Crane [3], the literature concerning the boundary layer flows past a stretching sheet has been in continuous growing. In fact Crane's problem has been considered for several other features such as viscoelasticity, heat and mass transfer, porosity, magnetic field etc. (see Rajagopal et al. [4], Mahapatra and Gupta [5], Cortell [6], Bachok et al. [7], Abbasbandy and Ghehsareh [8], Fang et al. [9], Hayat et al. [10], Mustafa et al. [11,12] etc.). On the other hand, a literature survey witnesses that the flow analysis over an exponentially stretching sheet has been scarcely presented. Combined heat and mass transfer in the boundary layer flow over an exponentially stretching sheet has been reported by Magyari and Keller [13]. Suction and heat transfer characteristics in the exponentially stretched flow has been studied by Elbashbe-shy [14]. Viscoelastic effects in the flow over an exponentially stretching sheet have been described by Khan and Sanjayanand [15]. Analytic solutions for flow and heat transfer over an exponentially stretching sheet have been provided by Sajid and Hayat [16]. Nadeem et al. [17] examined the flow and heat transfer of viscoelastic (second grade) fluid over an exponentially sheet in the presence of thermal radiation.
Nanofluid is a liquid suspended with nanometer-sized particles (diameter less than 50 nm) called nanoparticles. These nanoparticles are typically made of metals, oxides and carbides or carbon nanotubes. In the past, the concept of nanofluids has been used as a route to enhance the performance of heat transfer rate in liquids. Detailed review studies on nanofluids have been conducted by Daungthongsuk and Wongwises [18], Wang and Mujumdar [19,20] and Kakaç and Pramuanjaroenkij [21]. Natural convective boundary layer flow of nanofluid past a vertical flat plate has been studied by Kuznetsov and Nield [22]. The Cheng-Mincowcz problem for flow of nanofluid embedded in a porous medium has been considered by Nield and Kuznetsov [23]. Bachok et al. [24] examined the flow of nanofluid over a continuously moving surface with a parallel free stream. Flow of nanofluid over a linearly stretching sheet has been studied by Khan and Pop [25]. Finite element analysis for flow of nanofluid over a nonlinearly stretching sheet is presented by Rana and Bhargava [26]. Falknerskan problem for flow of nanofluid with with prescribed surface heat flux is investigated by Yacob et al. [27]. Makinde and Aziz [28] discussed the effect of convective boundary conditions on the flow of nanofluid over a stretching sheet. Analytic solutions for stagnation-point flow of nanofluid over a linearly stretching sheet are obtained by Mustafa et al. [29].
It is noticed that flow of nanofluid over an exponentially stretching sheet is never reported in the literature. Thus current work presents a theoretical study on the stagnation-point flow of nanofluid over an exponentially stretching sheet. The series expressions of velocity, temperature and nanoparticles concentration are developed by homotopy analysis method (HAM) developed by Liao [30]. This method is successfully applied to derive analytic solutions of variety of nonlinear problems [30][31][32][33][34][35]. The numerical solutions are obtained by the software MATLAB. Graphical results for various values of the parameters are presented to gain thorough insight towards the physics of the problem. The numerical values of reduced Nusselt number and reduced Sherwood number for different values of the parameters are also tabulated.

Mathematical Formulation
We investigate the laminar boundary layer flow of a nanofluid in the region of stagnation-point towards an exponentially stretching sheet situated at y~0. The x{ and y{ axis are taken along and perpendicular to the sheet and the flow is confined to y §0. The effects of Brownian motion and thermophoresis are also accounted. U w (x)~a exp(x=l) denotes the velocity of the sheet while the velocity of the external flow is U ? (x)~b exp(x=l). Let T w~T? zc exp(x=l) and C w~C? zd exp(x=l) be the temperature and nanoparticles concentration at the sheet where T ? and C ? denote the ambient temperature and concentration respectively. The boundary layer equations governing the conservation of mass, momentum, energy and nanoparticles volume fraction are (see Kuznetsov and Nield [22], Nield and Kuznetsov [23], Bachok et al. [24], Khan and Pop [25], Rana and Bhargava [26], Yacob et al. [27], Makinde and Aziz [28] and Mustafa et al. [29]) With the boundary conditions Where u and v are the velocity components along xand ydirections respectively, n f is the kinematic viscosity, a is the thermal diffusivity, D B is the Brownian motion coefficient, D T is the thermophoretic diffusion coefficient and t(~rc ð Þ p = rc ð Þ f ) is the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid. We introduce Inserting Eq. (7) into Eqs. (2)-(5) yield the following ordinary differential equations Here l is the velocity ratio, Pr is the Prandtl number, Le is the Lewis number, Nb is the Brownian motion parameter and Nt is the thermophoresis parameter. It is clear that x{coordinate can not be eliminated from Eqs. (9) and (10) because Nb and Nt are functions of x. Thus we look for the availability of local similarity solutions which permits us to investigate the behaviors of these parameters at a fixed location above the sheet. Nb~0 corresponds to the case when there is no thermal transport generated by the nanoparticles concentration gradients.
The skin friction coefficient C f , the local Nusselt number Nu and the local Sherwood number Sh are given by Using (7) in (13) one obtains where Re~U w L=n f is the Reynolds number and Re x~Uw x=n f denotes the local Reynolds number.

Homotopy analytic solution
Rule of solution expression and the involved boundary conditions direct us to choose the following initial guesses f 0 , h 0 and w 0 of f (g),h(g) and w(g) The auxiliary linear operators are chosen as If q[0,1 denotes the embedding parameter and B is the nonzero auxiliary parameter then the generalized homotopic equations are constructed as follows: H(g; q)j g~0~1 ,H(g; q)j g??~0 , ð21Þ in which the non-linear operators N f , N h and N w are N h F (g; q),H(g; q),W(g; q) ½ 1 Pr N q F (g; q),H(g; q),W(g; q) ½ L 2 W(g; q) Using Maclaurin's series about q are Flow of Nanofluid over a Moving Surface Eqs. (29)-(36) can be easily solved by using the symbolic computation software Mathematica for m~1,2,3:::: where Dx~10=L and L~20.

Numerical method
Eqs. (8)-(10) subject to the boundary conditions (11) have been solved numerically by using the built in function bvp4c of the software MATLAB. This software uses the higher order finite difference code that implements a collocation formula (see Shampine et al. [37] for more details). It will be seen shortly that numerical solutions are in a very good agreement with the homotopy solutions for all the values of the embedding parameters.

Numerical Results and Discussion
The representative results for velocity, temperature and nanoparticles concentration are provided graphically and in tabular form. There is a considerable increase in the velocity with an increase in velocity ratio l for some fixed values of parameters (see Fig. 7). It is evident from this figure that when lw1, the thickness of the boundary layer decreases with the increase in l.
Here the straining motion near the stagnation region increases so the acceleration of the external stream increases which causes a reduction in the boundary layer thickness and as a consequence   Flow of Nanofluid over a Moving Surface PLOS ONE | www.plosone.org the horizontal velocity increases. On the other hand, when lv1, the flow has an inverted boundary layer structure. Here the sheet velocity U w (x) exceeds the velocity of external stream U ? (x). It is also noticed that boundary layer is not formed when l~1. Fig. 8 is plotted to perceive the effects of Brownian motion and thermophoresis parameters on the temperature. There is a substantial increase in the temperature and the thermal boundary layer thickness with an increase in Nb and Nt. The growth in the thermal boundary layer thickness is compensated with smaller rate of heat transfer at the sheet. Fig. 9 portrays the behavior of Prandtl number Pr on the temperature h. An increase in Pr rapidly shifts the profiles towards the boundary causing a diminution in the thickness of thermal boundary layer. A bigger Prandtl number has a relatively lower thermal diffusivity. Thus an increase in Pr reduces conduction and thereby increases the variation in the thermal characteristics. As expected, the variation in the temperature is more pronounced for smaller values of Pr than its larger values. Fig. 10 depicts the effect of velocity ratio l on the temperature h. The temperature and the thermal boundary layer thickness decrease with an increase in l. Fig. 11 plots the concentration function versus g for different values of the Brownian motion parameter Nb. Here unlike the temperature h, concentration boundary layer reduces as Nb increases which thereby enhances the nanoparticles concentration at the sheet. Further we noticed that concentration w is only affected for the values of Nb in the range 0vNbƒ2. The influence of thermophoresis parameter Nt on the concentration boundary layer is noticed in Fig. 12. An abnormal increase in the concentration w is found for a weaker Brownian motion (Nb~0:2). In fact an over shoot in the concentration function occurs as we gradually increase Nt. On the other hand, when the effect of Brownian motion is increased i.e Nb changes from 0:2 to 1, there is a little increase in the concentration w with an increase in Nt. This outcome is attributed to the fact that an increase in Nt     appreciably enhances the mass flux due to temperature gradient which in turn rises the nanoparticles concentration. The behavior of Lewis number Le on the concentration field w is presented in Fig. 13. As Le gradually increases, this corresponds to a weaker molecular diffusivity and thinner concentration boundary layer. In accordance with [27] the variation in w with Le is prominent neat the stretching wall. Fig. 14 shows that the influence of l on the nanoparticles concentration w is virtually similar to that accounted for the temperature h.
Reduced Nusselt number Nur~({h 0 (0)) for different values of Nb is plotted versus Nt in the Fig. 15. It is observed that for a weaker thermophoretic effect, there is a significant decrease in the rate of heat transfer at the sheet with an increase in Nb. However when the strength of thermophoretic is increased i.e Nb changes from 0:1 to 2 the absolute decrease in Nur with an increase in Nt is negligible. This reduction actually occurs due to the excessive movement of nanoparticles from the stretching wall to the quiescent fluid. Fig. 16 shows the simultaneous effects of Nb and Nt on the reduced sherwood number Shr. There is a slight increase in Shr with an increase in Nb when the thermophoretic effect is weak. However this increase is significant when the thermophoretic effect intensifies. The variations of Nur and Shr with the velocity ratio l is sketched in the Figs. 17 and 18. The           dimensionless heat and mass transfer rates at the sheet increase when l is increased. In Table 1 the dimensionless velocity gradient on the sheet is approximated for various values of l. We observed that skin friction coefficient is reduced by assuming sufficiently large values of l. The numerical values of Nur and Shr corresponding to different values of Pr and Le have been given in Table 2. It is clear from this table that numerical and analytical solutions are in a very good agreement. We noticed earlier that increase in Pr and Le reduce the thermal boundary layer thickness and curves become steeper. The reduced Nusselt and Sherwood numbers, being proportional to the corresponding initial slopes, increase with an increase in Pr and Le respectively.

Conclusions
Flow of nanofluid in the region of stagnation-point towards an exponentially stretching sheet is studied. The developed mathematical problems have been solved for series solutions. A very good averaged residual error of about 10 {10 is achieved at only 15th-order of approximations in nearly all the cases. The numerical solutions are computed by the built-in solver bvp4c of the software MATLAB. Analytic and numerical solutions are found in excellent agreement for all the values of embedding parameters. It is observed that the velocity ratio l has a dual behavior on the momentum boundary layer. An increase in the strengths of Brownian motion and thermophoretic effects causes an appreciable increase in the temperature and the thermal boundary layer thickness. The current analysis for the case of regular fluid, which is not yet reported can be obtained by setting Nb~Nt~0.