Dtm-bf Method for the Flow and Heat Transfer over a Nonlinearly Stretching Sheet with Nanofluids

This paper investigates an analytical analysis for the flow and heat transfer in a viscous fluid over a nonlinear stretching sheet. The governing partial differential equations are transformed into coupled nonlinear differential equations by introducing a similarity transformation. The asymptotic analytical solutions are obtained by using differential transform method-basic functions (DTM-BF). Four types of nanofluids, namely Cu-water, Ag-water, 3 2 O Al-water and 2 TiO-water were studied. The influence of the nanoparticle volume fraction  , the nonlinear stretching parameter n , Prandtl number Pr, Eckert number Ec and different nanoparticles on the velocity and temperature are discussed and shown graphically. The comparison with the numerical results is presented and it is found to be in excellent agreement.


INTRODUCTION
In the past few years the flow over stretching surface has received considerable attention because of its many engineering applications.Crane [1] considered the steady two-dimensional flow of a Newtonian fluid driven by a sheet moving in its own plane with a velocity varying linearly with the distance from a fixed point.Ho et al. [2] identified the effects due to uncertainties in effective dynamic viscosity and thermal conductivity of nanofluid on laminar natural convection heat transfer in a square enclosure.Khan and Pop [3] investigated numerically the problem of laminar fluid flow resulting from the stretching of a flat surface in a nanofluid.Mahdy and Sameh [4] reported numerical analysis for laminar free convection over a vertical wavy surface embedded in a porous medium saturated with a nanofluid.
Due to many applications of nanofluids in technical process, the heat and mass transfer of nanofluids with a chemical reaction also caused more attention.The term 'nanofluid' was first proposed by Choi [5] to indicate a liquid suspension containing ultra-fine particles.Eastman [6] obtained an excellent assessment of nanofluid physics and developments.Kuznetsov and Nield [7] studied the influence of nanoparticles on natural convection boundary layer flow past a vertical plate by taking Brownian motion and thermophoresis into account.Besides, Mokmeli [8] and Xuan [9] explained that nanofluids clearly exhibit enhanced thermal conductivity, which gone up with increasing volumetric fraction of nanoparticles.Motivated by the above works, in this paper we present similarity solutions for the nonlinear problem of flow and heat transfer of nanofluid past a nonlinearly stretching sheet, which are then solved analytically using DTM-BF.This method was first proposed by Xiaohong Su et al. [10].In this paper, we can find that the approximate solution agrees very well with the numerical solution, which shows the reliability and validity of the present work.The effects of the governing parameters and different nanoparticles on the velocity and temperature are discussed by graph in detail.

MATHEMATICAL FORMULATION
Consider the two-dimensional steady laminar flow of viscous and incompressible nanofluid past a flat sheet.The fluid is a water based nanofluid containing different types of nanoparticles such as copper Cu, silver Ag, alumina  TiO .It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them.The thermophysical properties of the nanofluid are listed in Table 1.We choose the slit, from which the sheet is drawn, as the origin of the system.In this coordinate frame the x -axis is taken along the direction of the continuous stretching surface and the y -axis is measured normal to the surface of the sheet.
Under above assumptions, the steady, two-dimensional boundary layer equations for this fluid can be written as follow [11]: Subject to the following boundary conditions where v u, are the velocity components in the x and y directions, respectively.T is the temperature of the nanofluid,  T is the temperature of the fluid far from the sheet.a and n are parameters related to the surface stretching speed.The effective dynamic DTM-BF Method for Flow and Heat Transfer Over a Nonlinearly Stretching Sheet 17 viscosity of the nanofluid nf  , the effective density nf  and the heat capacitance of the nanofluid are given by [12] The thermal conductivity of nanofluids restricted to spherical nanoparticles is approximated [13]: where  is the solid volume fraction of nanoparticles, the subscripts f nf , and s denote the thermophysical properties of the nanofluid, base fluid and nano-solid particles, respectively.
Introducing the following similarity variables: The transformed momentum and energy equations together with the boundary conditions given by Eqs. ( 2)-( 5) can be written as where In equation ( 14), ) ( f is the original function and Combining equations ( 14) and (15), we obtain Theorems proposed by Rashidi, Mohimanian and Laraqito [14] in the transformation procedure are given below Taking differential transform of equation ( 10), the following equation can be obtained is the differential transforms of f .The transform of the boundary conditions are where  is a constant.
Using equation (18) and the iterative formula (17), ) (k F can be calculated.Then the following series solutions of the initial value problem is Using equation (10) and the boundary condition (12) and selecting the basic functions { as a linear combination of the basic functions where satisfies the boundary conditions (12) and is an attenuation parameter which to be determined.In practical applications, we will get a good precision when i N less than 5.For the case , then expending equation (20) in the following power series  The following equations can be obtained from the equations (19)(20) and ( 22) by comparing the coefficient of the same order of k It is easy to solve these equations successively, and then the series solution of the equation can be obtained

DISCUSSION OF THE SOLUTIONS
The boundary layer flow and heat transfer due to stretching vertical sheet have been investigated analytically.All the results obtained by the DTM-BF method are compared with the numerical results obtained by bvp4c with Matlab.Fig. 1 shows a comparison between analytical and numerical solutions for f f  , and g .Moreover, the results are also illustrated in Table 2.It is obvious that excellent agreement exists for all values considered.
The effects of the nonlinear stretching parameter n , Prandtl number Pr and Eckert number Ec on the velocity f f  , and g are plotted in Fig. 2. It is observed that as the nonlinear stretching parameter increases, f and f  decreases.Besides, Ec tends to increase the temperature profiles while Pr tends to decrease it.In addition, both of Eckert number and Prandtl number doesn't affect the velocity profile.TiO -water nanofluid, respectively.It is observed that increasing the volume fraction results in an increase in temperature.Besides, adding nanoparticle to the pure water decreases velocity profile while increases temperature profiles.Fig. 5  .It can be observed that the velocity and temperature distributions decrease gradually far away from the surface of the stretching sheet.Moreover, Al2O3-water nanofluid and TiO2-water nanofluid exhibits higher velocity and lower temperature than that of the other nanofluid concerned while Ag-water nanofluid exhibits lower velocity and higer temperature instead.Besides, Al2O3-water nanofluid and TiO2-water nanofluid have the similar profiles.Therefor, in the cooling intensification process, it is more suitable for selecting metal oxides such as aluminum oxide and titanium oxide than copper and silver.

CONCLUSION
In the present paper, we have investigated the problem of flow and heat transfer in a viscous nanofluid over a nonlinear stretching sheet.The momentum and energy boundary layer transfer characteristics for different parameters are discussed.Some important conclusion can be drawn.
(i) Variations of the dimensionless velocity and temperature are affected with the solid volume fraction of nanoparticles, the nonlinear stretching parameter, Eckert number and Prandtl number.f and f  decreases with an increase in the nonlinear stretching parameter, Ec tends to increase the temperature profiles while Pr tends to decrease it.
(ii) A high velocity can be obtained for lower solid volume concentration of nanoparticles since f  decrease with the increasing parameter  .Besides, the increasing volume fraction results in an increase in temperature.In addition, adding nanoparticle to the pure water decreases velocity profile while increases temperature profiles.
(iii) In the cooling intensification process, It is more suitable for selecting metal oxides nanofluids than the metal nanofluids since

Al
-water nanofluid and 2 TiOwater nanofluid exhibits higher velocity and lower temperature while Ag-water nanofluid exhibits lower velocity and higer temperature instead.

Figure 1 .
Figure 1.Comparison of numerical and analytical solutions for f f  , and g as

Figure 2 .
Figure 2. Effects of the parameters n , Pr and Ec for f f  , and g as 0  

Figure 3 .
Figure 3. Effects of the solid volume fraction  on f and f  for Cu and Ag as 3 .0 , 2 .6 Pr , 1    Ec n .

Figure 4 .Figure 5 .
Figure 4. Effects of the solid volume fraction  on temperature g for

Table 1 .
Thermophysical properties of water and nanoparticles.

Table 2 .
Comparison of