## 3 Numerical Analysis  
### Governing Equations  
Prior to the experiment, numerical analysis was performed using the COMSOL Multiphysics, CFD (Computational Fluid Dynamics) commercial software program. Numerical analysis was set under the same conditions as the experiment. To analyze changes according to the vertical arrangement of permanent magnets, numerical analysis was performed in 2D. The equations for continuity, momentum, and energy are as follows [15]:  
* Continuity: \[\
abla\cdot(\rho\overset{\rightharpoonup}{V})=0\] (1)
* Momentum: \[\
abla\cdot(\rho\overset{\rightharpoonup}{V}\overset{ \rightharpoonup}{V})=\
abla\cdot\mu\left(\left(\overset{\rightharpoonup}{V} \overset{\rightharpoonup}{V}\right)^{T}+\
abla\overset{\rightharpoonup}{V} \right)-\
abla p-\frac{2}{3}\mu(\
abla\cdot\overset{\rightharpoonup}{V})I+F\] (2)
* Energy: \[\
abla\cdot\left(\overset{\rightharpoonup}{V}(p+\rho E)\right)=\
abla\cdot(( \overset{\rightharpoonup}{\tau}\overset{\rightharpoonup}{V})+k\
abla T-\sum h _{j}\overset{\rightharpoonup}{J}_{j})+S_{h}\] (3) The last \(F\) term of momentum Equation (2) is the force term that corresponds to both the buoyancy force and the magnetic force. Therefore, the \(F\) term can be replaced with the sum of \(\rho\beta\big{(}T-T_{ref}\big{)}\) corresponding to buoyancy force and \((\overset{\rightharpoonup}{M}\cdot\
abla)\overset{\rightharpoonup}{B}\) corresponding to magnetic force. The final Equation (2) with these terms is as follows:
* Momentum: \[\
abla\cdot(\rho\overset{\rightharpoonup}{V}\overset{ \rightharpoonup}{V})=\
abla\cdot\mu\left(\left(\
abla\overset{ \rightharpoonup}{V}\right)^{T}+\
abla\overset{\rightharpoonup}{V} \right)-\
abla p-\frac{2}{3}\mu(\
abla\cdot\overset{\rightharpoonup}{V})I+ \rho\beta\Big{(}T-T_{ref}\Big{)}\] (4) \[+\left(\overset{\rightharpoonup}{M}\cdot\
abla\right)\overset{ \rightharpoonup}{B}\]  
Additionally, the expression for [PERSON]'s laws is as follows [16]:  
\[\overset{\rightharpoonup}{B}=\mu_{0}(\overset{\rightharpoonup}{M}+\overset{ \rightharpoonup}{H}) \tag{5}\]  
where \(B\) is the magnetic flux density, \(M\) is the magnetization, and \(H\) is the magnetic field strength. Furthermore, \(\mu_{0}\) is a constant value, meaning a permeability value in a vacuum state (=\(4\pi\times 10^{-7}\)); \(B\) and \(H\) as expressions for [PERSON]'s equation, respectively, are as follows [17]:  
\[\overset{\rightharpoonup}{\
abla}\cdot\overset{\rightharpoonup}{B}=0 \tag{6}\] \[\overset{\rightharpoonup}{\
abla}\times\overset{\rightharpoonup}{H} =0 \tag{7}\]  
Moreover, the relation for \(M\) and \(H\) in Equation (5) is expressed as follows:  
\[M=\chi_{m}\overset{\rightharpoonup}{H} \tag{8}\]  
In Equation (8), \(\chi_{m}\) is the magnetic susceptibility, and is defined as follows [18]:  
\[\chi_{m}=\frac{\chi_{0}}{1+\beta(T-T_{0})} \tag{9}\]  
In this expression, \(\beta\) means a thermal expansion coefficient, and it is confirmed that an inverse relationship between \(\chi_{m}\) and \(T\) is established. This is, high temperature ferrofluid responds loss to the external magnetic field, and low temperature fluid actively respondsto the magnetic field. Due to these characteristics, the convection phenomenon occurring in the magnetic field is called a thermomagnetic convection, and it promotes heat exchange.