Nomenclature

Greek Symbols

1 Introduction

The stretching sheets moving in nanofluid flows found useful applications in a wide range of manufacturing processes such as hot rolling,glass fiber production,melt-pinning,extrusion, manufacture of rubbers and plastics,and cooling of large metallic plates in bathes. Crane^[1] investigated the boundary layer flow over an elastic flat sheet. He achieved a precise solution for 2D Navier-Stokes equations. Then,some other researchers[2−7] presented solutions for various problems about the stretching issue.

Choi et al.^[8] showed the thermal conductivity of conventional liquids increases by adding the nanoparticles of metals. In recent years,some investigators have used nanofluids in the prob- lems of stretching sheets. Khan and Pop^[9] studied the heat transfer phenomena in the steady boundary layer nanofluid flow of a stretching sheet as the surface temperature was constant. They studied numerically the dependencies of the local Sherwood and local Nusselt numbers on the Prandtl number Pr,the Lewis number Le,the Brownian motion number Nb,and the ther- mophoresis number Nt. Steady flow of a nanofluid developed over a flat stretching sheet while it was moving in the uniform freestream was investigated by Bachok et al.^[10]. They obtained variations of the surface friction coefficient,the local Nusselt and the local Sherwood numbers with respect to the fluid flow conditions numerically. Their numerical solutions depended on the parameters presented in Khan and Pop’s paper^[9]. Hassani et al.^[11] studied analytically the boundary layer of flow over a long flat stretching sheet by applying the homotopy method. For many cases of the fluid flow problem,they confirmed the results obtained by Khan and Pop^[9]. However,they indicated that there were some disparities between their results and those reported by Khan and Pop^[9]. They found that for a large domain of the Nb numbers, increasing of the Pr number caused to decrease the reduced Nusselt number. However,there were conversely interesting results for a special value of the Nb that were interpreted in more details in their paper. Following this research work,Makinde and Aziz^[12] analyzed the problem subjected to a convective boundary condition instead of an isothermal condition. The main objective of this work is to study the characteristics of the thermal boundary layer of nanofluid flow over a stretching sheet subjected to constant and linear variations of surface temperature and nanoparticle volume fraction. The constant boundary condition was the same as those by Khan and Pop^[9] for validating the numerical work. Based on the authors’ knowledge,the nanofluid flow past a stretching sheet subjected to linear variations of both parameters over the surface has not been reported yet. These results may serve as a complement for the previous studies and provide useful information for applications.

2 Mathematical modeling

In this work,the steady state 2D boundary layer flow of a nanofluid past a flat stretching sheet is investigated numerically. The stretching sheet velocity equals u_w(x) = ax,where a has a constant value and x is the coordinate surface as shown in Fig. 1.

Two boundary conditions are assumed for this problem. One includes a constant tempera- ture and a constant nanoparticle volume fraction,called as Case 1 in this paper,and the other includes the variable temperature and nanoparticle volume fraction on the stretching surface, called as Case 2. The continuity and transport equations of the momentum,thermal energy, and nanoparticle volume fraction of nanofluids can be written as follows^[9]:

The boundary conditions for Case 1 are considered as follows:

while for Case 2,the boundary conditions are as follows: where a,m,and n are constants in the Eqs. (7) and (8). For simplifying the mathematical complexity of the problem,similar to the procedures in Refs. ^{[13, 14]} used for transforming the basic governing equations of fluid flow and heat transfer to a set of coupled non-linear ordinary differential equations,a similarity solution of Eqs. (1)-(5) subjected to the above boundary conditions can be defined in the following form: where the stream function Ψ is defined as For extracting the similarity solution,it is considered that in the outer flow (inviscid),the pressure is P = P₀ (constant). By substituting the variables defined in Eq. (3) into Eqs. (2)-(5) and taking into account two different boundary conditions,the following ordinary differential equations are observed for both cases.

Case 1 The equations considering the first boundary conditions (6) are

Case 2 The equations considering the second case of boundary conditions (7) are

Both sets of these equations are subjected to the following forms of boundary conditions:

Therefore,using Eq. (8),two different types of equations with the same boundary conditions mentioned above are derived. In Eq. (15),primes denote a differentiation with respect to η. Therefore,the fluid flow parameters are defined by

Based on the above quantities,the Sherwood number and Nusselt number are defined as where qw and qm are the heat and mass fluxes of the wall,respectively. Using variables defined in Eq. (8),the reduced Sherwood number and Nusselt number equations are obtained as

3 Results and discussion

Equations (9)-(11) and (12)-(14) subjected to the boundary conditions of Eq. (15) are solved numerically by using the finite difference method for two different cases. First of all,for vali- dating the procedure of the numerical solution,Eqs. (9)-(11) are solved while the Nb and Nt numbers are set to zero. The reduced Nusselt number,−θ′(0),achieved from this procedure is compared with those obtained by others^{[9, 15, 16]}. A good consistency between the results for all Pr numbers can be observed in Table 1 .

In Fig. 2,the profile for temperature function θ(η) is depicted versus η. The present results are compared with the results in Khan and Pop^[9]. It should be noticed that,the first case of the present study has the same conditions as Ref. ^[9]. As expected,the results are in good agreement with each other. It can be seen that,as Nb and Nt increase,the thickness of the thermal boundary layer also grows up. In addition,it is obvious that the thickness of the thermal boundary layer decreases as temperature and nanoparticle volume fraction at the stretching sheet are variables (Case 2). This decrease causes to increase the reduced Nusselt number,−θ′(0). Figure 3 also shows good agreement between the results of the present work and those of Khan and Pop^[9] on the concentration distribution of the nanoparticle volume fraction. The boundary layer thickness for the volume fraction function Φ(η) decreases as Nb increases. It also decreases as the temperature and nanoparticle volume fraction vary at the stretching sheet (Case 2).

Figure 4 indicates that the thickness of the thermal boundary layer decreases with the increase of Pr for both cases. Figure 5 shows that the boundary layer thickness for the volume fraction function is a decreasing function as the Le number increases. In all of the above conditions,the thicknesses of the thermal and volume fraction boundary layers decrease when the surface temperature and the volume fraction of the nanoparticle vary over the stretching sheet,which causes the reduced Nusselt and Sherwood numbers to increase.

Variation of the reduced Nusselt numbers versus Nt is given in Fig. 6. It can be observed that the present results for Case 1 are in good agreement with the results presented by Khan and Pop^[9]. Additionally,the reduced Nusselt number increases in Case 2. Also,it has a decreasing rate as the Nb and Nt numbers increase. In Fig. 7,the same results are obtained for Pr = 10.

In Fig. 8,the effects of Pr on the reduced Nusselt number are presented while Nb = 0.1. It can be seen that the reduced Nusselt number increases with increasing of Pr. Figure 9 shows for Nb = 0.5,the reduced Nusselt number increases at first,and then decreases with the increase of Pr.

In Fig. 10,while Nb equals 1,increasing of Pr number causes to decrease the reduced Nusselt number. The results are interesting because the reduced Nusselt number is an increasing function for Nb = 0.1 and a decreasing function for Nb = 1. These results almost confirm the results presented by Hassani et al.^[11]. Apparently,the reduced Nusselt number has a proportional relation with Pr for Nb = 0.1 and a reversed relation with Pr while Nb = 1. Variation of the reduced Nusselt number versus Nt with the increase of Le is shown in Fig. 11. It shows that the reduced Nusselt number decreases with the increase of Le.

For some selected values of the parameters,variation of the reduced Nusselt number versus Nt is demonstrated in Figs. 12-15. As can be observed in Fig. 12,the present results of Case 1 obtained for reduced Sherwood number,−Φ′(0),have very good agreement with those reported by Khan and Pop^[9]. The reduced Sherwood number increases with the increase of Nb and also is higher when the temperature and nanoparticle volume fraction vary at the stretching sheet.

Figure 13 shows that the same results are achieved for Pr = 10. It should be noted that the reduced Sherwood number is a decreasing function for Nb = 0.1 while for Nb > 0.1,it is almost constant. Also,the reduced Sherwood number increases as Pr increases,as shown in Fig. 14. For Pr = 1,the reduced Sherwood number decreases with the increase of Nt. However, for Pr > 1,it increases with the increase of Nt. In Fig. 15,the effects of Le on the reduced Sherwood number are depicted. The reduced Sherwood number is increased as Le increases. While Le is constant,the reduced Nusselt number is also constant but Nt increases.

4 Conclusions

In this work,the heat transfer of boundary layer nanofluid flow developed over a continuous flat stretching sheet moving in a quiescent flow is investigated numerically. The governing equa- tions are transformed to a boundary value problem in similarity variables and then the new set of equations is solved numerically. Two different cases are investigated which include stretch- ing sheets with constant temperature,nanoparticle volume fraction,variable temperature,and nanoparticle volume fraction. The novelty of this work is to use a linear variation of surface temperature and nanoparticle volume fraction in respect to previous works. A summary of the results can be listed as follows.

(i) The reduced Sherwood number and the reduced Nusselt number depend on the Prandtl number Pr,the Lewis number Le,the Brownian motion number Nb,and the thermophoresis number Nt.

(ii) The thickness of the thermal boundary layer is an increasing function of Nb and Nt. However,it is a decreasing function for Pr. The thickness of the boundary layer of the nanopar- ticle volume fraction is decreased as Nb or Le is increased.

(iii) Both boundary layer thicknesses of thermal and volume fraction are lower as temper- ature and nanoparticle volume fraction vary at the stretching sheet compared with Case 1. Therefore,higher values are caused for the reduced Nusselt and Sherwood numbers.

(iv) The reduced Nusselt number is decreased as the Nb,Nt,and Le numbers increase. However,there are three distinct regions as Pr increases based on the amount of Nb. For Nb = 0.1,the reduced Nusselt number increases with the increase of Pr. But,for Nb = 0.5, the reduced Nusselt number increases at first and then decreases. In contrast,for Nb = 1,the reduced Nusselt number decreases with the increase of Pr.

(v) The reduced Sherwood number increases with the increase of Nb. However,for Nb = 0.1, the reduced Sherwood number decreases as Nt increases. But,for Nt > 0.1,as Nt increases, the reduced Sherwood number is constant.

(vi) The reduced Sherwood number increases with the increase of Pr. For Pr = 1,the reduced Sherwood number decreases as Nt increases. For Pr > 1,it increases with the increase of Nt.

(vii) The reduced Sherwood number increases as Le increases. When Le is constant,the reduced Sherwood number is constant with the increase of Nt.