Numerical Study of Periodic Magnetic Field E ﬀ ect on 3D Natural Convection of MWCNT-Water / Nanoﬂuid with Consideration of Aggregation

: In this paper, a numerical study is performed to investigate the e ﬀ ect of a periodic magnetic ﬁeld on three-dimensional free convection of MWCNT (Mutli-Walled Carbone Nanotubes)-water / nanoﬂuid. Time-dependent governing equations are solved using the ﬁnite volume method under unsteady magnetic ﬁeld oriented in the x-direction for various Hartmann numbers, oscillation periods, and nanoparticle volume fractions. The aggregation e ﬀ ect is considered in the evaluation of the MWCNT-water / nanoﬂuid thermophysical properties. It is found that oscillation period, the magnitude of the magnetic ﬁeld, and adding nanoparticles have an important e ﬀ ect on heat transfer, temperature ﬁeld, and ﬂow structure.


Introduction
The control of the fluid flow and heat transfer via the magnetic damping effect can be an optimisation parameter in thermal and engineering systems such as microscope design, healthcare devices, biomechanics, and so on [1].
Izadi et al. [2] performed a numerical investigation on nanofluid's free convection inside a porous medium using the FEM (Finite Element Method). The authors considered the variable magnetic field and mentioned that, for higher Ra values, the heat transfer reduces considerably owing to the applied magnetic field. On the basis of the FEM, Hatami et al. [3] analysed the effects of Fe 3 O 4 nanoparticles and the variable magnetic field on natural convection. They mentioned that, owing to the induced Lorentz

Governing Equations
To simplify the resolution of the governing equations, the 3D vorticity vector potential formalism was used. In fact, it allows the elimination of the pressure gradient terms.
The vorticity and vector potential are expressed as follows: and To get the dimensionless governing equations, the variables ' Φ ,

Governing Equations
To simplify the resolution of the governing equations, the 3D vorticity vector potential formalism was used. In fact, it allows the elimination of the pressure gradient terms.
The vorticity and vector potential are expressed as follows: To get the dimensionless governing equations, the variables Φ , V and t , are divided by l·υ 0 ·B 0 , B 0 , σ·υ 0 ·B 0 , l 2 /α, α, α/l, and l 2 /α, respectively, and θ = ( 4 of 21 with The effective thermophysical properties are evaluated using the following expressions: -Heat capacitance: -Dynamic viscosity: Taking into account the fractal shape of aggregates [25], the modified nanoparticle volume fraction (volume fraction of the aggregates in water), φ ag , is expressed as follows: where • r a is the radii of the aggregates; • r p is the radii of the primary solid particles; • D is the fractal index depending on shear flow condition and type, size, and shape of the aggregates.
Thus, based on the Maron and Pierce model [26] and the expression of the modified volume fraction, the effective viscosity of the MWCNT-nanofluid can be expressed as follows: On the basis of the experimental work of Halelfadl et al. [27] for MWCNT with l p = 1.5 µm and d p = 9.2 nm, it was determined that D = 2.1, φ m = 0.0361, and r a /r p = 4.41.
-Thermal conductivity (Xue-model): On the basis of Nan's model [28] and taking into account the aggregates effect, the effective thermal conductivity of MWCNT is calculated using the following expressions: where • λ 11 is the transverse thermal conductivity; • λ 33 is the longitudinal thermal conductivity; • d p is the nanoparticle diameter; • l p is the nanoparticle length; • a k is the Kapitza radius; • R k is the Kapitza resistance (fixed at 8.83 × 10 −8 m 2 K/W); φ ag is volume fraction of the MWCNT in the aggregate.
-electrical conductivity: The local and average Nusselt numbers are evaluated using the following: As the applied magnetic field in this study is periodic, it is expected that Nu av will be periodic; thus, a time averaged Nusselt number (Nu t av ) is defined as follows: The solution is considered satisfactory if Equation (18) is satisfied in each time:

Boundary Conditions
The boundary conditions for the present problem are given as follows: Temperature: θ = 0 on the right wall, and θ = 1 on the left wall.

Solution Procedure, Grid Sensitivity Test, and Validation
The central-difference scheme based control volume method is used to discretize Governing Equations (4)- (8). The fully implicit procedure and central-difference scheme are used to treat the temporal derivatives and the convective terms, respectively.
The grid independency test was performed for Ha = 50, ϕ = 0.05, and τ p = 1. The tests were conducted for different grids. The time averaged Nusselt number was the sensitive parameter. The incremental increase between the grid 81 × 81 × 81 and 91 × 91 × 91 is only 0.333% (Table 2). For solution accuracy and time economy, a spatial mesh size of 81 × 81 × 81 is retained to perform all the simulations. The verification of the present code is firstly performed by comparing with results Okada and Ozoe [29], who proposed an experimental correlation (Equation (19)) allowing the evaluation of the Nusselt number as function of the Hartman and Grashof numbers for the case of gallium filled cubical cavity. Figure 2 shows an excellent concordance with the results of Okada and Ozoe [29]. Ozoe [29], who proposed an experimental correlation (Equation (19)) allowing the evaluation of the Nusselt number as function of the Hartman and Grashof numbers for the case of gallium filled cubical cavity. Figure 2 shows an excellent concordance with the results of Okada and Ozoe [29].
( ) In the case of nanofluids, a comparison was performed with the results of Jahanshahi et al. [30]. As presented in Figure 3, the comparison also shows a good agreement. In the case of nanofluids, a comparison was performed with the results of Jahanshahi et al. [30]. As presented in Figure 3, the comparison also shows a good agreement. Processes 2020, 6, x FOR PEER REVIEW 8 of 21

Results and Discussion
A finite volume based computational work was performed to investigate the three-dimensional natural convection of MWCNT-water/nanofluid filled cavity under variable magnetic field. While adding MWCNT particles enhances the heat transfer owing to high effective thermal conductivity, the application of a magnetic field opposes this enhancement. Thus, the simultaneous influence of a variable magnetic field and adding nanoparticles on the 3D flow and heat transfer is investigated.
The periodic magnetic field is considered along the x-direction. The Rayleigh number is fixed at Ra = 10 5 and the amplitude (A) is fixed at 0.5, so that the magnitude of the magnetic field oscillates between 0.5B0 and 1.5B0. Owing to the relatively low considered volume fractions, the single phase model is adopted. The ranges of Hartman number, MWCNT volume fraction, and oscillation period are (0 ≤ Ha ≤ 100), (0 ≤ φ ≤ 0.05), and (0.01 ≤ p τ ≤ 10), respectively.
In absence of a magnetic forces, the flow structure and temperature field are presented in Figure 4. The flow structure is shown via some particle trajectories. For φ = 0 (pure water), two central clockwise vortices characterize the flow structure. Adding MWCNT particles was

Results and Discussion
A finite volume based computational work was performed to investigate the three-dimensional natural convection of MWCNT-water/nanofluid filled cavity under variable magnetic field. While adding MWCNT particles enhances the heat transfer owing to high effective thermal conductivity, the application of a magnetic field opposes this enhancement. Thus, the simultaneous influence of a variable magnetic field and adding nanoparticles on the 3D flow and heat transfer is investigated.
The periodic magnetic field is considered along the x-direction. The Rayleigh number is fixed at Ra = 10 5 and the amplitude (A) is fixed at 0.5, so that the magnitude of the magnetic field oscillates between 0.5B 0 and 1.5B 0 . Owing to the relatively low considered volume fractions, the single phase model is adopted. The ranges of Hartman number, MWCNT volume fraction, and oscillation period are (0 ≤ Ha ≤ 100), (0 ≤ ϕ ≤ 0.05), and (0.01 ≤ τ p ≤ 10), respectively.
In absence of a magnetic forces, the flow structure and temperature field are presented in Figure 4. The flow structure is shown via some particle trajectories. For ϕ = 0 (pure water), two central clockwise vortices characterize the flow structure. Adding MWCNT particles was found to enhance the buoyancy forces and to increase the size of the vortices. The particle paths are shifted towards the isothermal walls and the vortices become adjacent to these walls.
It is to be noted that the maximum of the velocity magnitude is lowered by adding nanoparticles owing to the inter-particular friction caused by the increase of the viscosity. The maximal values of the magnitude of the velocity are located near to the active walls because of the important gradients of temperatures in these regions. Figure 4c presents the temperature field for Ha = 0. The iso-surfaces of temperature are vertically stratified at the core region and distorted near of the active walls. This phenomenon becomes more pronounced by adding the MWCNT nanoparticles with an increase of the temperature gradient. model is adopted. The ranges of Hartman number, MWCNT volume fraction, and oscillation period are (0 ≤ Ha ≤ 100), (0 ≤ φ ≤ 0.05), and (0.01 ≤ p τ ≤ 10), respectively.
In absence of a magnetic forces, the flow structure and temperature field are presented in Figure 4. The flow structure is shown via some particle trajectories. For φ = 0 (pure water), two central clockwise vortices characterize the flow structure. Adding MWCNT particles was found to enhance the buoyancy forces and to increase the size of the vortices. The particle paths are shifted towards the isothermal walls and the vortices become adjacent to these walls.
It is to be noted that the maximum of the velocity magnitude is lowered by adding nanoparticles owing to the inter-particular friction caused by the increase of the viscosity. The maximal values of the magnitude of the velocity are located near to the active walls because of the important gradients of temperatures in these regions. Figure 4c presents the temperature field for Ha = 0. The iso-surfaces of temperature are vertically stratified at the core region and distorted near of the active walls. This phenomenon becomes more pronounced by adding the MWCNT nanoparticles with an increase of the temperature gradient.    Figure 5 presents the 3D pathlines and velocity magnitude at different times of the oscillation period for Ha = 50 and τ p = 1. The results for pure water are presented on the left side and those of MWCNT-water nanofluid with ϕ = 0.05 are presented on the right. The interaction between the magnetic field and the fluid motion generates a Lorentz force that opposes the buoyancy forces, causing the damping effect. This damping is more important near the active walls, except near the corners. In fact, because of the electric insulation of the walls, the electric current is redirected, and thus the Lorentz force vanishes in these regions. In addition, a merging phenomenon occurs and the pathlines become characterized by only one clockwise vortex.
causing the damping effect. This damping is more important near the active walls, except near the corners. In fact, because of the electric insulation of the walls, the electric current is redirected, and thus the Lorentz force vanishes in these regions. In addition, a merging phenomenon occurs and the pathlines become characterized by only one clockwise vortex. The lateral flow (near of the walls) is also affected by the applied magnetic field and the longitudinal flow becomes less pronounced owing to the reduction of the 3D character (MHD bi-dimensionalization). The structure and the intensity of the flow change slightly during the oscillation period. In fact, the size and the shape of the vortexes change periodically with the time. Figure 6 shows the velocity vectors projection at z = 0.5 plan for φ = 0.05, Ha = 25, and A = 0.5 at different times and different oscillation periods. Owing to the magnetic effect, there is a change from a roll structure to a square structure; this conclusion was also mentioned by Al-Najem  The lateral flow (near of the walls) is also affected by the applied magnetic field and the longitudinal flow becomes less pronounced owing to the reduction of the 3D character (MHD bi-dimensionalization). The structure and the intensity of the flow change slightly during the oscillation period. In fact, the size and the shape of the vortexes change periodically with the time. Figure 6 shows the velocity vectors projection at z = 0.5 plan for ϕ = 0.05, Ha = 25, and A = 0.5 at different times and different oscillation periods. Owing to the magnetic effect, there is a change from a roll structure to a square structure; this conclusion was also mentioned by Al-Najem et al. [31]. It is seen that, for all chosen oscillation periods, the main flow structure changes periodically with time. In fact, vortexes' merging and division occur during the period for all considered τ p values. Thus, owing to the periodic magnitude of the magnetic field, the flow structure oscillates between one and two central vortexes. In addition to the number of vortexes, their shape also changes with time.
As presented in Figure 7, for higher Hartman numbers (Ha = 100), the merging-division phenomenon persists, but with a flow structure having a slope that approaches the vertical. The flows near the active walls are more attenuated owing to the magnetic damping. This result is also recalled by Ozoe and Okada [29], by mentioning that the induced Lorentz force opposes the flow perpendicular to the direction of the magnetic field. In fact, compared with the Ha = 25 case, in which the maximum values of velocity magnitude are near of the active walls, for Ha = 100, the maximums become near to the top and bottom walls. In opposition with all other τ p values, for τ p = 0.01 when the merging occurs at t = τ p 2 , the central symmetry disappears, in fact, the vortex is located in the highest region of the enclosure. Figure 8 presents a sequence of the temperature field for different Hartmann numbers. The vertical stratification described in Figure 4 is reduced by applying the magnetic (Ha = 50) and, for high values of Hartman number (Ha = 100), the stratification disappears completely. Owing to the suppression of the convective forces opposed by the Lorentz force, the temperature field becomes similar to the conductive regime with an important decrease of the temperature gradients near to the active walls. This conclusion is confirmed by the experimental work of Xu et al. [32]. Owing to their high thermal conductivity, the addition of MWCNT nanoparticles opposes the heat transfer diminution caused by the presence of the magnetic field. Thus, the reduction of the vertical stratification is less pronounced for the case of MWCNT-water compared with pure water. Similarly to the flow structure, the temperature field also changes during the period. For a better understanding of this change, the isotherms at the central plan ( Figure 9) for τ p = 1, ϕ = 0.05, and A = 0.5.
From this figure, it is seen that isotherms are clearly modified with time, especially at the core of the cavity. In fact, near of the active walls, the isotherms are almost similar during the period and different at the core region. It is to be noted that these changes are more pronounced for higher Hartman values owing to the increase of the extremum of the magnitude of the magnetic field.
high thermal conductivity, the addition of MWCNT nanoparticles opposes the heat transfer diminution caused by the presence of the magnetic field. Thus, the reduction of the vertical stratification is less pronounced for the case of MWCNT-water compared with pure water. Similarly to the flow structure, the temperature field also changes during the period. For a better understanding of this change, the isotherms at the central plan ( Figure 9  From this figure, it is seen that isotherms are clearly modified with time, especially at the core of the cavity. In fact, near of the active walls, the isotherms are almost similar during the period and different at the core region. It is to be noted that these changes are more pronounced for higher Hartman values owing to the increase of the extremum of the magnitude of the magnetic field.  Figure 10 illustrates the effect of adding nanoparticles on isotherms at the central plan for τ p = 0.1 and A = 0.5 (flood (ϕ = 0.05) and line (ϕ = 0)). As seen from the figure, that addition of nanoparticle to base fluid has an important effect on temperature distribution. Isotherms become steeper with the increasing Ha number. This means that heat transfer turns to conduction mode. As mentioned above, the isotherms' structure varies during the period, especially in the central zones for all Hartman values and nanoparticles concentrations. Figure 11 presents the effect of the time-periodic magnetic field magnitude on the average Nusselt number at a certain value of Hartman numbers and nanoparticles volume fraction. As the magnetic field is periodic, it is obvious that the results show the fluctuation in amplitude of the average Nusselt number around a certain value. By decreasing the oscillation period (τ p ), the fluctuation decreases, especially for high Hartman values (Figure 11), but it does not become zero. It is to be noted that the average value around which the Nusselt number fluctuates decreases by increasing τ p ; this is an interesting result when the purpose is to reduce the heat transfer. Obviously, the effects of the applied magnetic field and adding nanoparticles are opposite. In fact, the Nusselt number decreases with Ha and increases with ϕ.  Obviously, the effects of the applied magnetic field and adding nanoparticles are opposite. In fact, the Nusselt number decreases with Ha and increases with φ.

Conclusions
A computational work was conducted on heat transfer and fluid flow inside a three-dimensional closed space under a variable magnetic field. The main findings are as follows: • The addition of MWCNT particles enhances the convective heat transfer, increases the vortices' sizes, and promotes the vertical stratification of the temperature field. • Lower heat transfer is observed with the increasing Hartmann number and the variation of heat transfer is almost smooth. In other words, oscillation of heat transfer becomes very weak.

•
For the same Hartmann number, higher heat transfer and lower amplitude are formed at a higher nanoparticle volume fraction.

Conclusions
A computational work was conducted on heat transfer and fluid flow inside a three-dimensional closed space under a variable magnetic field. The main findings are as follows: • The addition of MWCNT particles enhances the convective heat transfer, increases the vortices' sizes, and promotes the vertical stratification of the temperature field.