# Rayleigh-Bénard Convection of Paramagnetic Liquid under a Magnetic Field from Permanent Magnets

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## Abstract

**:**

## 1. Introduction

## 2. Computational Methods

#### 2.1. Schematic Model

_{h}(=2.0) and top plate is cooled at T

_{c}(=1.0), both isothermally. Sidewalls of the computational domain are considered as the periodic boundary for the simplicity. In the presence of the gravity, the working fluid is driven by buoyancy.

#### 2.2. Lattice Boltzmann Heat and Fluid Flow Simulation

**x**= (X, Y) and time t are below.

_{α}. τ

_{c}is the relaxation time of g. In the density distribution function (Equation (1)), external force term F

_{α}is added [21,22]. In the preset study, the force term contains buoyancy F

_{b}and magnetothermal force F

_{m}.

_{0}is the magnetic permeability in vacuum, χ

_{mm0}is the mass magnetic susceptibility at the reference temperature T

_{0}, and ρ the density. In the above equation, the constant C is defined as follows.

_{0}in Equation (8) is presumed at 293.15 K. The computational dimensionless numbers are the Prandtl number Pr, the Rayleigh number Ra, and the dimensionless magnetic induction γ. Definitions of these parameters are as follows.

_{f}represents the relaxation time used in Equation (1). The computational dimensionless parameters are as follows. The Prandtl number is fixed at 13.0 which corresponds to the gadolinium nitrate aqueous solution of 15.5 wt. % which is the same fluid of [20]. The Rayleigh number is fixed at 10

^{4}which corresponds to the temperature difference ΔT of 0.415 K when the gap between hot and cold is presumed at 10 mm. The dimensionless applied magnetic induction γ is varied from 0 to 10.0. For the reference, γ = 1.0 corresponds to 1.01 Tesla by employing the same characteristic length as temperature. Please note that γ = 10.0 is out of available magnetic induction by commercial permanent magnets.

^{4}is evaluated by the domain resolutions of 300 × 52, 600 × 102, and 900 × 152 with uniform mesh. The resulted average Nusselt number on the hot wall is 2.6168 (300 × 52), 2.6175 (600 × 102), and 2.6177 (900 × 152), respectively. The reference value of the average Nusselt number at Ra = 10

^{4}is 2.661 [24]. RBC by LBM [25] has similar value. Therefore, grid resolution of 600 × 102 nodes is employed both for the computational time and precision.

#### 2.3. Magnetic Field Simulation

^{2}is also enhanced at the magnet junction due to a large value of |b| in a small area.

## 3. Results and Discussion

#### 3.1. Heat and Fluid Flow with and without Magnetic Field

^{4}. Since the results are symmetric regardless of the magnitude of γ, left half of figures (0 ≤ X ≤ 3.0) are extracted. When the magnetic field is absent (Figure 4a), the flow pattern becomes symmetrical and corresponding temperature field is formed. This is the typical phenomenon of RBC.

^{2}induced near the magnet junction toward the magnet as reported by our group [20]. As suggested in Equation (7), the magnetothermal force is resulted from the multiplication of ∇b

^{2}and temperature difference. When the cold fluid descends near the magnet junction, the local temperature difference becomes large near the magnet, then the magnetothermal force becomes remarkable as shown in Figure 5b–d. This induced force overlaps the gravity and accelerates the downward flow near the magnet junction, resulting in the locally-enhanced convection. Indeed, as the dimensionless magnetic induction γ increases (Figure 5b–d), the magnitude of the force becomes large and the symmetrical circulating flow is shifted toward the magnet. Due to the strong attracting force toward the heated wall, the working fluid near the junction receives more heat than the other area of the hot wall. Therefore, the upward buoyant flow emerges as soon as the fluid escapes from the magnetic force near the side of the pair magnet. This results in the roll cell thinning as observed in Figure 4. To compensate the roll cell thinning near the magnet, the cells away from the magnet are stretched horizontally.

#### 3.2. Effect on Heat Transfer

#### 3.3. Roll Cell Shifting by Magnet Location

^{4}without the magnetic field (same as Figure 4a). A series of snapshots is shown in Figure 8b–d. Interestingly, relatively cold fluid at the bottom of the magnet junction is pulled toward the junction, and then the roll cells are shifted towards it. Consequently, the descending flow is shifted to the magnet junction and correspondingly, the roll cells are shaped. Finally, the heat and fluid flows attain the steady state convection. This is also due to the periodic boundary condition at the sidewalls of the computational domain. Although this is an ideal situation, these results suggest that a locally strong magnetic field can control the position of the convection cells.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic model of the computation. The computational domain (fluid layer) is consisted by 600 × 102 nodes. The single block magnet is presumed 20 × 20 nodes. The distance of pair magnet is 300 nodes. (

**b**) arrangement of the pair magnets.

**Figure 4.**Streaklines and isotherms at Ra = 10

^{4}(left half of the computational domain). As the dimensionless magnetic induction γ increases, the roll cells near the magnet become thin. (

**a**) γ = 0.0 (no magnetic field); (

**b**) γ = 1.0; (

**c**) γ = 5.0; (

**d**) γ = 10.0.

**Figure 5.**Magnetothermal force on the working fluid at Ra = 10

^{4}. Vector length is respectively resized corresponding to the magnitude of the force. As the dimensionless magnetic induction γ increases, the magnetothermal force becomes stronger. This results in the thinning of roll cells. (

**a**) ∇B

^{2}; (

**b**) γ = 1.0; (

**c**) γ = 5.0; (

**d**) γ = 10.0.

**Figure 6.**Local Nusselt number on the hot wall at Ra = 10

^{4}(left half of the computational domain). (

**a**) 0 ≤ γ ≤ 1.0; (

**b**) 1.0 ≤ γ ≤ 10.0.

**Figure 8.**Transition of heat and fluid flow by different magnet location at Ra = 10

^{4}and γ = 1.0. (

**a**) 0 timestep; (

**b**) 33 × 10

^{4}timesteps; (

**c**) 78 × 10

^{4}timesteps; (

**d**) 267 × 10

^{4}timesteps (steady state obtained).

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**MDPI and ACS Style**

Wada, K.; Kaneda, M.; Suga, K.
Rayleigh-Bénard Convection of Paramagnetic Liquid under a Magnetic Field from Permanent Magnets. *Symmetry* **2020**, *12*, 341.
https://doi.org/10.3390/sym12030341

**AMA Style**

Wada K, Kaneda M, Suga K.
Rayleigh-Bénard Convection of Paramagnetic Liquid under a Magnetic Field from Permanent Magnets. *Symmetry*. 2020; 12(3):341.
https://doi.org/10.3390/sym12030341

**Chicago/Turabian Style**

Wada, Kengo, Masayuki Kaneda, and Kazuhiko Suga.
2020. "Rayleigh-Bénard Convection of Paramagnetic Liquid under a Magnetic Field from Permanent Magnets" *Symmetry* 12, no. 3: 341.
https://doi.org/10.3390/sym12030341