Appl. Math. Mech. -Engl. Ed. DOI 10.1007/s10483-015-1977-6 c

Shanghai University and Springer-Verlag Berlin Heidelberg 2015

Applied Mathematics and Mechanics (English Edition)

Second-order slip MHD flow and heat transfer of nanofluids with thermal radiation and chemical reaction∗ Jing ZHU1,† , Liu ZHENG1 , Liancun ZHENG1 , Xinxin ZHANG2 1. School of Mathematica and Physics, University of Science and Technology Beijing, Beijing 100083, China; 2. School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China

Abstract The effects of the second-order velocity slip and temperature jump boundary conditions on the magnetohydrodynamic (MHD) flow and heat transfer in the presence of nanoparticle fractions are investigated. In the modeling of the water-based nanofluids containing Cu and Al2 O3 , the effects of the Brownian motion, thermophoresis, and thermal radiation are considered. The governing boundary layer equations are transformed into a system of nonlinear differential equations, and the analytical approximations of the solutions are derived by the homotopy analysis method (HAM). The reliability and efficiency of the HAM solutions are verified by the residual errors and the numerical results in the literature. Moreover, the effects of the physical factors on the flow and heat transfer are discussed graphically. Key words nanofluid, velocity slip, temperature jump, homotopy analysis method (HAM), heat and mass transfer, magnetohydrodynamic (MHD) flow Chinese Library Classification O241.81 2010 Mathematics Subject Classification

34A25, 76D10, 80A20

Nomenclature u, v, a, b, c, l, T, η, θ, φ, µ, ν, B0 , M, R,

velocity components along the x-axis; velocity components along the y-axis; constants; reference length of the sheet; temperature inside the boundary layer; similarity variable; dimensionless temperature; dimensionless concentration; dynamic viscosity; kinematic viscosity of the fluid; constant magnetic flux density; magnetic parameter; radiation parameter;

P r, Ec, C, D, σ, ρ, Kn , λ, Cp , fw , knf , ρnf , Tw ,

Prandtl number; Eckert number; fluid concentration; coefficient of the mass diffusivity; electrical conductivity; fluid density; Knudsen number; molecular mean free path; effective heat capacity; suction/injection parameter; thermal conductivity of the nanofluid; density of the nanofluid; uniform temperature of the fluid;

∗ Received Sept. 16, 2014 / Revised Jan. 21, 2015 Project supported by the National Natural Science Foundation of China (Nos. 51276014 and 51476191) and the Fundamental Research Funds for the Central Universities (No. FRF-BR-12-004) † Corresponding author, E-mail: [email protected]

2 Cw , Le, NT ,

1

Jing ZHU, Liu ZHENG, Liancun ZHENG, and Xinxin ZHANG uniform concentration of the fluid; Lewis number; thermophoresis parameter;

NB , K1 , K2 , L1 , L2 ,

Brownian motion parameter; velocity slip parameters; temperature jump parameters.

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 thickness 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 convection 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 unconventional 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 derivatives. 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 Al2 O3 past a permeable stretching sheet. Moreover, the combined effects of

Second-order slip MHD flow and heat transfer of nanofluids

3

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-Al2 O3 /water-Cu can be calculated as follows: ρnf = (1 − ϕ)ρf + ϕρs ,

(1)

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

knf . (ρCp )nf

(2)

The effective heat capacity Cp of the nanofluid is (ρCp )nf = (1 − ϕ)(ρCp )f + ϕ(ρCp )s .

(3)

The thermal conductivity of the nanofluid knf satisfies ks + 2kf − 2ϕ(kf − ks ) knf = . kf ks + 2kf + ϕ(kf − ks )

(4)

The effective dynamic viscosity of the nanofluid is µnf =

µf 5

(1 − ϕ) 2

,

(5)

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: 5

ϕ1 = (1 − ϕ) 2 ,

(6)

ϕ2 = 1 − ϕ + ϕ

ρs , ρf

(7)

ϕ3 = 1 − ϕ + ϕ

(ρCp )s . (ρCp )f

(8)

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 Uw = ax, 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 Al2 O3 , whose thermophysical properties[2] are shown in Table 1. The velocity slip model used in this paper is valid, and is given by Uslip =

2  3 − ǫl3 3 1 − l2  ∂u 1  4 2(1 − l2 )  ∂ 2 u − λ − l + λ 2, 3 ǫ 2 Kn ∂y 4 Kn2 ∂y

(9)

4

Jing ZHU, Liu ZHENG, Liancun ZHENG, and Xinxin ZHANG

where Uslip is the velocity slip at the wall, l = min

 1  ,1 , Kn

and ǫ is the momentum accommodation coefficient satisfying 0 6 ǫ 6 1.

Fig. 1

Table 1

Physical model and coordinate system of problem

Thermo-physical properties of water and nanoparticles[2] ρ/(kg·m−3 )

Cp /(J·kg−1 · k−1 )

k/(W·m−1 · K−1 )

Pure water

997.1

4 179

0.613

Cu

8 933

385

401

Al2 O3

3 970

765

40

The governing boundary layer equations for the problem are ∂u ∂v + = 0, ∂x ∂y u

(10)

∂u ∂u ∂ 2 u σB02 +v =ν 2 − u ∂x ∂y ∂y ρnf

(11)

with the boundary conditions   u(x, 0) = Uw + Uslip ,

(12)

  u(x, ∞) → U∞ = 0,

(13)



v(x, 0) = Vw ,



v(x, ∞) = 0,

where ν is the kinematic viscosity defined by ν=

µnf . ρnf

Second-order slip MHD flow and heat transfer of nanofluids

5

The mathematical analysis of the problem is simplified by introducing the following dimensionless variables:  r √ a    y, ψ = aνxf (η), η=   ν (14)   ∂ψ ∂ψ   , v=− . u = ∂y ∂x

Substituting Eq. (14) into Eqs. (10)–(11), we can obtain the following ordinary differential equation with respect to the dimensionless variable η: f ′′′ + ϕ1 ϕ2 (f f ′′ − f ′2 ) − ϕ1 M f ′ = 0.

Then, the boundary conditions in Eqs. (12)–(13) become  Vw   f (0) = − √ ,   aν      r a ′′ a ′ f (0) = 1 + K f (0) + K2 f ′′′ (0),  1   ν ν       ′ f (∞) = 0,

(15)

(16)

where M is the magnetic number defined by

M=

σB02 , ρf a

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-Al2 O3 /water-Cu can be written as follows: u

∂T ∂T +v ∂x ∂y

= αnf

∂ 2T 1 ∂qr µnf  ∂u 2 σB02 2 − + + u 2 ∂y (ρC)f ∂y (ρCp )nf ∂y (ρCp )nf

 ∂C ∂T DT  ∂T 2  + , + τ DB ∂y ∂y T∞ ∂y  ∂T ∂2T   + L2 2 ,  T (x, 0) = Tw + L1 ∂y ∂y    T (x, ∞) → T∞ ,

(17)

(18)

where T∞ is the temperature far away from the sheet, DB and DT are the Brownian diffusion coefficient and the thermophoresis diffusion coefficient, respectively, and Tw = T∞ + b

 x 2 l

.

6

Jing ZHU, Liu ZHENG, Liancun ZHENG, and Xinxin ZHANG

Substituting θ(η) defined by θ(η) =

T − T∞ Tw − T∞

into Eqs. (17) and (18) yields  4  1 1 + R θ′′ + P rkϕ3 (f θ′ − 2f ′ θ) + P rEcf ′′2 3 ϕ1 +

1 P rM Ecf ′2 + P rNB φ′ θ′ + P rNT θ′2 ϕ1

= 0,

where

(19)

 r a ′ a   θ (0) + L2 θ′′ (0),  θ(0) = 1 + L1 ν ν    θ(∞) = 0,

(20)

 ν  Pr = ,   α  nf      3  4σ ∗ T∞    R= ,  ∗  3k knf        Uw2 Ec = , Cp (Tw − T∞ )         (ρCp )nf DB (Cw − C∞ )   NB = ,   (ρC)f ν         (ρCp )nf DT (Tw − T∞ )   NT = . (ρC)f ν

2.4 Mass transfer analysis The concentration boundary layer equation for the nanofluid is u

∂C ∂C ∂ 2C DT ∂ 2 T +v = DB 2 + , ∂x ∂y ∂y T∞ ∂y 2

(21)

and the boundary conditions are   C(x, 0) = Cw , 

C(x, ∞) → C∞ ,

where Cw is the concentration of the fluid defined by  x 2 . Cw = C∞ + C l Substituting the equation φ(η) =

C − C∞ Cw − C∞

(22)

Second-order slip MHD flow and heat transfer of nanofluids

7

into Eqs. (21) and (22), we get  NT ′′  φ′′ − 2Lef ′φ + Lef φ′ + θ = 0,    NB   φ(0) = 1,       φ(∞) = 0,

where

Le =

3

(23)

ν . DB

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 f0 =

2k1 − 3k2 2k1 − 3k2 −η e + ηe−η + fw − , 1 + k1 − k2 1 + k1 − k2

 1 + m1 − 2m2 −η  e ,  θ0 = ηe−η + 1 + m1 − m2   φ0 = e−η ,

(24)

(25)

where  r a   k = K1 ,    1 ν

The linear operators are

r    a   m1 = L 1 , ν

The auxiliary linear operators are

a k2 = K2 , ν a m2 = L 2 . ν

 Lf = f ′′′ − f ′ ,      Lθ = θ′′ − θ′ ,      L = φ′′ − φ′ . φ

 Lf (C1 + C2 e−η + C3 ηe−η ) = 0,      Lθ (C5 e−η + C6 ηe−η ) = 0,      Lφ (C7 + C8 e−η ) = 0,

where Ci (i = 1, 2, · · · , 8) are constants.

(26)

(27)

8

Jing ZHU, Liu ZHENG, Liancun ZHENG, and Xinxin ZHANG

The zeroth-order deformation equations are constructed as follows:

(1 − q)Lf (F (η; q) − f0 (η)) = qhf Hf (η)Nf (F (η; q)),

(28)

(1 − q)Lθ (Θ(η; q) − θ0 (η)) = qhθ Hθ (η)Nθ (Θ(η; q)),

(29)

(1 − q)Lφ (Φ(η; q) − φ0 (η)) = qhφ Hφ (η)Nφ (Φ(η; q)).

(30)

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 Lf (fm (η) − χm fm−1 (η)) = qhf Hf (η)Rm (η),

(31)

Lθ (θm (η) − χm θm−1 (η)) = qhθ Hθ (η)Sm (η),

(32)

Lφ (φm (η) − χm φm−1 (η)) = qhφ Hφ (η)Gm (η)

(33)

with the boundary conditions  fm (0) = 0,        f ′ (∞) = 0, m

(34)

  r   a ′′′ a ′′  ′   fm (0) = K1 f (0) + K2 fm (0), ν m ν  r a ′′ a ′    θm (0) = L1 θm (0) + L2 θm (0), ν ν    θm (∞) = 0,

  φm (0) = 0, 

φm (∞) = 0,

(35)

(36)

Second-order slip MHD flow and heat transfer of nanofluids

9

where     0, m 6 1,    χ =  m    1, m > 2,     1 ∂ m F (η; q)    fm = , m! ∂η m q=0  1 ∂ m Θ(η; q)   θm = ,   m! ∂η m q=0

4

(37)

(38)

 m    φm = 1 ∂ Φ(η; q) . m! ∂η m q=0

Convergence of HAM solutions

Liao[16] has pointed out that the convergence and its speed depend on the auxiliary parameters hf , hθ , and hφ to a great extent. It is straightforward to choose a proper value of h which ensures that the solutions are convergent. From Figs. 2–4, we can obtain the valid ranges as follows:  − 1.2 6 hf 6 −0.5,      − 0.4 6 hθ 6 −0.2,      − 0.8 6 h 6 −0.3. φ

Fig. 2

h-curve of f ′′ (0) when M = Re = Ec = P r = K = 1

Fig. 3

h-curve of θ′ (0) when M = Re = Ec = P r = K = 1

We can also use the residual error to help us to find the proper h. In this paper, we define the residual error Em,f [17] as follows: Z ∞ Em,f = (f ′′′ + ϕ1 ϕ2 (f f ′′ − f ′2 ) − ϕ1 M f ′ )2 dη. (39) 0

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).

10

Jing ZHU, Liu ZHENG, Liancun ZHENG, and Xinxin ZHANG

Fig. 4

h-curve of φ′ (0) when M = Re = Ec = P r = K = 1

Table 2

Fig. 5

Logarithms of residual errors with HAM approximation order m

Values of |f ′′ (0)| for various M and ϕ with fw = k1 = k2 = 0 |f ′′ (0)|

M

0.0

0.5

1.0

2.0

5

Cu

ϕ

Al2 O3

Present

Ref. [2]

Present

Ref. [2]

0.05

1.122 106

1.108 920

1.019 749

1.005 377

0.10

1.187 811

1.174 746

1.013 445

0.998 772

0.15

1.212 103

1.208 862

0.997 010

0.981 845

0.20

1.231 542

1.218 043

0.962 053

0.955 919

0.05

1.298 111

1.292 102

1.210 541

1.204 412

0.10

1.334 919

1.328 249

1.183 002

1.175 484

0.15

1.347 680

1.339 554

1.147 384

1.138 892

0.20

1.339 309

1.330 356

1.105 146

1.095 444

0.05

1.455 668

1.452 361

1.378 211

1.374 930

0.10

1.469 857

1.465 763

1.332 973

1.328 901

0.15

1.463 551

1.458 582

1.281 763

1.276 766

0.20

1.439 812

1.433 898

1.225 251

1.219 104

0.05

1.729 626

1.728 872

1.665 112

1.664 356

0.10

1.709 041

1.707 892

1.593 160

1.591 984

0.15

1.673 096

1.671 398

1.517 077

1.515 336

0.20

1.623 683

1.621 264

1.437 316

1.434 799

Results and discussion

Figures 6–11 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

Second-order slip MHD flow and heat transfer of nanofluids

Fig. 6

Variations of f (η) with k1 at R = P r = M = Ec = 1 and fw = 0

Fig. 7

Variations of f (η) with k2 at R = P r = M = Ec = 1 and fw = 0

Fig. 8

Variations of θ(η) with k1 at R = P r = M = Ec = 1 and fw = 0

Fig. 9

Variations of θ(η) with k2 at R = P r = M = Ec = 1 and fw = 0

Fig. 10

Variations of φ(η) with k1 at R = P r = M = Ec = 1 and fw = 0

Fig. 11

Variations of φ(η) with k2 at R = P r = M = Ec = 1 and fw = 0

11

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

12

Jing ZHU, Liu ZHENG, Liancun ZHENG, and Xinxin ZHANG

Fig. 12

Variations of θ(η) with m1 at R = P r = M = Ec = 1 and fw = 0

Fig. 13

Variations of θ(η) with m2 at R = P r = M = Ec = 1 and fw = 0

when both M and fw increase. However, the skin friction coefficient decreases when the firstorder velocity slip parameter k1 increases, while increases when the second-order velocity slip parameter k2 increases. Table 3

Comparison of skin friction coefficient |f ′′ (0)| for various k1 with fw = M = k2 = 0 Ref. [4]

|f ′′ (0)| Ref. [18]

Present

0.0

1.000 0

1.000 000

1.000 000

0.1

0.872 1

0.872 082

0.875 758

0.2

0.776 4

0.776 377

0.779 134

0.5

0.591 2

0.591 195

0.592 776

2.0

0.284 0

0.283 981

0.285 827

k1

Table 4

Results of skin friction coefficient |f ′′ (0)| for various fw , M , k1 , and k2

M

fw

k1

k2

|f ′′ (0)|

1.0

0.5

0.0

0.0

1.689 86

0.5

0.5

0.0

0.0

1.513 54

1.5

0.5

0.0

0.0

1.852 54

1.0

0.0

0.0

0.0

1.416 78

1.0

0.2

0.0

0.0

1.520 90

1.0

0.7

0.0

0.0

1.815 55

1.0

0.5

0.4

0.0

0.958 25

1.0

0.5

0.7

0.0

0.732 75

1.0

0.5

1.0

0.0

0.595 05

1.0

0.5

0.4

0.1

1.149 33

1.0

0.5

0.4

0.2

1.480 67

1.0

0.5

0.4

0.3

2.372 75

Figures 14 and 15 show the effects of the parameter fw on the velocity and the temperature field. From these figures, we can see that when fw increases, the velocity increases. This means

Second-order slip MHD flow and heat transfer of nanofluids

13

that the boundary layer thickness increases when fw increases. However, the temperature profiles follow an opposite trend.

Fig. 14

Velocity profiles for various fw when M = R = P r = Ec = 1

Fig. 15

Temperature profiles for various fw when M = R = P r = Ec = 1

Figure 16 shows the effects of both the radiation parameter R and the magnetic field parameter 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 NB and the thermophoresis parameter NT on the local Nusselt number |θ′ (0)|. From the figure, we can see that when both the Brownian motion parameter NB and the thermophoresis parameter NT increase, the heat transfer rate on the surface decreases.

Fig. 16

Variations of local Nusselt number |θ′ (0)| with R for different M

Fig. 17

Variations of local Nusselt number |θ′ (0)| with NT for different NB

Figure 18 draws the variations of the local Sherwood number |φ′ (0)| in response to the changes in NB and NT . The graph shows that the local Sherwood number |φ′ (0)| increases when NB increases, while decreases when NT 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

14

Jing ZHU, Liu ZHENG, Liancun ZHENG, and Xinxin ZHANG

Fig. 18

Variations of |φ′ (0)| with NT for different NB

Table 5

Results of local Nusselt number |θ′ (0)| and Sherwood number |φ′ (0)| for different values of P r, Ec, R, k1 , k2 , m1 , and m2 at Le = 5, M = 1, fw = 0, NB = 0.5, and NT = 0.5

Fig. 19

Concentration profiles for various values of Le when R = P r = M = 1 and fw = 0

Pr

R

Ec

k1

k2

m1

m2

|θ ′ (0)|

|φ′ (0)|

1.00

0.2

1.0

1.0

0.33

1.0

0.33

0.215 3

1.571 6

0.72

0.2

1.0

1.0

0.33

1.0

0.33

0.217 4

1.569 2

1.50

0.2

1.0

1.0

0.33

1.0

0.33

0.208 2

1.574 3

5.00

0.2

1.0

1.0

0.33

1.0

0.33

0.144 0

1.574 7

1.00

0.5

1.0

1.0

0.33

1.0

0.33

0.218 3

1.572 6

1.00

0.7

1.0

1.0

0.33

1.0

0.33

0.218 9

1.573 1

1.00

1.0

1.0

1.0

0.33

1.0

0.33

0.219 0

1.573 8

1.00

0.2

1.5

1.0

0.33

1.0

0.33

0.126 2

1.575 6

1.00

0.2

2.0

1.0

0.33

1.0

0.33

0.032 9

1.576 9

1.00

0.2

2.5

1.0

0.33

1.0

0.33

0.016 5

1.577 2

1.00

0.2

1.0

0.0

0.33

1.0

0.33

0.581 0

2.526 7

1.00

0.2

1.0

0.4

0.33

1.0

0.33

0.259 8

2.322 3

1.00

0.2

1.0

0.7

0.33

1.0

0.33

0.142 1

1.825 0

1.00

0.2

1.0

1.0

0.00

1.0

0.33

0.250 6

1.367 5

1.00

0.2

1.0

1.0

0.10

1.0

0.33

0.237 1

1.420 1

1.00

0.2

1.0

1.0

0.20

1.0

0.33

0.228 4

1.477 7

1.00

0.2

1.0

1.0

0.33

0.0

0.33

0.364 5

1.572 9

1.00

0.2

1.0

1.0

0.33

0.5

0.33

0.270 7

1.572 2

1.00

0.2

1.0

1.0

0.33

1.5

0.33

0.178 5

1.571 4

1.00

0.2

1.0

1.0

0.33

1.0

0.00

0.243 6

1.569 4

1.00

0.2

1.0

1.0

0.33

1.0

0.50

0.202 1

1.572 8

1.00

0.2

1.0

1.0

0.33

1.0

0.70

0.187 4

1.574 3

Second-order slip MHD flow and heat transfer of nanofluids

15

|φ′ (0)| with the velocity and thermal slip parameters by fixing other governing parameters. 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, NB , and NT . From the table, we can see that |φ′ (0)| increases when Le and NB increase, while decreases when the thermophoresis parameter increases. Table 6

Results of local Nusselt number |θ′ (0)| and Sherwood number |φ′ (0)| for different values of Le, NB , and NT at M = 1, fw = 0, R = 0.2, k1 = 1, k2 = 1/3, m1 = 1, and m2 = 1/3

Le

NB

NT

|θ ′ (0)|

|φ′ (0)|

5

0.2

0.2

0.250 1

1.571 7

10

0.2

0.2

0.246 5

2.391 2

15

0.2

0.2

0.243 3

2.997 4

20

0.2

0.2

0.235 0

3.501 6

5

0.1

0.2

0.264 4

1.224 0

5

0.3

0.2

0.239 6

1.762 5

5

0.4

0.2

0.229 1

1.881 5

5

0.5

0.2

0.218 2

1.962 4

5

0.2

0.1

0.254 7

1.881 6

5

0.2

0.3

0.249 8

1.367 6

5

0.2

0.5

0.244 1

1.117 0

6

Conclusions

The effects of various parameters, such as the magnetic parameter M , the radiation parameter 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.

References [1] Das, K. Slip flow and convective heat transfer of nanofluids over a permeable stretching surface. Computers and Fluids, 64, 34–42 (2012) [2] Turkylimazoglu, M. Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids. Chemical Engineering Science, 84, 182–187 (2012) [3] Turkylimazoglu, M. and Pop, I. Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect. International Journal of Heat and Mass Transfer, 59, 167–171 (2013)

16

Jing ZHU, Liu ZHENG, Liancun ZHENG, and Xinxin ZHANG

[4] Ibrahim, W. and Shankar, B. MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions. Computers and Fluids, 75, 1–10 (2013) [5] Nandy, S. K. and Mahapatra, T. R. Effects of slip and heat generation/absorption on MHD stagnation flow of nanofluid past a stretching/shrinking surface with convective boundary conditions. International Journal of Heat and Mass Transfer, 64, 1091–1100 (2013) [6] Sahoo, B. Effects of slip, viscous dissipation and Joule heating on the MHD flow and heat transfer of a second grade fluid past a radially stretching sheet. Applied Mathematics and Mechanics (English Edition), 31(2), 159–173 (2010) DOI 10.1007/s10483-010-0204-7 [7] Zhu, J., Zheng, L. C., and Zhang, Z. G. Effects of slip condition on MHD stagnation-point flow over a power-law stretching sheet. Applied Mathematics and Mechanics (English Edition), 31(4), 439–448 (2010) DOI 10.1007/s10483-010-0404-z [8] Mansur, S., Ishak, A., and POP, I. Flow and heat transfer of nanofluid past stretching/shrinking sheet with partial slip boundary conditions. Applied Mathematics and Mechanics (English Edition), 35(11), 1401–1410 (2014) DOI 10.1007/s10483-014-1878-7 [9] Liao, S. J. The Proposed Homopoty Analysis Technique for the Solution of Nonlinear Problems (in Chinese), Ph. D. dissertation, Shanghai Jiao Tong University, Shanghai (1992) [10] Yabushita, K., Yamashita, M., and Tsubo, K. An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. Journal of Physics, A: Mathematical and Theoretical, 40, 8403–8416 (2007) [11] Marinca, V. and Herisanu, N. Application of optional homotopy asymptotic method for solving nonlinear equations arising in heat transfer. International Communications in Heat and Mass Transfer, 35, 710–715 (2008) [12] Marinca, V. and Herisanu, N. Application of optional homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Applied Mathematics Letters, 22, 245–251 (2009) [13] Zhao, M. M. The Further Discussion for Homotopy Analysis Method and Their Modification (in Chinese), Ph. D. dissertation, Lanzhou University, Lanzhou (2009) [14] Niu, Z. A one-step optional homotopy analusis method for nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 15, 2026–2036 (2010) [15] Zhu, W. Extension and Implementation of the Homotopy Analysis Method (in Chinese), Ph. D. dissertation, East China Normal University, Shanghai (2011) [16] Liao, S. J. Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman Hall/CRC, Boca Raton (2003) [17] Fan, T. Applications of Homotopy Analysis Method in Boundary Layer Flow and Nanofluid Flow Problems (in Chinese), Ph. D. dissertation, Shanghai Jiao Tong University, Shanghai (2012) [18] Hayat, T. and Qasim, M. MHD flow and heat transfer over permeable stretching sheet with slip conditions. International Journal for Numerical Methods in Fluids, 66, 963–975 (2011)