Effects of Second-Order Velocity Slip and the Different Spherical Nanoparticles on Nanofluid Flow

Abstract

The paper theoretically investigates the heat transfer of nanofluids with different nanoparticles inside a parallel-plate channel. Second-order slip condition is adopted due to the microscopic roughness in the microchannels. After proper transformation, nonlinear partial differential systems are converted to ordinary differential equations with unknown constants, and then solved by homotopy analysis method. The residual plot is drawn to verify the convergence of the solution. The semi-analytical expressions between $N{u}_B$ and $N_{BT}$ are acquired. The results show that both first-order slip parameter and second-order slip parameter have positive effects on $N{u}_B$ of the MHD flow. The effect of second-order velocity slip on $N{u}_B$ is obvious, and $N{u}_B$ in the alumina–water nanofluid is higher than that in the titania–water nanofluid. The positive correlation between slip parameters and $N_{dp}$ is significant for the titania–water nanofluid.

1. Introduction

Modern industrial applications are expected to achieve higher heat transfer rates, so how to improve the heat transfer performance of heat exchanger becomes the main problem concerned by researchers. Meanwhile, microchannels have many applications such as automobile cooling systems and electronic devices in micro-sized cooling systems. Li et al. [1] and Duan et al. [2] studied the heat transfer rates of nanofluid in microchannels.

To study the flow of nanofluid, homogeneous flow models and dispersion models have been proposed. In 2006, Buongiorno [3] showed that the dispersed effects can be completely ignored due to the size of nanoparticles, and Brownian diffusion and thermophoresis are important in nanofluids. Based on the above analysis, he proposed that the homogeneous models are more appropriate for predicting the heat transfer coefficient. By using this model, Yang et al. [4] studied the variation of forced convection transport with temperature jump in continuous flow and slip flow regimes. F. Hedayati et al. [5] studied the variation of $Ti{O}_2-H_{2}O$ nanofluid mixing convection within vertical microchannel of nanoparticle migration and asymmetric heating. R.S.Andhare et al. [6] studied pressure drop characteristics of a flat plate manifold microchannel heat exchanger. O.D. Makinde et al. [7] studied MHD variable viscosity reacting flow with thermophoresis and radiative heat transfer. A.Malvandi et al. [8] discussed effects of nanoparticle migration on alumina–water nanofluid.

Boundary conditions are critical to the model; initially, the common velocity slip is the Maxwell [9] slip condition. Kou et al. [10] studied the effects of wall slip and temperature jump on heat and mass transfer characteristics of evaporative films. A.A. Avramenko et al. [11] investigated mixed convection in a circular microchannel with the slip boundary conditions. As micro/nanotechnology develops, the size of micro/nanodevices are getting smaller and smaller. The Navier slip condition will break down at higher shear rates. In 1997, Thompson [12] developed a nonlinear slip model based on the first-order slip model proposed by Maxwell. However, many researchers found that the model could not predict the flow at a high $Kn$ number. The values calculated by the second-order slip boundary condition are closer to the experimental data. Beskok and Karniadakis [13] improved a second-order slip conditions. Based on Beskok and Karniadakis, Wu [14] improved the slip condition. Zhu et al. [15] and Almutairi et al. [16] described the effects of second-order velocity slip.

However, as a result of the migration of nanoparticles under second-order slip condition and the influence of different nanoparticles, the heat transfer of nanofluids is limited. Besides, there is little attention paid to the analytic solution [17]. In this paper, the overall goal is to study the fully developed convection of nanofluids in a parallel plate channel theoretically. Two water-based nanofluids, containing alumina and titania nanoparticles, respectively, are considered. The governing partial differential equations are transformed into ordinary differential equations with an unknown constant by using similar variables, which are solved by the homotopy analysis method (HAM).

2. Mathematical Analysis

Considering a stable, incompressible, laminar flow in a parallel-plate channel with a uniform magnetic field, the upper wall of the parallel plate channel remains insulated, while the lower wall receives a constant cooling heat flow. Taking parallel to the wall as the x-axis and perpendicular to the wall as the y-axis, a two-dimensional coordinate frame is established. Nanofluids have been studied using an improved two-component heterogeneous model. Hence, the mass, momentum, thermal energy, and nanoparticle fraction equations of the flow system can be expressed as follows:

$$\begin{array}{cc}\hfill & {\partial }_i\left(\rho U_i\right)=0\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & {\partial }_t\left(\rho U_i\right)+{\partial }_j\left(\rho U_i{U}_j\right)=-{\partial }_{i}P+{\partial }_j\mu ({\partial }_i{U}_j+{\partial }_j{U}_i)-{\sigma }_0{B}_0^2{U}_i\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & {\partial }_t\left(\rho cT\right)+{\partial }_i\left(\rho c{U}_{i}T\right)={\partial }_i\left(k{\partial }_{i}T\right)+\rho c(D_B{\partial }_i\varphi +\frac{D_T}{T_C}{\partial }_{i}T){\partial }_{i}T+Q_0(T-T_w)-\frac{\partial q_r}{\partial y}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\partial }_t\left(\varphi \right)+{\partial }_i\left(U_i\varphi \right)={\partial }_i(D_B{\partial }_i\varphi +\frac{D_T}{T_C}{\partial }_{i}T)\hfill \end{array}$$

(4)

when the nanoparticle volume fractions are different, $\rho$, $\mu$, k, and c also change. The expressions are as follows:

$$\begin{array}{c}\mu \left(\varphi \right)=\left\{\begin{array}{c}{\mu }_{bf}(1+39.11\varphi +533.9{\varphi }^2),Alumina--water\hfill \\ {\mu }_{bf}(1+5.45\varphi +108.2{\varphi }^2),Titania--water\hfill \end{array}\right.\hfill \end{array}$$

(5)

$$\begin{array}{c}k\left(\varphi \right)=\left\{\begin{array}{c}{k}_{bf}(1+7.47\varphi ),Alumina--water\hfill \\ k_{bf}(1+2.92\varphi -11.99{\varphi }^2),Titania--water\hfill \end{array}\right.\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \rho =\varphi {\rho }_p+(1-\varphi ){\rho }_{bf},c=\frac{\varphi {\rho }_p{c}_p+(1-\varphi ){\rho }_{bf}{c}_{bf}}{\rho }\hfill \end{array}$$

(7)

where p stands for particle and $bf$ stands for base fluid. Moreover, the thermal physical properties of $A{l}_2{O}_3$ nanoparticles, $Ti{O}_2$ nanoparticles, and the base fluid (water) are also analyzed as follows:

$c_{p_{bf}}=4182$ J/kgK, $k_{bf}=0.597$ W/mK, ${\rho }_{bf}=998.2$ kg/m${}^3$, ${\mu }_{bf}=9.93*10$${}^{-4}$ kg/ms

$c_{p_{A{l}_2{O}_3}}=773$ J/kgK, $k_{A{l}_2{O}_3}=36$ W/mK, ${\rho }_{A{l}_2{O}_3}=3380$ kg/m${}^3$

$c_{p_{Ti{O}_2}}=385$ J/kgK, $k_{Ti{O}_2}=8.4$ W/mK, ${\rho }_{Ti{O}_2}=4175$ kg/m${}^3$

Based on material performance of a typical water-based nanofluid with alumina (titania/water) nanoparticles, the coefficients of Equation (3) can be calculated [3] by scale analysis. Scale analysis indicates that the heat conduction term is about 1000 times more than virtue of nanoparticle diffusion. Actually, heat transfer in connection with the diffusion of nanoparticles $\rho c(D_B{\partial }_i\varphi +\frac{D_T}{T_C}{\partial }_{i}T){\partial }_{i}T$ can be neglected in comparison with heat conduction and convection. When the flow velocity is very low, the Re is very small. Therefore, compared with viscous resistance [8], inertia effect can be ignored. Assuming hydrodynamically and thermally fully developed conditions, Equations (1)–(4) can be simplified as follows [18]:

$$\begin{array}{cc}\hfill & -\frac{dP}{dx}+\frac{d}{dy}(\mu \left(\varphi \right)\frac{dU}{dy})-{\sigma }_0{B}_0^{2}U=0\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \frac{\partial }{\partial y}(k(\varphi )\frac{\partial T}{\partial y})+Q_0(T-T_w)-\frac{\partial q_r}{\partial y}=0\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \frac{\partial }{\partial y}(D_B\frac{\partial \varphi }{\partial y}+\frac{D_T}{T_C}\frac{\partial T}{\partial y})=0\hfill \end{array}$$

(10)

Radiant heat flux $q_r$ is described by Rosseland approximation [17] as follows:

$$\begin{array}{cc}\hfill & q_r=-\frac{4{\sigma }^{*}}{3\delta }\frac{\partial T^4}{\partial y}\hfill \end{array}$$

(11)

Assuming that temperature difference is small enough in the flow, using Taylor series to expand $T^4$, and ignoring the higher-order terms, $T^4$ can be expressed as a linear function [19]. The approximate expression is as follows:

$$\begin{array}{cc}\hfill & T^4\cong 4T_{\infty }^{3}T-3T_{\infty }^4\hfill \end{array}$$

(12)

The following appropriate transformations are:

$$\begin{array}{cc}\hfill & \eta =\frac{y}{H},u=\frac{U}{U_m},H{a}^2=\frac{\sigma B_0^2{H}^2}{{\mu }_w},\theta =\frac{k_w(T-T_w)}{q_{w}H}\hfill \\ \hfill & \sigma =\frac{(\mathrm{d}p/\mathrm{d}x)}{U_m/H^2},N_{BT}=\frac{D_B}{D_T}\frac{k_w{T}_C}{q_{w}H},\gamma =\frac{Q_0{q}_w{H}^3}{k_w},\alpha =\frac{\mathrm{d}P/\mathrm{d}x}{\left({\mu }_{bf}{u}_B\right)/H^2}\hfill \end{array}$$

(13)

Equations (8)–(10) can be reduced to:

$$\begin{array}{cc}\hfill & \mu \left(\varphi \right)\frac{d^{2}u}{d{\eta }^2}+\frac{d\mu \left(\varphi \right)}{d\eta }\frac{du}{d\eta }-H{a}^{2}u-\alpha =0\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & k\left(\varphi \right)\frac{d^2\theta }{d{\eta }^2}+\frac{dk\left(\varphi \right)}{d\eta }\frac{\mathrm{d}\theta }{\mathrm{d}\eta }+\gamma \theta +\frac{16{\sigma }^{*}}{3k^{*}}\frac{{\mathrm{d}}^2\theta }{\mathrm{d}{\eta }^2}=0\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & N_{BT}{(1+\gamma \theta )}^2\frac{\partial \varphi }{\partial \eta }-\varphi \frac{\partial \theta }{\partial \eta }=0\hfill \end{array}$$

(16)

3. Boundary Conditions

At micro- or nanoscale, the slip boundary condition can be used to predict accurately. In current investigations, the most common velocity slip is the Maxwell [9] slip condition. The Maxwell expression is:

$$\begin{array}{cc}\hfill & \overrightarrow{u_{slip}}=-\frac{2-\beta }{\beta {\mu }_w}\xi \overrightarrow{\tau }-\frac{3}{4}\frac{N_{Pr}(\delta -1)}{\delta p}\overrightarrow{q}\hfill \end{array}$$

(17)

where $\overrightarrow{\tau }=S·(n·\Pi )$, $\overrightarrow{q}=\overrightarrow{Q}·S$.

Beskok and Karniadakis [13] improved the second-order slip conditions:

$$\begin{array}{cc}\hfill & u_s-u_w=\frac{2-{\sigma }_v}{{\sigma }_v}[Kn{(\frac{\partial u}{\partial n})}_s+\frac{K{n}^2}{2}{(\frac{{\partial }^{2}u}{\partial n^2})}_s]\hfill \end{array}$$

(18)

where $(\frac{\partial }{\partial n})$ shows gradients normal to the wall surface. Based on Beskok and Karniadakis, Wu [14] improved the slip condition in detail:

$$\begin{array}{cc}\hfill & u_{slip}=\frac{2}{3}(\frac{3-\omega l^3}{\omega }-\frac{3}{2}\frac{1-l^2}{Kn})\lambda \frac{\partial u}{\partial y}-\frac{1}{4}[l^4+\frac{2}{K{n}^2}(1-l^2)]{\lambda }^2\frac{{\partial }^{2}u}{\partial y^2}=A\frac{\partial u}{\partial y}+B\frac{{\partial }^{2}u}{\partial y^2}\hfill \end{array}$$

(19)

where $l=min[\frac{1}{Kn},1]$. The expression of velocity boundary condition is as follows:

$$\begin{array}{cc}\hfill & y=0:U=N_1\frac{\partial U}{\partial y}+N_2\frac{{\partial }^{2}U}{\partial y^2}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & y=H:U=-N_1\frac{\partial U}{\partial y}-N_2\frac{{\partial }^{2}U}{\partial y^2}\hfill \end{array}$$

(21)

The other boundary conditions are as follows:

$$\begin{array}{cc}\hfill & y=0:-k_w\frac{\partial T}{\partial y}=q_w,\frac{\partial \varphi }{\partial y}=-\frac{D_T}{D_B}\frac{1}{T_C}\frac{\partial T}{\partial y}\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & y=H:\frac{\partial T}{\partial y}=0,\frac{\partial \varphi }{\partial y}=-\frac{D_T}{D_B}\frac{1}{T_C}\frac{\partial T}{\partial y}\hfill \end{array}$$

(23)

Substituting Equation (13) into Equations (20)–(23), the boundary conditions are as follows:

$$\begin{array}{cc}\hfill & \eta =0:u={\lambda }_1\frac{\partial u}{\partial \eta }+{\lambda }_2\frac{{\partial }^{2}u}{\partial {\eta }^2},\frac{\partial \theta }{\partial \eta }=1,\theta =0,\varphi ={\varphi }_w\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & \eta =1:u=-{\lambda }_1\frac{\partial u}{\partial \eta }-{\lambda }_2\frac{{\partial }^{2}u}{\partial {\eta }^2}\hfill \end{array}$$

(25)

In actual applications, the mass flow rate is specified through the channels. Therefore, the average fluid velocity is introduced:

$$U_m=\frac{{\int }_0^{H}Udy}{{\int }_0^{H}dy}$$

Dimensionless variables can be obtained as follows:

$$\begin{array}{cc}\hfill & {\int }_0^{1}ud\eta =1\hfill \end{array}$$

(26)

The average of the parameters on the cross section can be calculated using the following formula [20]:

$$\langle \Gamma \rangle =\frac{1}{A}{\int }_0^1\mathrm{d}A={\int }_0^1\Gamma \mathrm{d}\eta$$

Further, ${\theta }_B$ and ${\varphi }_B$ can be worked out as follows:

$$\begin{array}{cc}\hfill & {\theta }_B=\frac{<\rho cu\theta >}{<\rho cu>},{\varphi }_B=\frac{<u\varphi >}{<u>}\hfill \end{array}$$

(27)

According to the bulk properties and hydraulic diameter of nanofluids, the Nusselt number can be assessed as [21]:

$$\begin{array}{cc}\hfill & N{u}_B=\frac{hH}{k_B}=\frac{1}{{\theta }_B}\frac{k_w}{k_B}\hfill \end{array}$$

(28)

The non-dimensional pressure drop can be defined as:

$$\begin{array}{cc}\hfill & N_{dp}=\frac{-(\mathrm{d}p/\mathrm{d}x)}{\left({\mu }_{bf}{u}_B\right)/H^2}=-\alpha \hfill \end{array}$$

(29)

In addition, the semi-analytical relationship between $N{u}_B$ and $N_{BT}$ in the alumina–water nanofluid can be obtained as:

$$\begin{array}{cc}\hfill & a_1=2.1494(-248469-551.028N_{BT}+72.1567N_{BT}^2-0.0184761N_{BT}^3)\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & b_1=-68012.1+2498.11N_{BT}+45.6722N_{BT}^2-1.80232N_{BT}^3+0.000305721N_{BT}^4\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & c_1=\frac{(-0.00184879+0.000522536N_{BT}-0.0000613978N_{BT}^2+1.57212\times {10}^{-8}{N}_{BT}^3)}{-0.0600415+0.0000151093N_{BT}}\hfill \\ \hfill & \times 7.47+1\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & N{u}_{B_{A{l}_2{O}_3}}=\frac{a_1}{b_1{c}_1}\hfill \end{array}$$

(33)

The semi-analytical relation between $N{u}_B$ and $N_{BT}$ in the titania–water nanofluid can be obtained as:

$$\begin{array}{cc}\hfill & a_2=2.1494(-202053-426.941N_{BT}+46.4139N_{BT}^2-0.00159493N_{BT}^3)\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & b_2=-20516.7+623.982N_{BT}+7.42706N_{BT}^2-0.334142N_{BT}^3+7.1142\times {10}^{-6}{N}_{BT}^4\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & c_2=\frac{(-0.00148734+0.000369449N_{BT}-0.0000394933N_{BT}^2+1.35712\times {10}^{-9}{N}_{BT}^3)}{-0.0488208+1.73602\times {10}^{-6}{N}_{BT}}\hfill \\ \hfill & \times 7.47+1\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & N{u}_{B_{Ti{O}_2}}=\frac{a_2}{b_2{c}_2}\hfill \end{array}$$

(37)

4. Application of HAM

In this article, to obtain the series solutions, we adopt homotopy analysis method (HAM). HAM is one of the well-known semi-analytical methods for solving various types of linear and nonlinear differential equations (ordinary as well as partial). This method is based on coupling of the traditional perturbation method and homotopy in topology. By this method, one may obtain an exact solution or a power series solution which converges in general to the exact solution. HAM consists of the convergence control parameter, which controls the convergent region and rate of convergence of the series solution. We select the initial guess solutions:

$$\begin{array}{cc}\hfill & u_0\left(\eta \right)=-0.1+\eta -{\eta }^2\hfill \\ \hfill & {\theta }_0\left(\eta \right)=\eta -2{\eta }^2\hfill \\ \hfill & {\varphi }_0\left(\eta \right)={\varphi }_B\hfill \end{array}$$

(38)

What calls for special attention is that the boundary condition (26) is not yet used, which can be used to determine the unknown parameter ${\alpha }_{k-1}$. For example, when $k=1$, we are able to obtain $u_1\left(\eta \right)$ and its integration with $\eta$ in the range $[0,1]$, which is the function of ${\alpha }_0$. Using the boundary condition (26), we obtain:

$$\begin{array}{cc}\hfill & {\alpha }_0\left(\eta \right)=1\hfill \end{array}$$

(39)

In this way, $u_k\left(\eta \right),{\theta }_k\left(\eta \right),{\varphi }_k\left(\eta \right),{\alpha }_k\left(\eta \right)$ can be successively worked out one after another according to the order $k=0,1,2,...$. At mth-order, we obtain:

$$\begin{array}{cc}\hfill & u\left(\eta \right)=u_0\left(\eta \right)+\sum _{k=1}^m{u}_k\left(\eta \right)\hfill \\ \hfill & \theta \left(\eta \right)={\theta }_0\left(\eta \right)+\sum _{k=1}^m{\theta }_k\left(\eta \right)\hfill \\ \hfill & \varphi \left(\eta \right)={\varphi }_0\left(\eta \right)+\sum _{k=1}^m{\varphi }_k\left(\eta \right)\hfill \\ \hfill & \alpha ={\alpha }_0+\sum _{k=1}^m{\alpha }_k\hfill \end{array}$$

(40)

The auxiliary linear operators are:

$$\begin{array}{cc}\hfill & L_u=\frac{{\mathrm{d}}^{2}u}{\mathrm{d}{\eta }^2},L_{\theta }=\frac{{\mathrm{d}}^2\theta }{\mathrm{d}{\eta }^2},L_{\varphi }=\frac{\mathrm{d}\varphi }{\mathrm{d}\eta }\hfill \end{array}$$

(41)

The properties of the auxiliary linear operator are as follows:

$$\begin{array}{cc}\hfill & L_u[C_1+C_2\eta +C_3{\eta }^2]=0,L_{\theta }[C_4+C_5\eta +C_6{\eta }^2]=0,L_{\varphi }[C_7+C_8\eta ]=0\hfill \end{array}$$

(42)

where $C_i,i=1,...,8$ are constants.

Next, construct the mth-order deformation equation as follows:

$$\begin{array}{cc}\hfill & L_u[u_m\left(\eta \right)-{\chi }_m{u}_{m-1}\left(\eta \right)]=q{h}_u{R}_m\left(\eta \right)\hfill \\ \hfill & L_{\theta }[{\theta }_m\left(\eta \right)-{\chi }_m{\theta }_{m-1}\left(\eta \right)]=q{h}_{\theta }{R}_m\left(\eta \right)\hfill \\ \hfill & L_{\varphi }[{\varphi }_m\left(\eta \right)-{\chi }_m{\varphi }_{m-1}\left(\eta \right)]=q{h}_{\varphi }\left(\eta \right)R_m\left(\eta \right)\hfill \end{array}$$

(43)

5. Convergence of the HAM Solutions

Liao [22] showed that the values of auxiliary parameters $h_u$, $h_{\theta }$, and $h_{\varphi }$ can adjust and control the convergence of the series solutions. Directly selecting the appropriate values of $h_u$, $h_{\theta }$, and $h_{\varphi }$ ensures the convergence of the series solutions. Figure 1 and Figure 2 give the respective valid ranges of $h_{\theta }$, $h_{\varphi }$. The valid ranges are as follows:

$$\left\{\begin{array}{c}0\le h_u\le 0.4\hfill \\ -0.8\le h_{\theta }\le -0.35\hfill \\ -3\le h_{\varphi }\le 0.1\hfill \end{array}\right.$$

In addition, one way to find the appropriate $h_u$, $h_{\theta }$, and $h_{\varphi }$ is to utilize the residual error. In this article, the residual error $E_{m,t}$ [23] is defined as follows:

$$\begin{array}{cc}\hfill & E_{m,\theta }={\int }_0^{1}k{\theta }^{″}+k^{\prime }{\theta }^{\prime }+\gamma \theta +\frac{16{\sigma }^{*}}{3k^{*}}{\theta }^{″}d\eta \hfill \end{array}$$

(44)

Using the square residual error function, it is found that the residual error becomes more and more accurate as the order of HAM approximation increases (Figure 3). Finally, $\alpha$ of the HAM solution agrees well with the BVPh 2.0 solution (

Table 1. Comparison of HAM results with BVPh2.0 results.

${\mathit{\varphi }}_{\mathit{B}}$	$\mathit{\alpha }$
	BVPh2.0	HAM	Relative Error(%)
0.01	−0.00286834	−0.00286673	0.05624769
0.02	−0.00396349	−0.00394892	0.36760532
0.03	−0.00527046	−0.00529243	0.33803511
0.04	−0.00678941	−0.00671253	1.13235171

). BVPh 2.0 is a free software package for nonlinear boundary-value and eigenvalue problems based on HAM. In addition, it serves to show that the current results are in accordance with the results given by Yang et al. [24] (

Table 2. Comparison of HAM results with those of C. Yang et al. [24].

${\mathit{N}}_{\mathit{BT}}$	${\mathit{Nu}}_{\mathit{B}}$
	Yang et al. [24]	HAM	Relative Error(%)
0.1	7.26823	7.26679	0.01981
0.2	7.55883	7.55889	0.00079
0.3	7.69768	7.69418	0.04547
0.4	7.79492	7.79163	0.04225
0.5	7.85227	7.85200	0.00344
0.6	7.90000	7.90338	0.04278
0.7	7.94920	7.94526	0.04956
0.8	7.95957	7.95947	0.00126
0.9	7.97313	7.97791	0.05995
1	8.04496	8.04478	0.00224
2	8.12940	8.12983	0.00529
10	8.21841	8.21630	0.02567

) greatly.

6. Results and Discussion

The effects of $N_{BT}$, ${\lambda }_2$ and ${\lambda }_1$ on the nanoparticle velocity $u/u_B$, the nanoparticle volume fraction $\varphi /{\varphi }_B$, temperature profiles $\theta /{\theta }_B$, and Nusselt number $N{u}_B$ are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. In these figures, $\eta =1$ corresponds to the adiabatic wall, whereas $\eta =0$ corresponds to the cooled wall.

The slip parameter characterizes slip resistance at the surface. The first-order velocity slip parameters ${\lambda }_1$ and second-order velocity slip parameters ${\lambda }_2$ affect the flow and heat. Figure 4, Figure 5 and Figure 6 depict the effects of second-order velocity slip ${\lambda }_2$ on $u/u_B$, $\varphi /{\varphi }_B$, and $\theta /{\theta }_B$. Figure 7, Figure 8 and Figure 9 illustrate the effects of first order velocity slip ${\lambda }_1$ on $u/u_B$, $\varphi /{\varphi }_B$, and $\theta /{\theta }_B$. Figure 4 and Figure 7 show that $u/u_B$ is lower near the walls and peaks near the middle of the microchannel. As Figure 4 reveals, the increase in ${\lambda }_2$ causes momentum to build up in the core area, with the velocity profile becoming more uniform as the slip parameters decrease. Figure 7 shows that an increase in ${\lambda }_1$ results in the momentum accumulation at the core region. The second-order slip condition shows a prominent effect on the velocity profile $u/u_B$ in Figure 4 and Figure 7. Assuming that mass flows are constant, in order to satisfy continuity, they must increase in the core region as the magnitude of the velocities at the boundary decreases. Meanwhile, in Figure 5 and Figure 8, the temperature profile $\theta /{\theta }_B$ decreases and then increases toward the upper wall. The titania–water nanofluid temperature changes more gently than that of the alumina–water nanofluid. The minimum of the temperature profile is increasing and shifts toward the upper wall with increasing ${\lambda }_2$. Figure 8 shows no significant variation in the dimensionless temperature for ${\lambda }_1$. In addition, with increasing ${\lambda }_2$ or ${\lambda }_1$, the volume fraction $\varphi /{\varphi }_B$ of nanoparticles shows an increasing trend in Figure 6 and Figure 9. Hence, s more uniform distribution of the volume fraction emerges.

Figure 10, Figure 11 and Figure 12 plot the effects of $N_{BT}$ on $u/u_B$, $\varphi /{\varphi }_B$, and $\theta /{\theta }_B$. Apparently, on the cooling wall, the concentration of nanoparticles is higher; at the adiabatic wall, the nanoparticle concentration is lower. Hence, the trend of the nanoparticle motion is moving from the adiabatic wall toward the cooled wall; accordingly, an uneven distribution of nanoparticles is constructed. This motion makes the viscosity near the cold wall much greater than that near the adiabatic wall, thus increasing the velocity near the adiabatic wall and decreasing the velocity near the cold wall. Therefore, the velocity profile deforms and its peak moves toward the adiabatic wall. As a result, at higher values of $N_{BT}$, $\varphi /{\varphi }_B$ becomes more uniform, which can be observed in Figure 12. At higher value of $N_{BT}$, with momentum enhanced, the heat transfer rate of the cooling wall also increases. Therefore, the increase of $N_{BT}$ gives rise to an increase in the temperature gradient of the cooling wall, as shown in Figure 11.

Figure 13 depicts the effect of ${\lambda }_2$ on the Nusselt number $N{u}_B$. One thing to note is that the growth trend for $N{u}_B$ with the increase in ${\lambda }_2$ comes from the momentum accumulation near the wall. Therefore, the second-order slip parameter plays a positive role in $N{u}_B$ of the MHD flow. As a result, compared with Navier’s condition, under second-order slip conditions, nanofluids transfer heat more efficiently. It also must be stated that the effect of the slip parameters on $N{u}_B$ is quite protensive; it rests with the type of nanoparticle. In alumina–water nanofluid, the sensitivity on $N{u}_B$ is much higher than that of titania–water nanofluid, since the corresponding $N{u}_B$ of alumina–water nanofluid is higher than that of titania–water nanofluid. Figure 14 depicts the ${\varphi }_B$ on the $N{u}_B$. It can be obtained that increasing ${\varphi }_B$ leads to a decrease $N{u}_B$ because the increasing ${\varphi }_B$ can increase the thermal conductivity and viscosity near walls.

Table 3. $N_{dp}$ with different ${\lambda }_2$, ${\lambda }_1$, $N_{BT}$, and ${\varphi }_B$ when $Ha$ = 0.

${\mathit{\lambda }}_2$	${\mathit{\lambda }}_1$	${\mathit{N}}_{\mathit{BT}}$	${\mathit{\varphi }}_{\mathit{B}}$	Types of Fluids
				${\mathit{Al}}_2{\mathit{O}}_3$-Water	${\mathit{TiO}}_2$-Water
0.1	0.1	0.5	0.01	0.000127706	0.000129222
0.2				0.000131924	0.014154800
	0.2			0.000179880	0.020517000
		10		0.000127722	0.000127916
			0.04	0.000119804	0.000127917

gives $N_{dp}$ with different ${\lambda }_2$, ${\lambda }_1$, $N_{BT}$, and ${\varphi }_B$ when $Ha$ = 0, respectively. It can be concluded that both one-slip parameter and second-order slip parameter have positive correlation with the pressure drop ratio of the nanofluid to base fluid $N_{dp}$. As ${\lambda }_2$ or ${\lambda }_1$ decreases, the frictional forces on the walls diminish because of the velocity jumps at the walls. However, the positive correlation is significant to titania–water nanofluid. Because of a slight increase in viscosity at the wall, $N_{BT}$ has a minor positive effect on $N_{dp}$ and ${\varphi }_B$ has a minor negative effect on $N_{dp}$.

Table 4. $N{u}_B$ with different ${\lambda }_2$, ${\lambda }_1$, $N_{BT}$, and ${\varphi }_B$ when $Ha$ = 0.

${\mathit{\lambda }}_2$	${\mathit{\lambda }}_1$	${\mathit{N}}_{\mathit{BT}}$	${\mathit{\varphi }}_{\mathit{B}}$	Types of Fluids
				${\mathit{Al}}_2{\mathit{O}}_3$-Water	${\mathit{TiO}}_2$-Water
0.1	0.1	0.5	0.01	5.62714	4.93726
0.2				8.57429	7.38838
	0.2			8.95671	7.71938
		10		8.69342	7.40811
			0.04	8.56113	7.37139

gives $N{u}_B$ with different ${\lambda }_2$, ${\lambda }_1$, $N_{BT}$, and ${\varphi }_B$ when $Ha$ = 0, respectively. It can be deduced that the velocity gradient at the wall of the microchannel increases. The slip velocity increases with the increase of velocity gradient. Thus, momentum closer to the wall increases and causes convective heat transfer to rise.

7. Conclusions

In this paper, we conduct a theoretical study on the heat transfer of alumina/water and titania/water nanofluids in a parallel-plate channel. We discuss the effects of Brownian motion and thermophoresis. Their effects are characterized by the ratio of the Brownian to thermophoretic diffusion coefficients $N_{BT}$. Moreover, The second-order velocity slip condition is considered. Analytic solutions are obtained by HAM. The main conclusions of this paper can be drawn as follows:

a

The semi-analytical relation between $N{u}_B$ and $N_{BT}$ is obtained.

b

Both first-order slip parameter and second-order slip parameter have positive effects on $N{u}_B$ of the MHD flow, but nanofluids can transfer heat more efficiently with a second-order slip condition than with a Navier’s condition.

c

In the alumina–water nanofluid, $N{u}_B$ is higher than that of titania–water nanofluid.

d

The positive correlation between slip parameters and $N_{dp}$ is significant for the titania-water nanofluid.