Influence of Induced Magnetic Field on Free Convection of Nanofluid Considering Koo-Kleinstreuer-Li (KKL) Correlation

Abstract

In this paper, the influence of induced magnetic field on free convection of Al₂O₃-water nanofluid on permeable plate by means of Koo-Kleinstreuer-Li (KKL) model is reported. Impact of Brownian motion, along with the properties of nanofluid, are also taken into account. The resulting equations are solved utilizing Runge-Kutta integration method. Obtained results are examined for innumerable energetic parameters, namely Al₂O₃ volume fraction, suction parameter, and Hartmann and magnetic Prandtl numbers. Results indicate that the velocity profile reduces with rise of the suction parameter and magnetic Prandtl and Hartmann numbers but it increases with addition of nanoparticles. Shear stress enhances with rise of suction parameter, magnetic Prandtl and Hartmann numbers. Temperature gradient improves with augment of suction parameter.

1. Introduction

Magnetohydrodynamic (MHD) free convection has several applications, such as combustion modeling, geophysics, fire engineering, etc. In recent decades, nanotechnology has been presented as a new passive technique for heat transfer improvement. MHD nanofluid natural convection in a tilted wavy cavity has been presented by Sheremet et al. [1]. They illustrated that a change of titled angle causes convective heat transfer to be enhanced. 3D MHD free convective heat transfer was examined by Sheikholeslami and Ellahi [2] using Lattice Boltzmann method (LBM). Their results showed that Lorentz forces cause the temperature gradient to reduce. Ismael et al. [3] investigated the influence of Lorentz forces on nanofluid flow in an enclosure with moving walls. Their outputs indicated that the impact of Lorentz forces reduces with direction of magnetic field. Sheikholeslami and Ellahi [4] utilized LBM to study Fe₃O₄-water flow, with the aim of drug delivery. They concluded that the velocity gradient reduces with the rise of magnetic number. The influence of non-uniform Lorentz forces on nanofluid flow style has been studied by Sheikholeslami Kandelousi [5]. He concluded that improvement in heat transfer reduces with rise of Kelvin forces. A new model for nanofluid on peristaltic flow was presented by Tripathi and Beg [6]. They reported different behavior for nanofluid temperature profiles with changing temperature. Kouloulias et al. [7] presented an experimental analysis for free convection of nanofluid. They showed that greater nanoparticle volume fraction leads to higher Rayleigh numbers.

The influence of thermal radiation on magnetohydrodynamic nanofluid motion has been reported by Sheikholeslami et al. [8]. They concluded that the nanofluid concentration gradient augments with the rise of the radiation parameter. Mineral oil-based nanofluids have been utilized in natural convection by Peña et al. [9]. MHD Fe₃O₄-water flow in a wavy cavity with moving wall has been investigated by Sheikholeslami and Chamkha [10]. The influence of Lorentz forces on forced convective heat transfer has been examined by Sheikholeslami et al. [11]. They illustrated that a greater Reynolds number has a more sensible effect on Kelvin forces. Akbar and Khan [12] investigated the impact of magnetic field on nanofluid motion in an asymmetric channel. Hakeem et al. [13] studied the influence of Lorentz forces on various nanofluids by means of second order slip flow mode. They showed that a unique solution exists for this problem for high Hartman number values. Several researchers have investigated about this subject [14,15,16,17,18,19,20,21,22].

In almost all the previous papers, the authors neglected the induced magnetic field. However, in various physical states it is necessary to consider this effect in governing equations. This assumption is considered in order to simplify the mathematical analysis of the problem. Furthermore, the induced magnetic field produces its own magnetic field in the fluid; therefore, it can amend the original magnetic field. Also, nanofluid motion in the magnetic field produces mechanical forces which change the motion of motion. Ghosh et al. [23] reported the impact of induced Lorentz forces on temperature profile. Unsteady magnetohydrodynamic flow on a cone has been investigated by Vanita and Kumar [24]. Beg et al. [25] examined the impact of induced magnetic field on boundary layer flow. The influence of atherosclerosis on hemodynamics of stenosis has been forecasted by Nadeem and Ijaz [26]. They showed that the velocity gradient on the wall of titled arteries reduces with augment of Strommers number.

The chief end of this paper is to illustrate the influence of induced magnetic field on nanofluid hydrothermal treatment between two vertical plates. To obtain outputs, Runge-Kutta method is selected. The impacts of the suction parameter, magnetic Prandtl and Hartmann numbers, volume fraction of nanofluid on temperature, and induced magnetic, velocity and current density profiles are examined.

2. Problem Statement

Al₂O₃-water fluid through two vertical permeable sheets is investigated as illustrated in Figure 1. The boundary conditions are clear in this figure. The variables are only the function of y because plates are infinite. Velocity and magnetic field vectors are considered as $\overrightarrow{v}=\left[u,v_0,0\right]$ and $\overrightarrow{b}=\left[b_x,b_0,0\right]$ respectively. The governing equations and boundary conditions can be obtained as follows:

$${\mathsf{\upsilon }}_{nf}\frac{d^{2}u}{d{y}^2}+\frac{{\mathsf{\mu }}_e{b}_0}{{\mathsf{\rho }}_{nf}}\frac{d{b}_x}{dy}+g{\mathsf{\beta }}_{nf}\left(T-T_0\right)+v_0\frac{du}{dy}=0$$

(1)

$$\frac{1}{{\mathsf{\mu }}_e{\mathsf{\sigma }}_{nf}}\frac{d^2{b}_x}{d{y}^2}+b_0\frac{du}{dy}+v_0\frac{d{b}_x}{dy}=0$$

(2)

$$\frac{k_{nf}}{{\left(\mathsf{\rho }{C}_p\right)}_{nf}}\frac{d^{2}T}{d{y}^2}+v_0\frac{dT}{dy}=0$$

(3)

$$b_x\left(0\right)=0,u\left(0\right)=0,\frac{dT}{dy}\left(0\right)=-\frac{q}{k_f}$$

(4)

$$\frac{d{b}_x}{dy}\left(h\right)=0,u\left(h\right)=0,T\left(h\right)=T_0$$

(5)

${\left(\mathsf{\sigma }\right)}_{nf}$, ${\left(\mathsf{\rho }{C}_p\right)}_{nf}$, ${\left(\mathsf{\rho }\mathsf{\beta }\right)}_{nf}$ and $\left({\mathsf{\rho }}_{nf}\right)$ can be introduced as [3]:

$$\begin{array}{lll}\frac{{\mathsf{\sigma }}_{nf}}{{\mathsf{\sigma }}_f}=1+\frac{3\left({\mathsf{\sigma }}_p/{\mathsf{\sigma }}_f-1\right)\mathsf{\varphi }}{\left({\mathsf{\sigma }}_p/{\mathsf{\sigma }}_f+2\right)-\left({\mathsf{\sigma }}_p/{\mathsf{\sigma }}_f-1\right)\mathsf{\varphi }},& & {\left(\mathsf{\rho }\mathsf{\beta }\right)}_{nf}=(1-\mathsf{\varphi }){\left(\mathsf{\rho }\mathsf{\beta }\right)}_f+{\left(\mathsf{\rho }\mathsf{\beta }\right)}_p\mathsf{\varphi },\\ {(\mathsf{\rho } C_p)}_{nf}=\mathsf{\varphi }{(\mathsf{\rho } C_p)}_p+{(\mathsf{\rho } C_p)}_f(1-\mathsf{\varphi }),& & {\mathsf{\rho }}_{nf}=\mathsf{\varphi }{\mathsf{\rho }}_p+{\mathsf{\rho }}_f(1-\mathsf{\varphi })\end{array}$$

(6)

$\left(k_{nf}\right)$ and $\left({\mathsf{\mu }}_{nf}\right)$ are obtained according to Koo-Kleinstreuer-Li (KKL) model [27]:

$$\begin{array}{l}{k}_{nf}=\frac{3\left(k_p/k_f-1\right)\mathsf{\varphi }}{-\left(k_p/k_f-1\right)\mathsf{\varphi }+\left(k_p/k_f+2\right)}+1+5\mathsf{\varphi }\times {10}^4{c}_{p,f}{g}^{\prime }(d_p,T,\mathsf{\varphi }){\mathsf{\rho }}_f\sqrt{\frac{{\mathsf{\kappa }}_{b}T}{{\mathsf{\rho }}_p{d}_p}}\\ g^{\prime }\left(d_p,T,\mathsf{\varphi }\right)=\\ \left(a_1+a_{2}Ln\left(d_p\right)+a_{5}Ln{\left(d_p\right)}^2+a_{3}Ln\left(\mathsf{\varphi }\right)+a_4\text{ln}\left(d_p\right)Ln\left(\mathsf{\varphi }\right)\right)Ln\left(T\right)+\\ \left(a_6+a_{7}Ln\left(d_p\right)+a_{10}Ln{\left(d_p\right)}^2+a_{8}Ln\left(\mathsf{\varphi }\right)+a_9\text{ln}\left(d_p\right)Ln\left(\mathsf{\varphi }\right)\right)\\ R_f=d_p\left(1/k_{p,eff}-1/k_p\right),R_f=4\times {10}^{-8}k{m}^2/W\end{array}$$

(7)

$${\mathsf{\mu }}_{nf}=\frac{{\mathsf{\mu }}_f}{{\left(1-\mathsf{\varphi }\right)}^{2.5}}+\frac{k_{Brownian}}{k_f}\times \frac{{\mathsf{\mu }}_f}{\text{Pr}}$$

(8)

All needed coefficients and properties are illustrated in

Table 1. Constants of Al₂O₃ − Water [27].

Coefficient Values	Al2O3 − Water
$a_1$	52.813
$a_2$	6.115
$a_3$	0.695
$a_4$	4.1 × 10−2
$a_5$	0.176
$a_6$	−298.198
$a_7$	−34.532
$a_8$	−3.922
$a_9$	−0.235
$a_{10}$	−0.999

and

Table 2. Properties of water and Al₂O₃ [27].

Material	$\mathsf{\rho }$ (kg/m3)	Cp (j/kg⋅k)	k (W/m⋅k)	$\mathsf{\beta }$ × 105 (K−1)	dp (nm)	$\mathsf{\sigma }$ (Ω⋅m)−1
Pure water	997.1	4179	0.613	21	-	0.05
Al2O3	3970	765	25	0.85	47	1 × 10−10

[27].

Dimensionless parameters are presented as:

$$\begin{array}{l}U=\frac{\mathsf{\upsilon }u}{g\mathsf{\beta }{h}^2\Delta T},B=\sqrt{\frac{{\mathsf{\mu }}_e}{{\mathsf{\rho }}_f}}{\mathsf{\upsilon }}_f{b}_x{\left(g{\mathsf{\beta }}_f{h}^2\Delta T\right)}^{-1},\mathsf{\theta }=\Delta T^{-1}\left(T-T_0\right),\Delta T=qh/k_f,Y=\frac{y}{h}\\ \text{Pr}={\left(\mathsf{\rho }{C}_p\right)}_f\frac{{\mathsf{\mu }}_f}{k_f},Pm={\mathsf{\mu }}_e{\mathsf{\upsilon }}_f{\mathsf{\sigma }}_f,Ha=\frac{B_{0}h}{{\mathsf{\upsilon }}_f}\sqrt{\frac{{\mathsf{\mu }}_e}{\mathsf{\rho }}},V_0=\frac{v_{0}h}{{\mathsf{\upsilon }}_f}\end{array}$$

(9)

Finally, the dimensionless governing equations are

$$\frac{d^{2}U}{d{Y}^2}+\frac{Ha}{A_2}\frac{dB}{dY}+\frac{A_1{A}_6}{A_2}\mathsf{\theta }+\frac{A_1}{A_2}{V}_0\frac{dU}{dY}=0$$

(10)

$$\frac{d^{2}B}{d{Y}^2}+A_5{V}_{0}Pm\frac{dB}{dY}+A_{5}HaPm\frac{dU}{dY}=0$$

(11)

$$\frac{d^2\mathsf{\theta }}{d{Y}^2}+\frac{A_3}{A_4}{V}_0\text{Pr}\frac{d\mathsf{\theta }}{dY}=0$$

(12)

$$B\left(0\right)=0,U\left(0\right)=0,\frac{d\mathsf{\theta }}{dY}\left(0\right)=-1$$

(13)

$$\frac{dB}{dY}\left(1\right)=0,U\left(1\right)=0,\mathsf{\theta }\left(1\right)=0$$

(14)

Induced current density can be defined:

$$J=-dB/dY$$

(15)

$C_{\mathrm{f}}$ and $Nu$ can be expressed as:

$$C_{\mathrm{f}}=\frac{A_2}{A_1}{U}^{\prime }(0),Nu=A_4/\mathsf{\theta }(0).$$

(16)

3. Runge-Kutta Method

In Runge-Kutta method, at first the following definitions are applied: $x_2=U,x_1=Y,x_3=U^{\prime }$, $x_4=B,x_5=B^{\prime },x_6=\mathsf{\theta },x_7={\mathsf{\theta }}^{\prime }$. The final system and initial conditions are:

$$\left(\begin{array}{l}{x_1}^{\prime }\\ {x_2}^{\prime }\\ {x_3}^{\prime }\\ {x_4}^{\prime }\\ {x_5}^{\prime }\\ {x_6}^{\prime }\\ {x_7}^{\prime }\end{array}\right)=\left(\begin{array}{c}1\\ x_3\\ \frac{1}{A_2}\left[-Ha{x}_5-A_1{A}_6{x}_6-A_1{V}_0{x}_3\right]\\ x_5\\ -A_5{V}_0{x}_{5}Pm-A_5{x}_{3}HaPm\\ x_7\\ -\frac{A_3}{A_4}{V}_0{x}_7\text{Pr}\end{array}\right)$$

(17)

$$\left(\begin{array}{c}0\\ 0\\ u_1\\ 0\\ u_2\\ u_3\\ -1\end{array}\right)=\left(\begin{array}{l}{x}_1\\ x_2\\ x_3\\ x_4\\ x_5\\ x_6\\ x_7\end{array}\right)$$

(18)

Equations (17) and (18) are solved utilizing fourth order Runge-Kutta method. According to $U\left(1\right)=0,B^{\prime }\left(1\right)=0,\mathsf{\theta }\left(1\right)=0$, unknown initial conditions can be obtained by Newton’s method.

4. Results and Discussion

Steady two-dimensional nanofluid hydrothermal treatment between two parallel vertical permeable plates is studied considering induced magnetic field. The Runge-Kutta integration scheme is utilized to solve this problem. MAPLE code has been validated by comparison with a previously published paper [28].

Table 3. Comparison of skin friction tension over the upper wall between the present results and previous work. H_a, Hartmann number; V₀, suction parameter; Pr, Prandtl number.

V0	Pr	Prm	Ha	Sarveshanand and Singh [21]	Present Work
0.5	0.7	0.5	5	0.016	0.015
0.75	0.7	0.5	5	0.011	0.011
1	0.7	1	5	0.018	0.018
1	0.015	0.5	0.5	2.695	2.700

indicates good accuracy of present code. The influences of important parameters such as magnetic Prandtl number $\left(Pm\right)$, Hartmann number $\left(Ha\right)$, suction parameter $\left(V_0\right)$ and nanoparticle volume fraction $\left(\mathsf{\varphi }\right)$ on flow style are examined.

Impact of $\mathsf{\varphi }$ on induced magnetic field, current density, temperature and velocity distributions is shown in Figure 2. As volume fraction of nanofluid augments, nanofluid velocity and temperature are enhanced due to an increase in fluid motion by adding nanoparticles. Induced current density increases with an augment of $\mathsf{\varphi }$ while the opposite behavior is shown for induced magnetic field. Influence of suction parameter on hydrothermal behavior is depicted in Figure 3. Velocity, temperature and induced current density decreases, with an augment of suction parameter while induced magnetic field enhances with rise of $V_0$. Therefore, this parameter can be considered as control parameter for engineering designs.

Figure 4 depicts the impacts of Lorentz forces on induced magnetic field, induced current density and velocity distributions. As Lorentz forces augments, the back flow appears and in turn velocity of nanofluid decreases. In addition, it can be seen that the maximum velocity point shifts to the hot wall. Induced magnetic field decreases with rise of magnetic field strength but induced current density is enhanced with the rise of Lorentz forces. Influence of $Pm$ on induced magnetic field, induced current density and velocity profiles is depicted in Figure 5. Without the magnetic field, the shape of the velocity profiles is parabolic but in the existence of the magnetic field its shape changes to being flattened. The nanofluid motion and induced magnetic field reduces with an augment of $Pm$. Induced current density rises with augment of Pm.

Influences of magnetic Prandtl number $\left(Pm\right)$, Hartmann number $\left(Ha\right)$, suction parameter $\left(V_0\right)$ and nanoparticle volume fraction $\left(\mathsf{\varphi }\right)$ on skin friction coefficient are depicted in Figure 6 and Figure 7. According to these data, a correlation is presented for skin friction coefficient as follows:

$$\begin{array}{l}{C}_f=0.25335-0.34173V_0-0.011146Ha-0.12279\mathsf{\varphi }\\ +0.0076966V_0 Ha+0.079691V_0 Pm-2.54355V_0 \mathsf{\varphi }\\ +0.00154943Ha Pm-0.0424Ha \mathsf{\varphi }-1.15754Pm \mathsf{\varphi }\\ +0.11737V_0^2+0.0001179H{a}^2+0.033354P{m}^2+13.69282{\mathsf{\varphi }}^2\end{array}$$

(19)

It can be concluded that $C_f$ has reverse relationship with all active parameters except for $\mathsf{\varphi }$. Figure 8 shows the influence of $V_0$ and $\mathsf{\varphi }$ on Nusselt number. In addition, a good correlation has been presented for the Nusselt number as follows:

$$\begin{array}{l}Nu=0.80342+4.92333V_0+1.72177\mathsf{\varphi }-3.50195V_0\mathsf{\varphi }\\ +1.822V_0^2-0.75963{\mathsf{\varphi }}^2\end{array}$$

(20)

As suction parameter $\left(V_0\right)$ and nanoparticle volume fraction $\left(\mathsf{\varphi }\right)$ increase, temperature gradient increases. Therefore, Nu is enhanced with enhancement of $V_0, \mathsf{\varphi }$.

5. Conclusions

The influence of induced magnetic field on nanofluid motion and forced convection between two vertical permeable plates is investigated. To solve coupled equations, Runge-Kutta method is utilized. The influence of different dimensionless parameters on induced magnetic field, velocity and temperature distributions are considered. Results illustrate that current density augments with a rise of volume fraction of nanofluid and Hartmann and magnetic Prandtl numbers, while it is reduced with a rise in the suction parameter. As Lorentz force increases, velocity and induced magnetic field are reduced and maximum velocity point shifts to the left side.