# Boundary layer flow of nanofluid over an exponentially stretching surface

- Sohail Nadeem
^{1}Email author and - Changhoon Lee
^{2}

**7**:94

**DOI: **10.1186/1556-276X-7-94

© Nadeem and Lee; licensee Springer. 2012

**Received: **26 July 2011

**Accepted: **30 January 2012

**Published: **30 January 2012

## 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 *h*-curve. 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 stretching## 1 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 non-Newtonian fluids over linear and nonlinear stretching surfaces [2–10]. 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 [17–22]. 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 [24–39]. The convergence of HAM solution has been discussed by plotting *h*-curve. The effects of pertinent parameters of nanofluid have been discussed through graphs.

## 2 Formulation of the problem

*x*-axis is taken along the stretching surface in the direction of the motion and y-axis is normal to it. The plate is stretched in the

*x*-direction 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.

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.

*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

*L*

_{ i }(

*i*= 1 - 3) are

*C*

_{1}to

*C*

_{7}are constants. From Equations (7)

*to*(9), we can define the following zeroth-order deformation problems

*ħ*

_{1},

*ħ*

_{2}, and

*ħ*

_{3}denote the non-zero auxiliary parameters,

*H*

_{1},

*H*

_{2}and

*H*

_{3}are the non-zero auxiliary function (

*H*

_{1}=

*H*

_{2}=

*H*

_{3}= 1) and

*p*varies from 0 to 1, then $\widehat{f}\left(\eta ,p\right)$, $\widehat{\theta}\left(\eta ,p\right)$, $\u011d\left(\eta ,p\right)$ vary from initial guesses

*f*

_{0}(

*η*),

*θ*

_{0}(

*η*) and

*g*

_{0}(

*η*) to the final solutions

*f*(

*η*),

*θ*(

*η*) and

*g*(

*η*), respectively. Considering that the auxiliary parameters

*ħ*

_{1},

*ħ*

_{2}and

*ħ*

_{3}are so properly chosen that the Taylor series of $\widehat{f}\left(\eta ,p\right)$, $\widehat{\theta}\left(\eta ,p\right)$ and $\u011d\left(\eta ,p\right)$ expanded with respect to an embedding parameter converge at

*p*= 1, hence Equations (17)-(19) become

in which ${a}_{m,0}^{0}$, ${a}_{m,n}^{k}$, ${A}_{m,n}^{k}$, ${F}_{m,n}^{k}$ are the constants and the numerical data of above solutions are shown through graphs in the following section.

## 4 Results and discussion

*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

*ħ*-curves 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 non-dimensional 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).

## Declarations

### 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 R31-2008-000-10049-0.

## Authors’ Affiliations

## References

- Sakiadis BC: Boundary layer behavior on continuous solid surfaces: I Boundary layer equations for two dimensional and axisymmetric flow.
*AIChE J*1961, 7: 26–28.View Article - Liu IC: Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to a transverse magnetic field.
*Int J Heat Mass Transf*2004, 47: 4427–4437.View Article - Vajravelu K, Rollins D: Heat transfer in electrically conducting fluid over a stretching surface.
*Int J Non-Linear Mech*1992, 27(2):265–277.View Article - Vajravelu K, Nayfeh J: Convective heat transfer at a stretching sheet.
*Acta Mech*1993, 96(1–4):47–54.View Article - 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.
*Int J Heat Mass Transf*2003, 40: 47–57.View Article - 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.
*Phys Lett A*2006, 357: 298–305.View Article - 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.
*Int J Heat Mass Transf*2007, 50: 991–996.View Article - Nadeem S, Hussain A, Malik MY, Hayat T: Series solutions for the stagnation flow of a second-grade fluid over a shrinking sheet.
*Appl Math Mech Engl Ed*2009, 30: 1255–1262.View Article - Nadeem S, Hussain A, Khan M: HAM solutions for boundary layer flow in the region of the stagnation point towards a stretching sheet.
*Comm Nonlinear Sci Numer Simul*2010, 15: 475–481.View Article - Afzal N: Heat transfer from a stretching surface.
*Int J Heat Mass Transf*1993, 36: 1128–1131.View Article - Magyari E, Keller B: Heat and mass transfer in the boundary layer on an exponentially stretching continuous surface.
*J Phys D Appl Phys*1999, 32: 577–785.View Article - Sanjayanand E, Khan SK: On heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet.
*Int J Therm Sci*2006, 45: 819–828.View Article - Khan SK, Sanjayanand E: Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet.
*Int J Heat Mass Transf*2005, 48: 1534–1542.View Article - Bidin B, Nazar R: Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation.
*Eur J Sci Res*2009, 33: 710–717. - Nadeem S, Hayat T, Malik MY, Rajput SA: Thermal radiations effects on the flow by an exponentially stretching surface: a series solution.
*Zeitschrift fur Naturforschung*2010, 65a: 1–9. - Nadeem S, Zaheer S, Fang T: Effects of thermal radiations on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface.
*Numer Algor*2011, 57: 187–205.View Article - Bachok N, Ishak A, Pop I: boundary Layer flow of nanofluid over a moving surface in a flowing fluid.
*Int J Therm Sci*2010, 49: 1663–1668.View Article - Choi SUS: Enhancing thermal conductivity of fluids with nanoparticle. In
*Developments and Applications of Non-Newtonian Flows*Edited by: Siginer DA, Wang HP. 1995, 66: 99–105. ASME FED 231/MD ASME FED 231/MD - Khanafer K, Vafai K, Lightstone M: Buoyancy driven heat transfer enhancement in a two dimensional enclosure utilizing nanofluids.
*Int J Heat Mass Transf*2003, 46: 3639–3653.View Article - Makinde OD, Aziz A: Boundary layer flow of a nano fluid past a stretching sheet with a convective boundary condition.
*Int J Therm Sci*2011, 50: 1326–1332.View Article - 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/s00231–011–0773–0 doi:10.1007/s00231-011-0773-0 - Hojjat M, Etemad SG, Bagheri R: Laminar heat transfer of nanofluid in a circular tube.
*Korean J Chem Eng*2010, 27(5):1391-*1396.View Article - Fan J, Wang L: Heat conduction in nanofluids: structure-property correlation.
*Int J Heat Mass Transf*2011, 54: 4349–4359.View Article - Liao SJ:
*Beyond Perturbation Introduction to Homotopy Analysis Method*. Boca Raton: Chapman & Hall/CRC Press; 2003.View Article - Abbasbandy S: The application of homotopy analysis method to nonlinear equations arising in heat transfer.
*Phys Lett A*2006, 360: 109–113.View Article - Abbasbandy S: Homotopy analysis method for heat radiation equations.
*Int Commun Heat Mass Transf*2007, 34: 380–387.View Article - Abbasbandy S, Tan Y, Liao SJ: Newton-homotopy analysis method for nonlinear equations.
*Appl Math Comput*2007, 188: 1794–1800.View Article - Abbasbandy S: Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method.
*Chem Eng J*2008, 136: 144–150.View Article - Abbasbandy S: Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method.
*Appl Math Model*2008, 32: 2706–2714.View Article - Tan Y, Abbasbandy S: Homotopy analysis method for quadratic Ricati differential equation.
*Comm Non-linear Sci Numer Simm*2008, 13: 539–546.View Article - Alomari AK, Noorani MSM, Nazar R: Adaptation of homotopy analysis method for the numeric-analytic solution of Chen system.
*Commun Nonlinear Sci Numer Simulat*2008. doi:10.1016/j.cnsns.2008.06.011 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.
*Commun Nonlinear Sci Numer Simul*2009, 14: 708–717.View Article - Chowdhury MSH, Hashim I, Abdulaziz O: Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems.
*Commun Nonlinear Sci Numer Simul*2009, 14: 371–378.View Article - Sami Bataineh A, Noorani MSM, Hashim I: On a new reliable modification of homotopy analysis method.
*Commun Nonlinear Sci Numer Simul*2009, 14: 409–423.View Article - Sami Bataineh A, Noorani MSM, Hashim I: Modified homotopy analysis method for solving systems of second-order BVPs.
*Commun Nonlinear Sci Numer Simul*2009, 14: 430–442.View Article - Sami Bataineh A, Noorani MSM, Hashim I: Solving systems of ODEs by homotopy analysis method.
*Commun Nonlinear Sci Numer Simul*2008, 13: 2060–2070.View Article - Sajid M, Ahmad I, Hayat T, Ayub M: Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet.
*Commun Nonlinear Sci Numer Simul*2008, 13: 2193–2202.View Article - Nadeem S, Hussain A: MHD flow of a viscous fluid on a non linear porous shrinking sheet with HAM.
*Appl Math Mech Engl Ed*2009, 30: 1–10.View Article - Nadeem S, Abbasbandy S, Hussain M: Series solutions of boundary layer flow of a Micropolar fluid near the stagnation point towards a shrinking sheet.
*Z Naturforch*2009, 64a: 575–582.

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.