SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Highly Accessed Nano Idea

Boundary layer flow of nanofluid over an exponentially stretching surface

Sohail Nadeem1* and Changhoon Lee2

Author Affiliations

1 Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad 44000, Pakistan

2 Department of Computational Science and Engineering, Yonsei University, Seoul, Korea

For all author emails, please log on.

Nanoscale Research Letters 2012, 7:94  doi:10.1186/1556-276X-7-94

The electronic version of this article is the complete one and can be found online at: http://www.nanoscalereslett.com/content/7/1/94

Received:26 July 2011
Accepted:30 January 2012
Published:30 January 2012

© 2012 Nadeem and Lee; licensee Springer.

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.


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.

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

Consider the steady two-dimensional flow of an incompressible nanofluid over an exponentially stretching surface. We are considering Cartesian coordinate system in such a way that 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 Uw = U0 exp (x/l). defined at y = 0. The flow and heat transfer characteristics under the boundary layer approximations are governed by the following equations

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M1">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M2">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M3">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M4">View MathML</a>


where (u, v) are the velocity components in (x, y) directions, ρf 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, DB is the Brownian diffusion coefficient and DT is the thermophoretic diffusion coefficient.

The corresponding boundary conditions for the flow problem are

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M5">View MathML</a>


in which U0 is the reference velocity, β(x) is the suction and injection velocity when β(x) > 0 and β(x) < 0, respectively, Tw and Tare the temperatures of the sheet and the ambient fluid, Cw, Care 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

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M6">View MathML</a>


Making use of transformations (6), Eq. (1) is identically satisfied and Equations (2)-(4) take the form

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M7">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M8">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M9">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M10">View MathML</a>



<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M11">View MathML</a>

The physical quantities of interest in this problem are the local skin-friction coefficient Cf, Nusselt number Nux and the local Sherwood number Shx, which are defined as

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M12">View MathML</a>


where Rex = Uwx/ν is the local Renolds number.

3 Solution by homotopy analysis method

For HAM solutions, the initial guesses and the linear operators Li (i = 1 - 3) are

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M13">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M14">View MathML</a>


The operators satisfy the following properties

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M15">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M16">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M17">View MathML</a>


in which C1 to C7 are constants. From Equations (7) to (9), we can define the following zeroth-order deformation problems

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M18">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M19">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M20">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M21">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M22">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M23">View MathML</a>


In Equations (17)-(22), ħ1, ħ2, and ħ3 denote the non-zero auxiliary parameters, H1, H2 and H3 are the non-zero auxiliary function (H1 = H2 = H3 = 1) and

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M24">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M25">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M26">View MathML</a>



<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M27">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M28">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M29">View MathML</a>


When p varies from 0 to 1, then <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M30">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M31">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M32">View MathML</a> vary from initial guesses f0 (η), θ0 (η) and g0 (η) to the final solutions f (η), θ (η) and g (η), respectively. Considering that the auxiliary parameters ħ1, ħ2 and ħ3 are so properly chosen that the Taylor series of <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M33">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M34">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M35">View MathML</a> expanded with respect to an embedding parameter converge at p = 1, hence Equations (17)-(19) become

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M36">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M37">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M38">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M39">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M40">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M41">View MathML</a>


The mth-order problems are defined as follow

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M42">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M43">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M44">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M45">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M46">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M47">View MathML</a>



<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M48">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M49">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M50">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M51">View MathML</a>


Employing MATHEMATICA, Equations (35)-(40) have the following solutions

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M52">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M53">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M54">View MathML</a>


in which <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M55">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M56">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M57">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/94/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/94/mathml/M58">View MathML</a> are the constants and the numerical data of above solutions are shown through graphs in the following section.

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 non-zero auxiliary parameters ħi (i = 1, 2, 3, h1 = h2 = h3), 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 vw. It is seen that with the increase in suction parameter vw, 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 vw. 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).

thumbnailFigure 1. h-Curve for velocity.

thumbnailFigure 2. Velocity for different values of suction and injection parameter.

thumbnailFigure 3. Variation of temperature for different values of Pr when Le = 2, h = -0.1, Nt = Nb = 0.5, vw= 1.

thumbnailFigure 4. Variation of temperature for different values of Nb when Le = 2, h = -0.1, Nt = 0.5, vw = 1, Pr = 2.

thumbnailFigure 5. Variation of temperature for different values of Le when h = -0.1, Nt = Nb = 0.5, vw = 1, Pr = 2.

thumbnailFigure 6. Variation of temperature for different values of Nt when Le = 2, h = -0.1, Nb = 0.5, vw = 1, Pr = 2.

thumbnailFigure 7. Variation of nanoparticle fraction g for different values of Nb when Le = 2, h = -0.1, Nt = 0.5, vw = 1, Pr = 2.

thumbnailFigure 8. Variation of nanoparticle fraction g for different values of Pr when Le = 2, h = -0.1, Nt = 0.5, vw = 1, Nb = 0.5.

thumbnailFigure 9. Variation of nanoparticle fraction g for different values of Le when Pr = 2, h = -0.1, Nt = 0.5, vw = 1, Nb = 0.5.

thumbnailFigure 10. Variation of nanoparticle fraction g for different values of Nt when Le = 2, h = -0.1, Nt = 0.5, vw = 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.


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.


  1. 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. OpenURL

  2. 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. OpenURL

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

    Int J Non-Linear Mech 1992, 27(2):265-277. OpenURL

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

    Acta Mech 1993, 96(1-4):47-54. OpenURL

  5. 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. OpenURL

  6. 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. OpenURL

  7. 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. OpenURL

  8. 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. OpenURL

  9. 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. OpenURL

  10. Afzal N: Heat transfer from a stretching surface.

    Int J Heat Mass Transf 1993, 36:1128-1131. OpenURL

  11. 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. OpenURL

  12. 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. OpenURL

  13. 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. OpenURL

  14. 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. OpenURL

  15. 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. OpenURL

  16. 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. OpenURL

  17. 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. OpenURL

  18. 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


  19. 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. OpenURL

  20. 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. OpenURL

  21. Bayat J, Nikseresht AH: Investigation of the different base fluid effects on the nanofluids heat transfer and pressure drop.

    Heat Mass Transf



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

    Korean J Chem Eng 2010, 27(5):1391-*1396. OpenURL

  23. Fan J, Wang L: Heat conduction in nanofluids: structure-property correlation.

    Int J Heat Mass Transf 2011, 54:4349-4359. OpenURL

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

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

    Phys Lett A 2006, 360:109-113. OpenURL

  26. Abbasbandy S: Homotopy analysis method for heat radiation equations.

    Int Commun Heat Mass Transf 2007, 34:380-387. OpenURL

  27. Abbasbandy S, Tan Y, Liao SJ: Newton-homotopy analysis method for nonlinear equations.

    Appl Math Comput 2007, 188:1794-1800. OpenURL

  28. 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. OpenURL

  29. Abbasbandy S: Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method.

    Appl Math Model 2008, 32:2706-2714. OpenURL

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

    Comm Non-linear Sci Numer Simm 2008, 13:539-546. OpenURL

  31. 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.



  32. 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. OpenURL

  33. 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. OpenURL

  34. 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. OpenURL

  35. 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. OpenURL

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

    Commun Nonlinear Sci Numer Simul 2008, 13:2060-2070. OpenURL

  37. 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. OpenURL

  38. 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. OpenURL

  39. 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. OpenURL