1 Introduction

Nowadays, the study on the heat transfer and nanofluid flow inside a annular pipe has been a topic of great interest. Nanofluids are significant for the production of nanostructured materials, whose sizes are below 100t nm, the engineering of complex fluids, and the cleaning oil from solid surfaces^[1] owing to their excellent wetting and spreading behaviors. Heat transfer is very important in the high-temperature processes like gas turbines, nuclear plants, thermal energy storage, etc. Some researchers^[2-4] have investigated the effects of the slip condition in nanofluids via molecular dynamics simulations.

The slip degree at the boundary depends on a number of interfacial parameters, including the strength of the thermal roughness of the interface, the liquid-solid coupling, and the liquid densities^[5]. A second-order slip condition, which appropriately represents the non-equilibrium region near the interface, is prescribed rather than the no-slip condition and the linear Navier slip condition. First, the conventional no-slip boundary conditions at the walls may not be accurate when the dimensions are reduced to microscale. Secondly, the linear Navier slip condition performs well when it is at a sufficiently low shear rate. However, at higher shear rates, when the slip length increases rapidly, the Navier slip condition will break down. Therefore, many researchers^[6-7] proposed the nonlinear slip conditions. Thirdly, due to the comparison between the calculation results and the experimental data, the values calculated by the second-order slip boundary condition are more close to the experimental data^[8]. Therefore, many scholars investigated the effects of the velocity slip condition on the flow and heat transfer with Nu_B. Zhu et al.^[9] studied the effects of the second-order velocity slip and nanoparticle migration on the Buongiorno nanofluid flow.

Originally, the proposed models are twofold, i.e., homogeneous flow models and dispersion models. In 2006, Buongiorno^[10] certified that the homogeneous models were more suitable to predict the nanofluid heat transfer coefficient. Simultaneously, the dispersion effect was completely negligible due to the nanoparticle size. Therefore, he proposed a two-component four equation non-homogeneous equilibrium model for the convective transport in nanofluids. On the basis of this model, Sheikholeslami ed al.^[11] studied the forced convection heat transfer in a semi-annulus under the influence of a variable magnetic field. Kasaeipoor et al.^[12] studied the convection of the Cu-water nanofluid in a vented T-shaped cavity in the presence of magnetic field.

Till now, a number of works have been studied on the fluid flow and heat transfer with asymmetric heating inside a annular pipe^[13-14]. However, very limited investigation has been given to the the heat transfer of nanofluids considering the nanoparticle migration under high-order slip boundary conditions, and there is no attention on the analytic solution. Hence, in the current research, a theoretical study of fully developed convection heat transfer of the nanofluid with a uniform magnetic field inside a annular pipe is presented based on the modified Buongiorno model^[10]. It is of particular interest to study the effects of a second-order slip condition and N_BT on the hydrodynamic and thermal characteristics of the system. The analytical approximations of the solutions are derived by the homotopy analysis method (HAM). The residual error curves and h-curves are verified to the accuracy and efficiency for the HAM solutions. Furthermore, the semi-analytical relation between Nu_B and N_BT are obtained.

2 Mathematical analysis

Consider an magnetohydrodynamics, laminar, and two-dimensional flow of a nanofluid inside a annular pipe with the second-order slip condition, which is subjected to different heat fluxes at the outer wall q₂ and the inner wall q₁ such that q₁ > q₂. A two-dimensional coordinate frame is selected, where the x-axis is aligned parallel and the r-axis is normal to the walls. A modified two-component heterogeneous model is employed for the nanofluid. Consequently, the basic incompressible conservation equations of the mass, momentum, thermal energy, and nanoparticle fraction can be expressed as follows:

(1)

(2)

(3)

(4)

where U_i represents the velocity components, T is the local temperature, p is the pressure, σ is the electric conductivity, B₀ is the uniform magnetic field strength, and D_B and D_T are the Brownian diffusion and the thermophoretic diffusion coefficients, respectively. ρ, µ, k, and C_p, depending on the nanoparticle volume fractions, are the density, the dynamic viscosity, the thermal conductivity, and the specific heat capacity of the nanofluid, respectively. The relations of Buongiorno^[10], which correlate the viscosity and the thermal conductivity of the nanofluid based on the experimental data of Pak and Cho^[15], are used. The expressions are

(5)

(6)

(7)

where bf stands for the base fluid, and p stands for the particle. The thermophysical properties of Al and TiO₂ nanoparticles and the base fluid-water are provided as follows:

Assuming the hydrodynamically and thermally fully developed conditions, Eqs. (1)-(4) can be simply reduced, i.e.,

(8)

(9)

(10)

The boundary conditions for this problem can be expressed as follows:

(11)

(12)

Regarding the nanoparticle continuity equation, it is obvious that the Brownian diffusion flux and the thermophoretic diffusion flux are cancelled out everywhere. Introduce the following non-dimensional parameters:

(13)

The radiative heat flux q_r is described by the Rosseland approximation^[16] such that

(14)

where σ^* and δ are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. We assume that the temperature differences within the flow are sufficiently small so that T⁴ can be expressed as a linear function after using the Taylor series to expand T⁴ about the free stream temperature T_∞ and neglecting the higher-order terms. The result is

(15)

Equations (8)-(10) can be reduced to

(16)

(17)

(18)

The boundary conditions are

(19)

(20)

where the average value of the parameters can be calculated over the cross-section by

Then, the bulk mean dimensionless temperature u_B, the bulk mean dimensionless temperature θ_B, and the bulk mean nanoparticle volume fraction Φ_B can be obtained as follows:

(21)

The dimensionless heat transfer coefficient C_htc at the inner and the outer walls can be defined, respectively, by

(22)

(23)

The total heat transfer ratio can be expressed as

(24)

and the non-dimensional pressure drop can be defined by

(25)

3 Application of HAM

In this paper, the HAM, which has been proved to be a strong and effective mathematical method to solve highly nonlinear problems, is employed to get the series solutions. For the analytical solution of Eqs. (16)-(20), using the HAM, we can select the following initial guess solutions:

(26)

The auxiliary linear operators are

(27)

The properties satisfied by the auxiliary linear operator are

(28)

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

The mth-order deformation equations are constructed as follows:

(29)

4 Convergence of HAM solutions

Professor Liao^[17] has pointed that the convergence rate of the approximation for the HAM solution strongly depends on the values of the auxiliary parameters h_u, h_θ, and h_Φ. It is straightforward to choose the proper values of h_u, h_θ, and h_Φ, which ensures that the solution series is convergent. Figures 1-3 give the respective valid ranges of h_u, h_θ, and h_Φ, respectively. From the figures, we can see that the valid ranges are

Besides, We can use the residual error to find the proper h_u, h_θ, and h_Φ. In this paper, we define the residual error E_{m, θ}^[18] by

(30)

Through following the square residual error function, Using BVPh2.0, the residual error in Fig. 4 shows that the higher the order of the HAM approximation is, the more accurate the result becomes. In addition, It can been seen that the present results agree well with those in Ref. [19] (see Table 1). Besides, the semi-analytical relation between Nu_B and N_BT can be obtained by

(31)

(32)

(33)

(34)

Figure 5 plots the semi-analytical relation between Nu_B and N_BT.

5 Results and discussion

For the Al-water nanofluid, when and , the ratio of the Brownian motion to the thermophoretic forces ranges from 0.1 to 10. Moreover, when

its effects on the solution is negligible (see Ref. [8]). Hence, in the paper, the results are obtained for γ=0.01. The effects of N_BT, λ₁, and λ₂ on the nanoparticle velocity u/u_B, the nanoparticle volume fraction Φ/Φ_B, the temperature profiles θ/θ_B, the total heat transfer rate C_htct, and the pressure drop N_p are shown in Table 2 and Figs. 6-17. In these figures, η=1 corresponds to the inner region of the microtube, whereas η=0 corresponds to the outer region.

When nanoparticles migrate, the viscosity and thermal conductivity distributions are mainly decided by the mutual effects of the Brownian diffusion and the thermophoresis. The Brownian diffusion is proportional to the concentration gradient, while the thermophoresis is proportional to the temperature. From Table 2, we can see that the nanoparticles migrate from the heated wall towards the colder wall at lower values of N_BT. This is because that the migration reduces the viscosity and the shear stress. Considering Eqs. (5) and (6), the thermal conductivities of the nanoparticles strongly depend on the volume fraction. Therefore, when N_BT increases, the thermal conductivity of the heated wall and pressure drop increases, while the temperature gradient of the heated wall decreases (see Fig. 6). The nanoparticle concentration at the higher values of N_BT can be observed from Fig. 7, which means that it becomes uniform.

The slip parameters λ₁ and λ₂ mean the amount of the slip velocity at the surface. The effects of the first-order and second-order velocity slip parameters λ₁ and λ₂ on , and are shown in Figs. 8-13. Because the mass flow rate is assumed to be constant, the velocity in the core region must decrease if it increases at the wall due to the continuity law (see Fig. 8). Apparently, when λ₁ increases, the velocities increase near the outer wall while decreases markedly near the inter wall (see Fig. 8). Figure 9 shows that a steeper temperature gradient at the walls is obtained. From Fig. 10, we can see that, the nanoparticle volume fraction has an increasing trend when λ₁ increases. From Fig. 11, we can see that, when λ₂ increases, the velocities increase. However, the temperature profile and the nanoparticle volume fraction profile show obvious differences when the second-order slip condition is considered (see Figs. 12 and 13). The exchanges of the momentum between the fluid layers lead to an increase in the heat transfer rate C_htct (see Fig. 14). However, an inverse trend can be observed for the pressure drop N_p (see Fig. 15). When λ₂ increases, both the nanoparticle volume fraction and the pressure drop N_p increase (see Figs. 13 and 17). From Figs. 14-17, we can see that, no matter increasing λ₁ or λ₂, the heat transfer rate C_htct has an increasing trend, meanwhile, the pressure drop N_p has an inverse trend. Hence, λ₁ and λ₂ are positive parameters in the current heat transfer system.

Figures 18 and 19 show the effects of the nanoparticles volume fraction Φ_B on the heat transfer coefficient C_htct and the pressure drop N_p for a range of N_BT. From Eq. (5), we can see that increasing the bulk nanoparticle concentration leads to an increase in the viscosity with nanoparticle concentration. Therefore, when the bulk nanoparticle concentration increases, the pressure drop along the channel increases obviously. From Fig. 19, we can see that the total heat transfer ratio increases when the nanoparticle volume fraction Φ_B increases. Figure 19 depicts the heat transfer variation with different values of the nanoparticle volume fraction Φ_B. When Φ_B increases, since there are more suspended particles, the heat transfer coefficient increases because of the increasing viscosity at the walls, which suppresses the convection rate.

The Hartmann number Ha is the ratio of the electromagnetic force. Figures 20 and 21 depict the heat transfer coefficient C_htct and the pressure drop N_p versus N_BT for different values of Ha. Figure 20 depicts the heat transfer variation with different values of Ha. From the figure, we can see that the total heat transfer ratio decreases when Ha increases, which means that in the presence of magnetic field, the advantage of nanofluids in heat transfer enhancement is reduced. An opposite phenomenon happens with an increase in Ha (see Fig. 21).

6 Conclusions

In this paper, the second-order velocity slip on the MHD flow and heat transfer of the nanofluid in an annulus is studied by the HAM. The analytic solutions are obtained through the HAM. The major findings of this paper can be assorted as follows:

(i) It can be observed that the nanoparticle concentration in the annulus is progressively uniform when N_BT increases.

(ii) Both the one-slip parameter and the second-order slip parameter have positive effects on the total heat transfer rate and the pressure drop of the MHD flow.

(iii) The semi-analytical relation between Nu_B and N_BT is obtained.

(iv) Increasing Φ_B increases the heat transfer coefficient C_htct.