Nomenclature
u,   velocity components along the x-axis;    v,   velocity components along the y-axis;
a, b, c,   constants;    l,   reference length of the sheet;
T,   temperature inside the boundary layer;    η,   similarity variable;
θ,   dimensionless temperature;    Φ,   dimensionless concentration;
μ,   dynamic viscosity;    ν,   kinematic viscosity of the fluid;
B₀,   constant magnetic flux density;    M,   magnetic parameter;
R,   radiation parameter;    Pr,   Prandtl number;
Ec,   Eckert number;    C,   fluid concentration;
D,   coefficient of the mass diffusivity;    σ,   electrical conductivity;
ρ,   fluid density;    Kn,   Knudsen number;
λ,   molecular mean free path;    C_p,   effective heat capacity;
f_w,   suction/injection parameter;    k_nf ,   thermal conductivity of the nanofluid;
ρ_nf ,   density of the nanofluid;    Tw,   uniform temperature of the fluid;
Cw,   uniform concentration of the fluid;    Le,   Lewis number;
N_T,   thermophoresis parameter;    N_B,   Brownian motion parameter;
K₁,K₂,   velocity slip parameters;    L₁,L₂,   temperature jump parameters.


1 Introduction

The no-slip boundary condition is known as the central tenet of the Navier-Stokes theory. However,the partial velocity slip and temperature jump boundary conditions may occur in microdevices when the fluid is particulate,e.g.,emulsions,suspensions,foams,and polymer solutions. Many scholars have investigated the effects of the velocity slip and temperature jump on flow and heat transfer. Das^[1] investigated the slip flow and convective heat transfer over a permeable stretching sheet,and pointed out that the thermal boundary layer thick- ness increased while the suction velocity decreased when the velocity slip and the injection velocity increased. Turkylimazoglu^[2] studied the heat and mass transfer characteristics of the magnetohydrodynamic (MHD) nanofluid flow over a permeable stretching/shrinking surface. Turkylimazoglu and Pop^[3] considered the heat and mass transfer of the unsteady natural con- vection flow of some nanofluids past a vertical infinite flat plate with the radiation effects. Ibrahim and Shankar^[4] analyzed the boundary layer flow and heat transfer over a permeable stretching sheet with the effects of the magnetic field,the slip boundary condition,and the thermal radiation. They concluded that the thermal boundary layer thickness decreased when the slip parameter and the Prandtl number increased. Nandy and Mahapatra^[5] analyzed the effects of the velocity slip and heat generation/absorption on the MHD stagnation-point flow and heat transfer in the presence of nanoparticle fractions. They found that the flow velocity, temperature field,and nanoparticle concentration profiles were strongly affected by the slip parameter. Sahoo^[6] considered the steady and laminar flow and heat transfer of an electrically conducting second-grade fluid over a stretching sheet with partial slip. Zhu et al.^[7] studied the effects of the slip condition on the MHD stagnation-point flow over a power-law stretching sheet. Mansur et al.^[8] considered the flow and heat transfer of a nanofluid with the partial slip boundary conditions.

It is well-known that most boundary layer transmission problems are described by a set of nonlinear partial differential equations. However,due to the strongly nonlinear and uncon- ventional nature of these problems,the solving processes are extraordinarily complex,and the effective solutions are hardly obtained. Recently,various methods have been tried to solve those problems. The homotopy analysis method (HAM) was originally put forward by Liao^[9]. It has been used and developed by many experts and scholars,and has been proved to be an effective mathematical method to solve weakly nonlinear problems. Yabushita et al.^[10] presented an analytic solution of projectile motion with the HAM. Marinca and Herisanu^{[11, 12]} proposed the Marinca optional HAM. Zhao^[13] modified the HAM for solving nonhomogeneous differential equations. Niu^[14] presented the one-step optional HAM for nonlinear differential equations. Zhu^[15] implemented the HAM,and developed a software package in the computer algebraic system MAPLE,which can solve ordinary differential equations,partial differential equations, coupled systems,and some special equations with undefined parameters and fractional deriva- tives.

In all the above mentioned studies,most numerical and experimental studies on flow and heat transfer were done with lower-order velocity slip and temperature jump conditions,and little attention has been given to the effects of higher-order velocity slip and temperature jump conditions on the flow of a nanofluid. Nanofluids have attracted considerable attention due to their significant applications. Nanofluids are produced by solid nanoparticles in a base fluid. The main purpose of this paper is to investigate the effects of the second-order velocity slip and temperature jump on the flow and heat transfer characteristics of a water-based nanofluid containing Cu and Al₂O₃ past a permeable stretching sheet. Moreover,the combined effects of other various parameters on the boundary layer flow and heat transfer are also examined.

2 Mathematical analysis 2.1 Nanofluid properties

The effective density of the nanofluid containing water-Al₂O₃/water-Cu can be calculated as follows:

where ϕ is the solid volume fraction of the nanoparticles. The thermal diffusivity of the nanofluid is

The effective heat capacity C_p of the nanofluid is

The thermal conductivity of the nanofluid k_nf satisfies

The effective dynamic viscosity of the nanofluid is

where nf represents the thermophysical properties of the nanofluids,and f and s represent the properties of the base fluid and the nano-solid particles,respectively. The parameters ϕ₁,ϕ₂, and ϕ₃ are defined as follows: 2.2 Flow analysis

Consider a two-dimensional steady state boundary layer flow of a nanofluid over a permeable stretching sheet. The stretching sheet velocity of the sheet is

where a is a constant. The coordinate system and the scheme of the problem are shown in Fig. 1.

The fluid is a water-based nanofluid containing different types of nanoparticles,i.e.,Cu and Al₂O₃,whose thermophysical properties^[2] are shown in Table 1. The velocity slip model used in this paper is valid,and is given by

where U_slip is the velocity slip at the wall, and ε is the momentum accommodation coefficient satisfying The governing boundary layer equations for the problem are with the boundary conditions where ν is the kinematic viscosity defined by

The mathematical analysis of the problem is simplified by introducing the following dimen- sionless variables:

Substituting Eq. (14) into Eqs. (10)-(11),we can obtain the following ordinary differential equation with respect to the dimensionless variable η:

Then,the boundary conditions in Eqs. (12)-(13) become where M is the magnetic number defined by K₁ is the first-order velocity slip parameter,and K₂ is the second-order velocity slip parameter. 2.3 Heat transfer analysis

The thermal boundary layer equation for the nanofluid containing water-Al₂O₃/water-Cu can be written as follows:

where T_∞ is the temperature far away from the sheet,D_B and D_T are the Brownian diffusion coefficient and the thermophoresis diffusion coefficient,respectively,and Substituting θ(η) defined by into Eqs. (17) and (18) yields where 2.4 Mass transfer analysis

The concentration boundary layer equation for the nanofluid is

and the boundary conditions are where Cw is the concentration of the fluid defined by

Substituting the equation

into Eqs. (21) and (22),we get where 3 Application of HAM

Due to the strongly nonlinear and unconventional nature of the above problems,we choose the HAM to get the approximately analytical solutions in this paper.

The initial approximations are

where

The linear operators are

The auxiliary linear operators are where C_i (i = 1,2,· · · ,8) are constants.

The zeroth-order deformation equations are constructed as follows:

To get the mth-order deformation equations,we first differentiate Eqs. (28)-(30) m-times (m = 1,2,3,· · · ) with respect to q at q = 0,and then divide the resulting expression by m!,so that with the boundary conditions where 4 Convergence of HAM solutions

Liao^[16] has pointed out that the convergence and its speed depend on the auxiliary param- eters h_f ,h_θ,and h_Φ to a great extent. It is straightforward to choose a proper value of h which ensures that the solutions are convergent. From Figs. 2,3,4,we can obtain the valid ranges as follows:

We can also use the residual error to help us to find the proper h. In this paper,we define the residual error E_m,f ^[17] as follows:

Figure 5 shows that when the order of the HAM approximation becomes higher,the residual error becomes smaller. This means that the higher the order of the HAM approximation is,the more accurate the result becomes. Moreover,it is seen that the present results are in excellent agreement with those given by Turkylimazoglu^[2] (see Table 2).

5 Results and discussion

Figures 6，7，8，9，10，11， display the variations of the velocity,the temperature,and the concentration with the first-order velocity slip parameter k₁ and the second-order velocity slip parameter k₂. We can conclude that,when k₁ increases,both the velocity and the temperature profiles decrease,while the concentration profile increases. When k₂ increases,both the velocity and the temperature profiles increase,while the concentration profile decreases. It also reveals that the thermal boundary layer thickness decreases when k₁ increases,while increases when k₂ increases.

Figures 12 and 13 illustrate the effects of the temperature jump parameters m₁ and m₂ on the temperature in the boundary layer. From the figures,we can see that the thermal boundary layer thickness decreases when m₁ and m₂ increase.

Tables 3 and 4 show the skin friction coefficients with different parameters. From these tables,we can conclude that the results agree well with those of Ibrahim and Shankar^[4] and Hayat and Qasim^[18]. Moreover,we can conclude that the skin friction coefficient increases when both M and f_w increase. However,the skin friction coefficient decreases when the first- order velocity slip parameter k₁ increases,while increases when the second-order velocity slip parameter k₂ increases.

Figures 14 and 15 show the effects of the parameter f_w on the velocity and the temperature field. From these figures,we can see that when f_w increases,the velocity increases. This means that the boundary layer thickness increases when f_w increases. However,the temperature profiles follow an opposite trend.

Figure 16 shows the effects of both the radiation parameter R and the magnetic field pa- rameter M on the local Nusselt number |θ′(0)|. The graph shows that the local Nusselt number increases when M increases,while decreases when R increases. Figure 17 shows the effects of both the Brownian motion parameter N_B and the thermophoresis parameter N_T on the local Nusselt number |θ′(0)|. From the figure,we can see that when both the Brownian motion parameter N_B and the thermophoresis parameter N_T increase,the heat transfer rate on the surface decreases.

Figure 18 draws the variations of the local Sherwood number |φ′(0)| in response to the changes in N_B and N_T. The graph shows that the local Sherwood number |φ′(0)| increases when N_B increases,while decreases when N_T increases. Figure 19 demonstrates the variations of the nanoparticle concentration with respect to the change in the Lewis number Le. From the figure,we can see that when Le increases,the concentration graph decreases,and the concentration boundary layer thickness decreases. This indicates that the concentration at the surface of a sheet decreases when Le increases.

Table 5 presents the results of the local Nusselt number |θ′(0)| and the Sherwood number From the table,we can see that the local Nusselt number |θ′(0)| decreases when both m₁ and m₂ increase. However,m₁ and m₂ have few effects on the Sherwood number |φ′(0)|. Moreover, the local Nusselt number |θ′(0)| decreases when the Eckert number Ec increases.

Table 6 illustrates the variations of the heat transfer rate and the mass transfer rate with different values of Le,N_B,and N_T. From the table,we can see that |φ′(0)| increases when Le and N_B increase,while decreases when the thermophoresis parameter increases.

6 Conclusions

The effects of various parameters,such as the magnetic parameter M,the radiation parame- ter R,the velocity slip parameters k₁ and k₂,the Eckert number Ec,the Lewis number Le,and the temperature jump parameters m₁ and m₂,in the boundary layer flow of a nanofluid past a permeable stretching sheet are studied through graphs and tables in detail. Some conclusions are drawn as follows:

(i) The boundary layer thickness decreases when M increases.

(ii) The thermal boundary layer thickness increases when the Eckert number Ec increases.

(iii) The concentration graph decreases and the concentration boundary layer thickness decreases when the Lewis number parameter Le increases.

(iv) The thermal boundary layer thickness decreases when k₁ increases,while increases when k₂ increases.

(v) The thermal boundary layer thickness decreases when m₁ and m₂ increase.