MHD Nanofluids in a Permeable Channel with Porosity

This paper introduces a mathematical model of a convection flow of magnetohydrodynamic (MHD) nanofluid in a channel embedded in a porous medium. The flow along the walls, characterized by a non-uniform temperature, is under the effect of the uniform magnetic field acting transversely to the flow direction. The walls of the channel are permeable. The flow is due to convection combined with uniform suction/injection at the boundary. The model is formulated in terms of unsteady, one-dimensional partial differential equations (PDEs) with imposed physical conditions. The cluster effect of nanoparticles is demonstrated in the C 2 H 6 O 2 , and H 2 O base fluids. The perturbation technique is used to obtain a closed-form solution for the velocity and temperature distributions. Based on numerical experiments, it is concluded that both the velocity and temperature profiles are significantly affected by ϕ . Moreover, the magnetic parameter retards the nanofluid motion whereas porosity accelerates it. Each H 2 O -based and C 2 H 6 O 2 -based nanofluid in the suction case have a higher magnitude of velocity as compared to the injections case.


Introduction
Heat transport in unsteady laminar flows has numerous real-world applications, particularly flows in a porous channel with permeable walls, which include medical devices, aerodynamic heating, chemical industry, electrostatic precipitation, petroleum industry, nuclear energy, and polymer technology. Based on this motivation, many researchers have considered the porous channel problem with suction and injection under different physical conditions. In earlier studies, Torda [1] studied the boundary layer flow with the suction/injection effect. Berman [2] derived an exact solution for the channel flow taking into consideration the uniform suction/injection at the boundary wall of the channel. The suction and injection and the combined effect of heat and mass transfer on a moving continuous flat surface were analyzed by Erickson et al. [3]. Alamri et al. [4] studied the Poiseuille flow of nanofluid in a channel under Stefan blowing and the second-order slip effect. Zeeshan et al. [5] reported analytical solutions for the Poiseuille flow of nanofluid in a porous wavy channel. Hassan et al. [6] investigated the flow of H 2 O based nanofluid on a wavy surface. Ellahi et al. [7] studied the boundary layer Poiseuille plan flow of kerosene oil based nanofluid fluid with variable thermal conductivity. Ijaz et al. [8] presented a comprehensive study on the interaction of nanoparticles in the flow of nanofluid in a finite symmetric channel. Some recent important and interesting studies can be found in [9][10][11][12].
Magnetohydrodynamic (MHD) is referred to as the magnetic properties of the fluids under the influence of an electromagnetic force. MHD flows have numerous applications in MHD bearings and MHD pumps. Many studies have been carried out on MHD flow in the literature. Abbas et al. [13] investigated the MHD flow of Maxwell fluid in a porous channel. The convective MHD flow of second-grade fluid was reported by Hayat and Abbas [14]. The effect of a transverse magnetic field on different flows in a semi-porous channel was presented by Sheikholeslami et al. [15]. Ravikumar et al. [16] studied three dimensional MHD due to the pressure gradient over the porous plate. Batti et al. [17] analyzed the heat transfer flow of nanofluid in a channel. They studied the effect of thermal radiation and the MHD effect by using Roseland's approximation, Ohm's law, and Maxwell equations. Ma et al. [18] study the MHD flow of nanofluid in a U-shaped enclosure using the Koo-Kleinstreuer-Li (KKL) correlation approximation for the effective thermal conductivity. Opreti [19] studied water-based silver nanofluid over a stretching sheet. They considered the effect of MHD, suction/injection, and heat generation/absorption in their study. Hosseinzadeh et al. [20] investigated the MHD squeezing flow of nonfluid in a channel. They presented analytical solutions by using similarity transformation and the perturbation technique. Narayana et al. [21] developed a mathematical model for the MHD stagnation point flow of Watler's-B fluid nanofluid.
Nano-sized particles of (Ag) nanoparticles inside H 2 O-based fluids are commonly known as silver-based nanofluids. The viscosity of the nanofluids containing metallic nanoparticles has a much higher thermal conductivity than the nanofluids containing metallic oxide and non-metallic nanoparticles. Because of this, the interest of researchers in investigating nanofluids containing metallic nanoparticles has increased recently. The first exact solutions for different types of nanofluid were developed by Loganathan et al. [22]. Qasim et al. [23] reported numerical solutions for MHD ferrofluid in a stretching cylinder. Amsa et al. [24] investigated nanofluid flow near a vertical plate containing five different nanoparticles. The radiative heat transfer in the natural convection flow of oxide nanofluid was studied by Das and Jana [25]. Dhanai et al. [26]. Numerically studied the MHD mixed convection flow of nanofluid in a cylindrical coordinate system. The MHD rotational flow of nanofluid taking into consideration the effect of a porous medium, thermal radiation, and the chemical reaction was presented by Reddy et al. [27]. For some other interesting studies, readers are referred to [28][29][30][31][32][33][34][35][36][37][38][39][40].
Motivated by the above-discussed literature, the present study focused on the MHD channel flow of nanofluid in a porous medium with the suction and injection effect. The flow of electrically conducting nanofluid is considered under the influence of a transverse magnetic field. The analytical solutions for the proposed model are developed by using the perturbation method. The solutions are numerically computed, and the influence of various flow parameters is studied graphically.

Problem Description
Consider a porous channel of a width, d, filled with incompressible H 2 O and C 2 H 6 O 2 based nanofluids with Ag nanoparticles. The channel walls are stationary with isothermal temperature conditions. The flow in the x-direction due to the temperature gradient is shown in Figure 1. Under the assumption of [11], the governing equations are as follows: together with the following physical conditions: together with the following physical conditions: ( ) where k b = 1.3807 × 10 −23 JK −1 and 300K > T > 325K are used, φ is the nanoparticles' volume fraction, and r c is the radius of gyration for a number of particles. The static part in the effective thermal conductivity is derived from Maxwell's [29] where the numerical values of the thermo-physical of base fluid and nanoparticles are given in Table 1 [11,31]. The radiative heat flux is given by: Substituting Equation (8) into Equation (2), gives: The dimensionless variables: Table 1. Thermo-physical properties of base fluid and nanoparticles. Are introduced into Equations (1) and (9), we get:

Nusselt Number
The dimensionless expression for the Nusselt number is given by:

Skin-Friction
From Equation (31), the skin friction is calculated as:

Results and Discussion
In this section, the graphs of the velocity and temperature for H 2 O and C 2 H 6 O 2 based nanofluids containing Ag nanoparticles were plotted for different values of volume fraction, φ, and buoyancy parameter, Gr, permeability parameter, K, magnetic parameter, M, and radiation parameter, N, for both cases of suction and injection. The thermophysical properties of the base fluids and Ag nanoparticles are mentioned in Table 1. For this purpose, Figures 2-19 were plotted. Figures 2-5 were prepared to study the effects of the velocity for the cases of suction and injection of Ag in H 2 O and C 2 H 6 O 2 based nanofluids, respectively. It was found that the velocity increases with increasing φ for both cases of suction and injection. However, no variation is observed in the velocity of Ag in C 2 H 6 O 2 based nanofluids in the case of injection. This behavior of velocity is found to be similar qualitatively to the results of Hajmohammadi et al. [32], however, they used Cu in water-based nanofluids.

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Ag in 2 H O and 2 6 2 C H O based nanofluids, respectively. It was found that the velocity increases with increasing  for both cases of suction and injection. However, no variation is observed in the velocity of Ag in 2 6 2 C H O based nanofluids in the case of injection. This behavior of velocity is found to be similar qualitatively to the results of Hajmohammadi et al. [32], however, they used Cu in water-based nanofluids.
Ag in 2 H O and 2 6 2 C H O based nanofluids, respectively. It was found that the velocity increases with increasing  for both cases of suction and injection. However, no variation is observed in the velocity of Ag in 2 6 2 C H O based nanofluids in the case of injection. This behavior of velocity is found to be similar qualitatively to the results of Hajmohammadi et al. [32], however, they used Cu in water-based nanofluids.         Figure 6 that the velocity of Ag in water-based nanofluids increases with the increase of Gr in the case of suction for Ag in water-based nanofluids while the velocity is decreased in the case of injection. The velocity in Figure 6, where Gr = 0, is not linear. However, the increasing values of Gr make it look like linear.

 
The effect of the magnetic parameter, M , on the velocity profile is studied in Figures 10 and 11.

 
The effect of the magnetic parameter, M , on the velocity profile is studied in Figures 10 and 11.
For the case of suction, the  The effect of the magnetic parameter, M, on the velocity profile is studied in Figures 10 and 11. For the case of suction, the Ag − H 2 O nanofluid's velocity profile decreases with increasing values of M. This effect is due to the Lorentz forces. Greater values of M correspond to stronger Lorentz forces, which reduces the nanofluid velocity. However, this trend reverses for the injection case.

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  Figure 11. Velocity profiles for different values of M of Ag in water based nanofluids when 0.1,

 
  Figure 11. Velocity profiles for different values of M of Ag in water based nanofluids when 0.1,   Figures 12 and 13 shows that the velocity profiles of Ag in Ag − H 2 O nanofluids increase with the decrease of N in the case of suction. This physically means that an increase in N increases the conduction, which in turn decreases the viscosity of nanofluids. Decreasing the viscosity of nanofluids increases the velocity of nanofluids. However, the effect is the opposite due to the suction, whereas no variation is observed for injection. However, the velocity of zero radiation is greater than the velocity of nanofluids with radiation.

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whereas no variation is observed for injection. However, the velocity of zero radiation is greater than the velocity of nanofluids with radiation.

 
The velocity profiles for different types of nanoparticles in water-based nanofluids are represented in Figure 14. It is clear that the velocity of Ag and C u in water-based nanofluids is

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whereas no variation is observed for injection. However, the velocity of zero radiation is greater than the velocity of nanofluids with radiation.

 
The velocity profiles for different types of nanoparticles in water-based nanofluids are represented in Figure 14. It is clear that the velocity of Ag and C u in water-based nanofluids is The velocity profiles for different types of nanoparticles in water-based nanofluids are represented in Figure 14. It is clear that the velocity of Ag and Cu in water-based nanofluids is greater than TiO 2 and Al 2 O 3 in water-based nanofluids. As mentioned in previous problems, different types of nanoparticles have different thermal conductivities and viscosities. It was concluded in our previous problems [31,32] that metallic nanoparticles, like Ag and Cu, had smaller velocities as compared to metallic oxide nanoparticles, like TiO 2 and Al 2 O 3 , due to high thermal conductivities and viscosities. However, the effect is the opposite to this problem because of the condition of permeable walls or suction. Due to these situations, different velocities have been observed.
The effect of φ in Ag − H 2 O nanofluids on the temperature profiles is shown in Figures 15 and 16 for the cases of suction and injection. It was found that the temperature of nanofluids increases with the increase of φ for the suction velocity whereas no significant variation is observed for the injection case. 14 different types of nanoparticles have different thermal conductivities and viscosities. It was concluded in our previous problems [31,32] that metallic nanoparticles, like Ag and C u , had smaller velocities as compared to metallic oxide nanoparticles, like 2 TiO and 2 3 Al O , due to high thermal conductivities and viscosities. However, the effect is the opposite to this problem because of the condition of permeable walls or suction. Due to these situations, different velocities have been observed.

 
The effect of  in nanofluids on the temperature profiles is shown in Figures 15 and   16 for the cases of suction and injection. It was found that the temperature of nanofluids increases with the increase of  for the suction velocity whereas no significant variation is observed for the injection case.  H O based nanofluids are plotted in Figure 19 for injection. It is observed that Cu in water-based nanofluid has the highest temperature followed by , Ag 2 3 Al O  Figure 19 for injection. It is observed that Cu in water-based nanofluid has the highest temperature followed by Ag, Al 2 O 3 , and TiO 2 in H 2 O based nanofluids. This is due to the higher thermal conductivities of copper followed by Ag, Al 2 O 3 , and TiO 2 in water-based nanofluids. Due to these situations, different velocities were observed. H O based nanofluids are plotted in Figure 19 for injection. It is observed that Cu in water-based nanofluid has the highest temperature followed by , Ag 2 3 Al O , and 2 TiO in 2 H O based nanofluids. This is due to the higher thermal conductivities of copper followed by , Ag 2 3 Al O , and 2 TiO in water-based nanofluids. Due to these situations, different velocities were observed.

Conclusions
The channel flow of nanofluids in a porous medium with permeable walls was studied. The focal point of this research was to study the influence of permeable walls on momentum and heat transfer. The permeable parameter, which physically corresponds to suction and injection, was incorporated in both the momentum and energy equations. Expressions for the velocity and temperature were obtained. The effects of various parameters, such as thermal Grashof number,

Conclusions
The channel flow of nanofluids in a porous medium with permeable walls was studied. The focal point of this research was to study the influence of permeable walls on momentum and heat transfer. The permeable parameter, which physically corresponds to suction and injection, was incorporated in both the momentum and energy equations. Expressions for the velocity and temperature were obtained. The effects of various parameters, such as thermal Grashof number, volume fraction, different types of nanoparticles, radiation, permeability, magnetic, suction, and injection, were studied in different plots. The concluding remarks are as follows:

1.
It was found that the velocity of nanofluids increases with an increase of the volume fraction, radiation, and permeability parameter in the case of suction whereas an opposite behavior was noted in the case of injection.

2.
The velocity of Ag nanofluids decreases with an increase of the magnetic parameter while the opposite behavior was noted in the case of injection. 3.
The temperature of Ag nanofluids was found to decrease with an increase of φ for the extraction of fluid from the walls whereas a very small change was observed in the case of injection. 4.
Finally, it was noticed that different types of nanoparticles have different effects on the velocity and temperature due to suction and injection.
Author Contributions: All the authors contributed equally to the conception of the idea, implementing and analyzing the experimental results, and writing the manuscript. volume fraction, different types of nanoparticles, radiation, permeability, magnetic, suction, and injection, were studied in different plots. The concluding remarks are as follows: 1. It was found that the velocity of nanofluids increases with an increase of the volume fraction, radiation, and permeability parameter in the case of suction whereas an opposite behavior was noted in the case of injection. 2.
The velocity of Ag nanofluids decreases with an increase of the magnetic parameter while the opposite behavior was noted in the case of injection.
3. The temperature of Ag nanofluids was found to decrease with an increase of  for the extraction of fluid from the walls whereas a very small change was observed in the case of injection.
4. Finally, it was noticed that different types of nanoparticles have different effects on the velocity and temperature due to suction and injection.
Author Contributions: All the authors contributed equally to the conception of the idea, implementing and analyzing the experimental results, and writing the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.