1 Introduction

In recent years, the applications of non-Newtonian power-law fluids have received many researchers' attention^[1]. For example, Soundalgekar and Ramana Murty^[2] investigated heat and mass transfer of the power-law fluid over a continuous moving sheet with a uniform surface motion. Howell et al.^[3], Kumari and Nath^[4] investigated the similar problem about the powerlaw fluid. However, in Kumari and Nath's work, they considered the influence of the magnetohydrodynamic (MHD) on the power-law fluid over a continuously stretching plate. Salama^[5] studied the effects of the suction or injection of the power-law steady flow on a stretching porous surface. Djordje et al.^[6] obtained the analytical solutions and gave the qualitative discussion for the boundary-layer flow of the power-law fluid over an arbitrary two-dimensional surface considering both steady and unsteady cases using the variational principle. Huang and Lin^[7] studied the influence of the local Nusselt number and the generalized Prandtl number for a power-law fluid over a plate, and two different boundary conditions of the energy equations, which were about the uniform surface temperature and the uniform surface heat flux, were considered, respectively. Mahmoud and Megahed^[8] also investigated analytically the similar problem and considered the viscous dissipation and heat generation or absorption. Sui et al.^[9] discussed analytically the heat transfer of the mixed convection of power-law fluid along an inclined surface over a moving conveyor by the homotopy analysis method (HAM).

However, above works did not consider the slip velocity between the flow and the plate. Accounting for the slip velocity effects, Afify et al.^[10] investigated numerically the mass and heat transfer of the two-dimensional MHD flow past a porous stretching sheet with the slip velocity. Zhu et al.^[11] investigated analytically the MHD stagnation-point flow with the powerlaw stretching surface and the slip velocity by the HAM. Shojaeian and Kosar^[12] studied the convective heat transfer in both Newtonian and non-Newtonian fluids between two parallel plates considering the slip velocity analytically. The nonlinear partial slip boundary also has been studied. For example, Roux^[13] analyzed the effects of the nonlinear partial slip boundary conditions in a bounded simply connected domain on an incompressible second grade fluid. Furthermore, some similar works also have been done^[14-17].

In order to control the fluid flow and its heat transfer, the magnetic field is often applied to many practical applications. Considering the transverse magnetic field, Nejad et al.^[18] investigated numerically the mixed convection flow of electrically conducting power-law fluids along a vertical wavy surface. Rajput et al.^[19] studied the power-law fluids over a stretching plate in the presence of a magnetic field. Using a quasi-linearization approach, Naikoti and Borra^[20] studied the effects of the magnetic field on the power-law fluids over a sheet with thermal dispersion. Some other similar problems also have been done, with the variable thermal conductivity considered. For example, Lin et al.^[21] studied the Maragoni boundary layer flow and heat transfer of power-law nanofluids over a porous medium.

Motivated by the above works, the aim of this article is to discuss the effects of nonlinear slip velocity and temperature jump on the velocity and temperature distributions of the nonNewtonian pseudo-plastic power-law fluid with the variable thermal conduction on the porous stretching sheet in the presence of the magnetic field. In this work, we consider that the temperature jump has the similar form as the slip velocity. Then, with the suitable similarity transformation, the semi-analytical solutions are obtained, and the results are discussed by graphs in detail.

2 Preliminaries

Consider the steady, laminar, incompressible boundary layer fluid with the velocity slip and the temperature jump on a continuously moving porous sheet with the constant velocity U in the pseudo-plastic power-law fluid at rest. Assume that u and v are the velocity components in the x-and y-directions, which are parallel and perpendicular to the surface, respectively. T is the temperature of the flow. A magnetic field B is applied in the y-direction. In the energy equation, we also consider the non-Newtonian effects, viscous dissipation and heat generation effects. Then, the governing equations of the boundary layer for the power-law non-Newtonian fluid are

(1)

(2)

(3)

where m is the kinematic viscosity, α is the thermal diffusivity, n is the power-law viscosity index, and c_p is the specific heat at constant pressure.

The corresponding boundary conditions are^[1]

(4)

(5)

where we assume that the nonlinear partial temperature jump is similar to the slip velocity. Here, T_w and T_∞ are the temperatures at the surface and infinity, respectively. v_w is the constant permeability velocity of the wall, and λ₁ and σ₁ are the slip and jump coefficients, respectively.

Introduce the stream function Ψ(x, y) and the corresponding dimensionless variables,

(6)

Substituting Eq. (6) into Eqs. (2)-(3) yields the following ordinary equations:

(7)

(8)

The parameters in the above equations are defined as follows:

(9)

(10)

where Pr, γ, and f_w are the local modified non-Newtonian Prandtl number, the local absorption parameter ( < 0) or the heat generation parameter (> 0), and the local permeability parameter which is larger than zero for suction and less than zero for injection. M² represents the magnetic parameter, and Ec is the Eckert number.

The boundary conditions (4) and (5) become

(11)

(12)

where is the local slip parameter, and is the local temperature jump parameter. According to the similarity transformation, the local Nusselt number and local skin-friction coefficient can be obtained as

(13)

(14)

Then, , and .

3 Numerical solutions and discussion

The nonlinear coupled ordinary differential equations (7) and (8) subject to the boundary conditions (9) and (10) can be solved numerically by bvp4c with MATLAB^[22-24]. During the computational process, the infinity is replaced by the finite point, and the initial guess and the number of the initial mesh need to be adjusted according to the variation of the physical parameters. Here, we assume that the maximum residual of the numerical solution is 10^-4. The comparison of our numerical results with previous results obtained by Mahmoud^[1] is shown in Table 1. Good agreement is obtained, which validates the precision of the numerical solutions.

In the following section, the effects of different physical parameters, such as the velocity slip parameter, the temperature jump parameter, the permeability parameter, and the magnetic field parameter on the dimensionless velocity and temperature distribution, are shown in Figs. 1-8. Moreover, in order to describe the effects of non-Newtonian pseudo-plastic power-law index clearly, for every figure we give the effects of physical parameters on the temperature and velocity for different power-law indices (n=0.8, and n=0.6) at the same time.

Figures 1 and 2 give the effects of the permeability parameter on the velocity and temperature. It can be observed that the boundary layer thicknesses of the velocity and temperature both increase with the decreasing permeability parameter, which also shows that the existing suction velocity at the plate makes the boundary layer stable. As the injection coefficient increases, the velocity boundary layer thickness and the temperature boundary layer thickness increase while the wall velocity gradient and the wall temperature gradient decrease. This also indicates that injection helps the flow penetrate more into the fluid. Furthermore, near the plate, the effects of different power-law indices are very small. With the increasing distance far away from the plate, the effects of the power-law index on the velocity distribution become more obvious. This phenomenon can also be found in the temperature distribution.

Figures 3 and 4 illustrate the effects of slip parameter on the velocity and temperature distribution. It can be found that decreasing the slip parameter leads to the increasing velocity at the wall. Furthermore, as the distance is far away from the plate, the effect of the slip parameter becomes smaller gradually. The slip velocity will prevent the increase in the velocity at the wall. However, it will promote the increase in the temperature at the wall. Figure 5 shows that the power-law index and the temperature jump parameter both have dramatic effects on the temperature distribution. The increasing temperature jump parameter promotes the decrease in the temperature. It is easy to understand that the temperature jump between the wall and the fluid means the thermal contact resistance which reduces the amount of heat exchange. Then, we can conclude that the slip velocity also has the similar influence on the velocity near the plate. Because of the existence of the temperature jump, different temperature jump parameters lead to different temperatures at the plate. Furthermore, with the increasing distance from the plate, the effect of the temperature jump parameter on the temperature becomes smaller.

The illustrations of different heat generation or absorption parameters on the temperature are given in Fig. 6. With the increasing heat generation, the temperature will increase. This point agrees with practice. Furthermore, near the wall of the plate, we can neglect the influence of the power-law index since there is almost no difference for different power-law indices. However, with the increasing distance far away from the plate, the effects of the power-law index become obvious. The temperature decreases with the increasing power-law index.

Since the Prandtl number and the Eckert number only exist in the energy equation, the effects of them on the temperature are discussed in Figs. 7 and 8. It can be found that increasing the Prandtl number leads to the decreasing temperature. Physically, increasing the Prandtl number causes a decrease in the temperature boundary layer thickness and an increase in the temperature gradient near the wall. The temperature will increase with the increasing Eckert number.

Figures 9 and 10 give the effects of the magnetic field on the velocity and temperature distribution, respectively. We can find that the boundary layers of velocity and temperature both become thinner with the increasing magnetic parameter. This also shows that the magnetic parameter can control the boundary layer. The existence of magnetic field produces the Lorentz force inside the fluid, and as a result, the increase in the magnetic parameter is equal to the increase in the Lorentz force. Furthermore, the magnetic field also has similar influence on the temperature distribution. That is to say, the influence of the magnetic field on the magnetic fluid changes the heat transfer performance dramatically.

Figures 11-14 illustrate the effects of the temperature jump parameter, the slip parameter, and the magnetic field parameter on the local Nusselt number and skin friction, respectively. It can be observed that, with the increasing temperature jump and slip parameter, the Nusselt number decreases. We can find that different power-law indices results in little difference in the Nusselt number, yet it shows the nonlinear characteristics against the temperature jump. However, the slip parameter has almost the linear effect on the Nusselt number. It can be found that, with the increasing magnetic field parameter, the skin friction at the wall will increase, no matter n=0.6 or n=0.8. However, for the Nusselt number, the trends are different obviously.

4 Conclusions

This paper investigates the effects of slip velocity and temperature jump on a non-Newtonian pseudo-plastic power-law fluid over a continuously moving permeable surface. The similar equations are solved numerically by bvp4c with MATLAB, and the results are discussed in detail for different power-law fluids. Some conclusions can be obtained as follows:

(ⅰ) With the increasing injection velocity, the boundary layer thicknesses of the velocity and temperature increase.

(ⅱ) The increasing temperature jump parameter leads to the decreasing temperature near the wall. With the increasing distance away from the plate, the effect of the temperature jump parameter becomes smaller.

(ⅲ) The increasing Prandtl number leads to the thinner temperature boundary layer.

(ⅳ) With the increasing magnetic field parameter, the velocity and temperature near the wall decrease. Furthermore, the boundary layer thicknesses of velocity and temperature become thinner.

(ⅴ) The local Nusselt number and skin friction decrease with the increasing temperature jump and slip parameters, respectively.