1 Introduction

Nanofluid is a term coined by Choi^[1] ,referring to dispersions of nanoparticles in the base fluids such as water,ethylene glycol,and propylene glycol. The capability of a nanofluid to enhance thermal conductivity was first reported by Masuda et al. ^[2] .Choi^[1] discovered that the addition of less than 1% of nanoparticles into the base fluid doubles the heat conductivity of the fluid. Other characteristics of nanofluid include less clogging of tube and the stability for long term as compared with micro- and millimeter sized particles ^[2,3,4,5,6] . Buongiorno ^[7] developed an explanation for the abnormal convective heat transfer enhancement in nanofluids. In his paper,it is identified that Brownian diffusion and thermophoresis are the two most important nanoparticle/base-fluid slip mechanisms. Nield and Kuznetsov ^[8,9] and Kuznetsov and Nield^[10,11] were among the firsts to employ the Buongiorno model in their studies. They identified that the similarity solutions depend on the four dimensionless parameters,namely,Lewis numberLe,buoyancy-ratio parameter Nr,Brownian motion parameter Nb,and thermophoresis parameter Nt. Khan and Pop ^[12] used the model of Nield and Kuznetsov ^[9] to study the boundary layer flow of a nanofluid past a stretching sheet with a constant surface temperature. On the other hand,Makinde and Aziz^[13] performed a similar study with a convective surface boundary condition.

The study of boundary layer flow due to a stretching sheet has garnered considerable interests ^{[14,15,16,17,18,19]} . Flow of a boundary layer over a stretching sheet is important in applications, such as extrusion,wire drawing,metal spinning,and hot rolling ^[14] . The pioneering study of stretching sheet by Crane ^[15] showed an exact analytical solution for the steady two-dimensional stretching of a surface in a quiescent fluid. His study has then become the benchmark for other researchers. Crane’s study was extended by Wang ^[16] to include both suction and slip where he found the solution and proved the uniqueness of the said solution for both two-dimensional stretching and axisymmetric stretching. On the other hand,the study on boundary layer flow due to a shrinking sheet has also attracted much attention. Miklavˇciˇ c and Wang ^[20] pioneered the study of flow on a shrinking sheet. They found that the vorticity over the shrinking sheet is not confined within a boundary layer. To confine the velocity of the shrinking sheet in the boundary layer,Miklavˇ ciˇ c and Wang ^[20] imposed an adequate suction on the boundary while Wang ^[21] considered a stagnation flow. Ever since,numerous studies emerge,investigating different aspects of this problem. Fang et al. ^[22] solved the viscous flow over a shrinking sheet analytically using a second order slip flow model. They found that the solution has two branches,or dual solutions,in a certain range of the parameters. Bhattacharyya et al. ^[23] observed that the velocity and thermal boundary layer thicknesses for the second solutions are always larger than those of the first solutions. Weidman et al. ^[24] ,Postelnicu and Pop ^[25] ,and Ro¸sca and Pop ^[26] performed a stability analysis to show that the upper branch solutions are stable and physically realizable while the lower branch solutions are not stable,and therefore, not physically possible.

During the recent years,it has been found that rarefied gas flows with slip boundary conditions are often encountered in the micro-scale devices and low-pressure situations ^[27] .The effects of slip conditions are also very important in technological applications such as in the polishing of artificial heart valves and the internal cavities. Many studies were conducted to include slip condition. Wang ^[28] reported that the partial slip between the fluid and the moving surface may occur in situations that the fluid is particulate such as emulsions,suspensions, foams,and polymers. Noghrehabadi et al. ^[29] showed that the slip parameter greatly influences the flow velocity,the surface shear stress,the local Nusselt number,and the local Sherwood number. Sharma et al. ^[30] found that the surface heat transfer of a nanofluid increases with the increasing velocity slip. Aman et al. ^[31] reported that while the velocity slip increases the surface heat transfer,the thermal slip reduces the heat transfer rate at the surface.

In the present paper,we study the boundary layer flow and heat transfer characteristics of a nanofluid past a permeable stretching/shrinking sheet with hydrodynamic and thermal slip boundary conditions. The paper is,in fact,an extension of Noghrehabadi et al. ^[29] to the case of a permeable shrinking sheet,which was not considered previously. Furthermore,we also impose adequate suction at the boundary that is necessary to confine the vorticity in the boundary layer. A closed form analytical solution has also been obtained when the velocity slip is absent. The dependency of the local Nusselt number and the local Sherwood number on five parameters,namely,the stretching/shrinking,velocity slip,thermal slip,Brownian motion, and thermophoresis parameters is the main focus of the investigation. Numerical solutions are presented graphically and in tabular form to show the effects of these parameters on the above mentioned quantities. It is worth pointing out to this end that in many industrial applications, such as power,manufacturing,and transportation,nanofluids are commonly used for cooling any sort of high energy device. However,these fluids have limited heat transfer capabilities due to their low heat transfer properties. Thus,effective cooling techniques are greatly needed. The characteristic feature of nanofluids is thermal conductivity enhancement,a phenomenon observed by Masuda et al. ^[2] . This phenomenon suggests the possibility of using nanofluids in advanced nuclear systems (Buongiorno and Hu^[32] ). Buongiorno ^[7] presented a comprehensive survey of convective transport in nanofluids,who says that a satisfactory explanation for the abnormal increase of the thermal conductivity and viscosity is yet to be found. He focused on the further heat transfer enhancement observed in convective situations. 2 Mathematical formulation

Consider a steady,two-dimensional flow of a viscous and incompressible fluid with hydrodynamic and thermal slip boundary conditions in the region y>0 driven by a permeable stretching/shrinking surface located at y= 0 in a nanofluid as shown in Fig. 1,where thexand yaxes are measured along and normal to the surface of the sheet,respectively. It is assumed that the stretching/shrinking velocity is U_w(x)=cx,where c>0 and c<0 correspond to stretching and shrinking sheets,respectively. It is also assumed that the uniform temperature and the uniform nanofluid volume fraction at the surface of the sheet are T_w and C_w,while the uniform temperature and the uniform nanofluid volume fraction far from the surface of the sheet (inviscid fluid) are T_∞ and C_∞,respectively. Further,it is assumed that the constant mass flux velocity is v₀ with v₀<0 for suctionandv₀>0 for injection or withdrawal of the fluid. Under these assumptions,the governing equations for the steady conservation of momentum, thermal energy,and nanoparticle volume fraction equations can be written as (Buongiorno ^[7] and Noghrehabadi ^[29] )

whereuandvare the velocity components along thex-andy-axes,respectively,pis the fluid pressure,Tis the fluid temperature,Cis the nanoparticle volume fraction,αis the thermal diffusivity of the fluid,ν is the kinematic viscosity,DBis the Brownian diffusion coefficient,andDTis the thermophoresis diffusion coefficient. Moreover,τ =(ρC)p/(ρC)f is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid with ρ_f and ρ_p being the density of the fluid and of the particles,respectively. In this paper,we employ the model proposed by Buongiorno ^[7] ,which paid attention to the effect of Brownian motion and thermophoresis on the flow and heat transfer of a nanofluid. The present model is subjected to the following boundary conditions ^[33] : whereLdenotes the slip length ^[29] ,His a proportionality constant,and σis the constant stretching/shrinking parameter with σ>0 for stretching and σ<0 for shrinking. The governing equations (1)−(5) subjected to the boundary conditions (6) can be expressed in a simpler form by introducing the following similarity transformations ^[29,34] : whereηis the similarity variable,and ψ is the stream function defined as and v=,which identically satisfies (1). By employing the boundary layer approximations and the similarity variables (7),(2)−(5) reduce to the following ordinary differential equations: and the boundary conditions (6) and (7) become where primes denote differentiation with respect toη.Further,Pris the Prandtl number,Leis the Lewis number,sis the mass flux parameter with s >0 forsuctionand s<0 for injection, respectively,Nb is the Brownian motion parameter,Ntis the thermophoresis parameter, λis the velocity slip parameter,and ξ is the thermal slip parameter,which are defined as We notice as in Ref. ^[10] that when Nb and Ntare zero,(8) and (9) involve just two dependent variables,namely f(η) and θ(η),and the boundary-value problem for these two variables reduces to the classical problem solved by Crane ^[15] and Wang ^[16] (the boundaryvalue problem for φ(η) then becomes ill-posed and is of no physical significance).

The physical quantities of interest are the skin friction coefficient Cf,the local Nusselt numberNuxand the local Sherwood number Sh_x,which are given by

where τ_w,q_w,and q_m are the surface shear stress,the surface heat flux,and the surface mass flux,respectively. Using similarity variables (7),we obtain Cf Re_x where Re_x=U_wx/ν is the local Reynolds number.

It is worth mentioning that for σ=−1 (shrinking sheet),(8) with the associated boundary conditions (11) becomes identical with (7) subjected to boundary conditions (8) from the papers by Fang and Zhang ^[35] ,and Fang et al. ^[22] for the case of first order slip. In fact,for λ=0and ξ= 0 (no slip condition),the boundary value problems (8)−(11) have the following closed form analytical solution:

where γ(>0) is given by the quadratic equation with the solution Thus,we have Hence,we can compare the numerical results for f^'' (0) with the analytical ones given by (18). 3 Results and discussion

The set of ordinary differential equations (8)-(10) subjected to boundary conditions (11) are solved numerically using a shooting method. In this method,the dual solutions are obtained by setting different initial guesses for the values of f^'' (0),θ^' (0),and φ^' (0). The asymptotic boundary conditions (11) at η=∞ are replaced by η=15,which is sufficient for all the velocity, temperature,and concentration profiles to vanish asymptotically. Throughout the paper,we have fixed Prandtl numberPrto be equal to 6.8,which is appropriate for a water-base nanofluid. Furthermore,suction parametersis taken in the range s>2 due to the fact that the solutions for a shrinking sheet may only exist if adequate suction is imposed ^[21] . In choosing the values of Brownian motion parameterNb,thermophoresis parameter Nt,Lewis number Le,we adhere to Buongiorno ^[7] who stated that Le≥ and Le·N_bt≥1,whereNbt is the ratio between the Brownian diffusivity coefficient and thermophoresis coefficient for particles with diameters of 1nm−100 nm. Furthermore,Noghrehabadi et al. ^[29] studied Nb and Nt in the range 0.1-0.5 andLein the range 1-25. In the present study,we consider Nbin the range 0.3-0.5 and Nt in the range 0.1-0.5 where Leis fixed at 2. These values are found to satisfy the rules laid out by Buongiorno ^[7] . Other values of parameters such as the velocity slip and thermal slip are arbitrarily selected and may represent the general trend. Numerical results for f^'' (0) are compared with analytical results obtained from (18) and are shown in Table 1,where they are found to be in excellent agreement,cementing the validity of the present numerical results.

Tables 2 and 3 and Figs. 2-5 show the variations of the local Nusselt number Nu_xRe_x^-1/2 (representing the heat transfer rate at the surface) and the local Sherwood number Sh_xRe_x^-1/2 (representing the mass transfer rate at the surface) with σ for different values of the velocity slip parameter λ,thermal slip parameter ξ,Brownian motion parameter Nb,and thermophoresis parameterNt. As can be seen,there are more than one solution obtained for a fixed value of σ.When σ is equal to a certain value σ= σ_c,where σ_c (<0) is the critical value of σ,there is only one solution,and when σ< σ_c,mputations have been performed until the point where the solution does not converge. The critical values σ_c are shown in the figures. Based on our computations,we found that σ_c=2.250 0,2.598 4,and 3.149 3 for λ=0,0.1,and 0.25 respectively. Although only the first solution is stable and physically realizable ^[24,25,26] ,the second solution is of mathematical interest as the differential equations are concerned. From these tables and figures,the local Nusselt number and the local Sherwood number change with the variations of λ,ξ,Nt,and Nb.Generally,while the local Nusselt number decreases asσincreases,the local Sherwood number increases with increasing σ. Further,according to Figs. 2−5,the local Nusselt number and the local Sherwood number change monotonically with the change in λ. The range of σfor which the solution exists increases as λincreases. For an increasing slip parameter at the boundary,the generation of vorticity for shrinking velocity is slightly reduced ^[23] . Therefore,with the imposed suction,that vorticity remains confined in the boundary layer region for larger shrinking velocity (i.e.,σ< 0). Hence,the steady solution is possible for some large values of σ.

Contrary to λ,σ_c remains unchanged for different values of ξ. This is clear from (8)-(11) where the thermal field does not affect the flow field. The increasingξyields a decreasing surface heat transfer (see Fig. 4) and encourages mass exchange efficiency at the surface (see Fig. 5). In both figures,we can see that the values change monotonically with the increasing stretching velocity. The effect occurring in surface heat transfer is greater than the effect observed in mass transfer at the surface. For an increase of 50% inξ,the local Nusselt number reduces by 30% while the mass exchange efficiency increases by only 6%. Despite the increasing mass transfer rate,the large drop in the heat transfer rate may prove to be detrimental to the performance of numerous heat transfer heat devices.

Tables 2 and 3 depict the effects of thermophoresis and Brownian motion parameter on the local Nusselt number and the local Sherwood number. It is seen from the tables that increasing the thermophoresis parameterNtis to reduce both the local Nusselt and Sherwood numbers. It is noted thatNthas more significant effect on the local Sherwood number than the local Nusselt number. Applying different Brownian motion parameterNbto the problem however produces several outcomes. As stated previously,in general,the local Nusselt number decreases with increasing σ. Nevertheless,it seems that for Nb=0.3 and Nb=0.4,the local Nusselt number increases with increasing σ. On the other hand,increasingNbresults in an increase in the local Sherwood number but reduces the surface heat transfer (i.e.,local Nusselt number). AsNb andNtincrease,the thermal boundary layer thickness increases too. The thickening boundary layer causes the local Nusselt number to drop. Furthermore,it is observed that for all values of parameters,we obtain θ^' (0)>0,which indicates that the heat is transferred from the hot surface to the cool fluid.

The samples of velocity,temperature,and concentration profiles are included in Figs 6-8. These profiles satisfy the far field boundary condition (11) asymptotically,which support the numerical results obtained besides supporting the existence of dual solutions shown in Figs 2-5. Figure 6 shows that for an increased λ,the velocity increases. However,in Fig. 7,it is observed that the temperature decreases as λincreases. Due to the dependency of concentration on the temperature field,it is expected that higher λprevents concentration from penetrating further, and hence the concentration decreases as λincreases as seen in Fig. 8.

4 Conclusions

The boundary layer flow of a nanofluid past a stretching/shrinking sheet with a hydrodynamic and thermal slip boundary conditions is studied. Numerical solutions to the governing equations are obtained using a shooting method. The values of the skin friction coefficient and the local Nusselt number decrease as the stretching/shrinking parameter increases. On the other hand,the local Sherwood number increases with increasing the stretching/shrinking parameter. The range of the stretching/shrinking parameter for which the solution exists increases as velocity slip parameter λincreases. The local Nusselt number diminishes as thermal slip parameterξincreases. As opposed to this,the local Sherwood number increases with increasing the thermal slip parameter. Increasing the thermophoresis parameter is to decrease the local Nusselt number and the local Sherwood number. Furthermore,an increase in the Brownian motion parameter results in increasing manner of the local Sherwood number,but decreases the local Nusselt number.

Acknowledgements The authors wish to thank the anonymous reviewers for their valuable comments and suggestions.