Abstract
The steady boundary layer flow of nanofluid over an exponential stretching surface is investigated analytically. The transport equations include the effects of Brownian motion parameter and thermophoresis parameter. The highly nonlinear coupled partial differential equations are simplified with the help of suitable similarity transformations. The reduced equations are then solved analytically with the help of homotopy analysis method (HAM). The convergence of HAM solutions are obtained by plotting hcurve. The expressions for velocity, temperature and nanoparticle volume fraction are computed for some values of the parameters namely, suction injection parameter α, Lewis number Le, the Brownian motion parameter Nb and thermophoresis parameter Nt.
Keywords:
nanofluid; porous stretching surface; boundary layer flow; series solutions; exponential stretching1 Introduction
During the last many years, the study of boundary layer flow and heat transfer over a stretching surface has achieved a lot of success because of its large number of applications in industry and technology. Few of these applications are materials manufactured by polymer extrusion, drawing of copper wires, continuous stretching of plastic films, artificial fibers, hot rolling, wire drawing, glass fiber, metal extrusion and metal spinning etc. After the pioneering work by Sakiadis [1], a large amount of literature is available on boundary layer flow of Newtonian and nonNewtonian fluids over linear and nonlinear stretching surfaces [210]. However, only a limited attention has been paid to the study of exponential stretching surface. Mention may be made to the works of Magyari and Keller [11], Sanjayanand and Khan [12], Khan and Sanjayanand [13], Bidin and Nazar [14] and Nadeem et al. [15,16].
More recently, the study of convective heat transfer in nanofluids has achieved great success in various industrial processes. A large number of experimental and theoretical studies have been carried out by numerous researchers on thermal conductivity of nanofluids [1722]. The theory of nanofluids has presented several fundamental properties with the large enhancement in thermal conductivity as compared to the base fluid [23].
In this study, we have discussed the boundary layer flow of nanofluid over an exponentially stretching surface with suction and injection. To the best of our knowledge, the nanofluid over an exponentially stretching surface has not been discussed so far. However, the present paper is only a theoretical idea, which is not checked experimentally. The governing highly nonlinear partial differential equation of motion, energy and nanoparticle volume fraction has been simplified by using suitable similarity transformations and then solved analytically with the help of HAM [2439]. The convergence of HAM solution has been discussed by plotting hcurve. The effects of pertinent parameters of nanofluid have been discussed through graphs.
2 Formulation of the problem
Consider the steady twodimensional flow of an incompressible nanofluid over an exponentially stretching surface. We are considering Cartesian coordinate system in such a way that xaxis is taken along the stretching surface in the direction of the motion and yaxis is normal to it. The plate is stretched in the xdirection with a velocity U_{w }= U_{0 }exp (x/l). defined at y = 0. The flow and heat transfer characteristics under the boundary layer approximations are governed by the following equations
where (u, v) are the velocity components in (x, y) directions, ρ_{f }is the fluid density of base fluid, ν is the kinematic viscosity, T is the temperature, C is the nanoparticle volume fraction, (ρc)_{p }is the effective heat capacity of nanoparticles, (ρc)_{f }is the heat capacity of the fluid, α = k/(ρc)_{f }is the thermal diffusivity of the fluid, D_{B }is the Brownian diffusion coefficient and D_{T }is the thermophoretic diffusion coefficient.
The corresponding boundary conditions for the flow problem are
in which U_{0 }is the reference velocity, β(x) is the suction and injection velocity when β(x) > 0 and β(x) < 0, respectively, T_{w }and T_{∞ }are the temperatures of the sheet and the ambient fluid, C_{w}, C_{∞ }are the nanoparticles volume fraction of the plate and the fluid, respectively.
We are interested in similarity solution of the above boundary value problem; therefore, we introduce the following similarity transformations
Making use of transformations (6), Eq. (1) is identically satisfied and Equations (2)(4) take the form
where
The physical quantities of interest in this problem are the local skinfriction coefficient C_{f}, Nusselt number Nu_{x }and the local Sherwood number Sh_{x}, which are defined as
where Re_{x }= U_{w}x/ν is the local Renolds number.
3 Solution by homotopy analysis method
For HAM solutions, the initial guesses and the linear operators L_{i }(i = 1  3) are
The operators satisfy the following properties
in which C_{1 }to C_{7 }are constants. From Equations (7) to (9), we can define the following zerothorder deformation problems
In Equations (17)(22), ħ_{1}, ħ_{2}, and ħ_{3 }denote the nonzero auxiliary parameters, H_{1}, H_{2 }and H_{3 }are the nonzero auxiliary function (H_{1 }= H_{2 }= H_{3 }= 1) and
Obviously
When p varies from 0 to 1, then
The mthorder problems are defined as follow
where
Employing MATHEMATICA, Equations (35)(40) have the following solutions
in which
4 Results and discussion
The numerical data of the solutions (45)(47), which is obtained with the help of Mathematica, have been discussed through graphs. The convergence of the series solutions strongly depends on the values of nonzero auxiliary parameters ħ_{i }(i = 1, 2, 3, h_{1 }= h_{2 }= h_{3}), which can adjust and control the convergence of the solutions. Therefore, for the convergence of the solution, the ħcurves is plotted for velocity field in Figure 1. We have found the convergence region of velocity for different values of suction injection parameter v_{w}. It is seen that with the increase in suction parameter v_{w}, the convergence region become smaller and smaller. Almost similar kind of convergence regions appear for temperature and nanoparticle volume fraction, which are not shown here. The nondimensional velocity f′ against η for various values of suction injection parameter is shown in Figure 2. It is observed that velocity field increases with the increase in v_{w}. Moreover, the suction causes the reduction of the boundary layer. The temperature field θ for different values of Prandtle number Pr, Brownian parameter Nb, Lewis number Le and thermophoresis parameter Nt is shown in Figures 3, 4, 5 and 6. In Figure 3, the temperature is plotted for different values of Pr. It is observed that with the increase in Pr, there is a very slight change in temperature; however, for very large Pr, the solutions seem to be unstable, which are not shown here. The variation of Nb on θ is shown in Figure 4. It is depicted that with the increase in Nb, the temperature profile increases. There is a minimal change in θ with the increase in Le (see Figure 5). The results remain unchanged for very large values of Le. The effects of Nt on θ are seen in Figure 6. It is seen that temperature profile increases with the increase in Nt; however, the thermal boundary layer thickness reduces. The nanoparticle volume fraction g for different values of Pr, Nb, Nt and Le is plotted in Figures 7, 8, 9 and 10. It is observed from Figure 7 that with the increase in Nb, g decreases and boundary layer for g also decreases. The effects of Pr on g are minimal. (See Figure 8). The effects of Le on g are shown in Figure 9. It is observed that g decreases as well as layer thickness reduces with the increase in Le. However, with the increase in Nt, g increases and layer thickness reduces (See Figure 10).
Figure 1. hCurve for velocity.
Figure 2. Velocity for different values of suction and injection parameter.
Figure 3. Variation of temperature for different values of Pr when Le = 2, h = 0.1, Nt = Nb = 0.5, v_{w}= 1.
Figure 4. Variation of temperature for different values of Nb when Le = 2, h = 0.1, Nt = 0.5, v_{w }= 1, Pr = 2.
Figure 5. Variation of temperature for different values of Le when h = 0.1, Nt = Nb = 0.5, v_{w }= 1, Pr = 2.
Figure 6. Variation of temperature for different values of Nt when Le = 2, h = 0.1, Nb = 0.5, v_{w }= 1, Pr = 2.
Figure 7. Variation of nanoparticle fraction g for different values of Nb when Le = 2, h = 0.1, Nt = 0.5, v_{w }= 1, Pr = 2.
Figure 8. Variation of nanoparticle fraction g for different values of Pr when Le = 2, h = 0.1, Nt = 0.5, v_{w }= 1, Nb = 0.5.
Figure 9. Variation of nanoparticle fraction g for different values of Le when Pr = 2, h = 0.1, Nt = 0.5, v_{w }= 1, Nb = 0.5.
Figure 10. Variation of nanoparticle fraction g for different values of Nt when Le = 2, h = 0.1, Nt = 0.5, v_{w }= 1, Pr = 2.
Competing interests
This is just the theoretical study, every experimentalist can check it experimentally with our consent.
Authors' contributions
SN done the major part of the article; however, the funding and computational suggestions and proof reading has been done by CL. All authors read and approved the final manuscript.
Acknowledgements
This research was supported by WCU (World Class University) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R312008000100490.
References

Sakiadis BC: Boundary layer behavior on continuous solid surfaces: I Boundary layer equations for two dimensional and axisymmetric flow.

Liu IC: Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to a transverse magnetic field.

Vajravelu K, Rollins D: Heat transfer in electrically conducting fluid over a stretching surface.

Vajravelu K, Nayfeh J: Convective heat transfer at a stretching sheet.

Khan SK, Subhas Abel M, Sonth Ravi M: Viscoelastic MHD flow, heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work.

Cortell R: Effects of viscous dissipation and work done by deformation on the MHD flow and heat transfer of a viscoelastic fluid over a stretching sheet.

Dandapat BS, Santra B, Vajravelu K: The effects of variable fluid properties and thermocapillarity on the flow of a thin film on an unsteady stretching sheet.

Nadeem S, Hussain A, Malik MY, Hayat T: Series solutions for the stagnation flow of a secondgrade fluid over a shrinking sheet.

Nadeem S, Hussain A, Khan M: HAM solutions for boundary layer flow in the region of the stagnation point towards a stretching sheet.

Magyari E, Keller B: Heat and mass transfer in the boundary layer on an exponentially stretching continuous surface.

Sanjayanand E, Khan SK: On heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet.

Khan SK, Sanjayanand E: Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet.

Bidin B, Nazar R: Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation.

Nadeem S, Hayat T, Malik MY, Rajput SA: Thermal radiations effects on the flow by an exponentially stretching surface: a series solution.

Nadeem S, Zaheer S, Fang T: Effects of thermal radiations on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface.

Bachok N, Ishak A, Pop I: boundary Layer flow of nanofluid over a moving surface in a flowing fluid.

Choi SUS: Enhancing thermal conductivity of fluids with nanoparticle.
In Developments and Applications of NonNewtonian Flows Edited by Siginer DA, Wang HP. 1995, 66:99105.
ASME FED 231/MD

Khanafer K, Vafai K, Lightstone M: Buoyancy driven heat transfer enhancement in a two dimensional enclosure utilizing nanofluids.

Makinde OD, Aziz A: Boundary layer flow of a nano fluid past a stretching sheet with a convective boundary condition.

Bayat J, Nikseresht AH: Investigation of the different base fluid effects on the nanofluids heat transfer and pressure drop.
Heat Mass Transf
doi:10.1007/s0023101107730

Hojjat M, Etemad SG, Bagheri R: Laminar heat transfer of nanofluid in a circular tube.

Fan J, Wang L: Heat conduction in nanofluids: structureproperty correlation.

Liao SJ: Beyond Perturbation Introduction to Homotopy Analysis Method. Boca Raton: Chapman & Hall/CRC Press; 2003.

Abbasbandy S: The application of homotopy analysis method to nonlinear equations arising in heat transfer.

Abbasbandy S: Homotopy analysis method for heat radiation equations.

Abbasbandy S, Tan Y, Liao SJ: Newtonhomotopy analysis method for nonlinear equations.

Abbasbandy S: Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method.

Abbasbandy S: Soliton solutions for the FitzhughNagumo equation with the homotopy analysis method.

Tan Y, Abbasbandy S: Homotopy analysis method for quadratic Ricati differential equation.

Alomari AK, Noorani MSM, Nazar R: Adaptation of homotopy analysis method for the numericanalytic solution of Chen system.
Commun Nonlinear Sci Numer Simulat 2008.
doi:10.1016/j.cnsns.2008.06.011

Rashidi MM, Domairry G, Dinarvand S: Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method.

Chowdhury MSH, Hashim I, Abdulaziz O: Comparison of homotopy analysis method and homotopyperturbation method for purely nonlinear fintype problems.

Sami Bataineh A, Noorani MSM, Hashim I: On a new reliable modification of homotopy analysis method.

Sami Bataineh A, Noorani MSM, Hashim I: Modified homotopy analysis method for solving systems of secondorder BVPs.

Sami Bataineh A, Noorani MSM, Hashim I: Solving systems of ODEs by homotopy analysis method.

Sajid M, Ahmad I, Hayat T, Ayub M: Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet.

Nadeem S, Hussain A: MHD flow of a viscous fluid on a non linear porous shrinking sheet with HAM.

Nadeem S, Abbasbandy S, Hussain M: Series solutions of boundary layer flow of a Micropolar fluid near the stagnation point towards a shrinking sheet.