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 Fe
3O
4-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 Fe
3O
4-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
Al
2O
3-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
and
respectively. The governing equations and boundary conditions can be obtained as follows:
,
,
and
can be introduced as [
3]:
and
are obtained according to Koo-Kleinstreuer-Li (KKL) model [
27]:
All needed coefficients and properties are illustrated in
Table 1 and
Table 2 [
27].
Dimensionless parameters are presented as:
Finally, the dimensionless governing equations are
Induced current density can be defined:
and
can be expressed as:
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 indicates good accuracy of present code. The influences of important parameters such as magnetic Prandtl number
, Hartmann number
, suction parameter
and nanoparticle volume fraction
on flow style are examined.
Impact of
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
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
. 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
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
. Induced current density rises with augment of
Pm.
Influences of magnetic Prandtl number
, Hartmann number
, suction parameter
and nanoparticle volume fraction
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:
It can be concluded that
has reverse relationship with all active parameters except for
.
Figure 8 shows the influence of
and
on Nusselt number. In addition, a good correlation has been presented for the Nusselt number as follows:
As suction parameter and nanoparticle volume fraction increase, temperature gradient increases. Therefore, Nu is enhanced with enhancement of .