## 2 Numerical Methods
### Governing Equations  
For fluids under low-vacuum conditions, the validity of fluid dynamic governing equation can be confirmed by the Knudsen number [34]. The definition of Knudsen number is shown in Equation (1),  
\[K_{n}=\frac{K_{b}T_{f}}{\sqrt{2}\pi d^{2}pL} \tag{1}\]  
where \(K_{n},K_{b},T_{f},d,p\) and \(L\) are Knudsen number, Boltzmann constant, fluid temperature, molecular diameter of the fluid, total pressure of fluid and characteristic length of the research object, respectively. In this study, \(L\) is the height of the magnet unit (\(H\)). Based on the parameters in this study, the maximum Knudsen number calculated from Equation (1) is \(1.03\times 10^{-4}\), which is less than the reference threshold of 0.01. Therefore, the air in this study can be treated as a continuum. For studies on compressible viscous transient flow in natural convection, the continuity equation and the momentum equation can be expressed in the form of Equations (2) and (3), respectively. \[\frac{\partial}{\partial t}\bigg{(}\int_{V}\rho_{f}dV\bigg{)}+\oint_{A}\rho_{ f}\mathbf{v}\cdot d\mathbf{a}=0 \tag{2}\]  
\[\frac{\partial}{\partial t}\bigg{(}\int_{V}\rho_{f}\mathbf{v}dV\bigg{)}+\oint_{ A}\rho_{f}\mathbf{v}\times\mathbf{v}\cdot d\mathbf{a}=-\oint_{A}p\mathbf{I} \cdot d\mathbf{a}+\oint_{A}\mathbf{e}d\mathbf{a}+\int_{V}\mathbf{f}_{b}dV \tag{3}\]  
where \(\rho_{f}\), \(t\), \(\mathbf{v}\), \(\mathbf{a}\), \(\mathbf{I}\), and \(\mathbf{e}\) represent the fluid density, reference time frame, velocity vector, surface vector, identity matrix and viscous stress tensor, respectively. \(\mathbf{f}_{b}\) is the buoyancy source term defined as \(\Big{(}\rho_{f}-\rho_{r}\Big{)}\mathbf{g}\), \(\rho_{r}\) is the reference density and \(\mathbf{g}\) is the gravitational acceleration vector. This study also involves the energy transfer process and therefore the energy equation needs to be applied to satisfy the conservation of energy requirement. The energy equations for the fluid and solid are shown in Equations (4) and (5), respectively,  
\[\frac{\partial}{\partial t}\int_{V}\rho_{f}EdV+\oint_{A}\rho_{f}H \mathbf{v}\cdot d\mathbf{a}=-\oint_{A}\mathbf{q}\cdot d\mathbf{a}+\oint_{A} \mathbf{e}\cdot\mathbf{v}d\mathbf{a}+\int_{V}\mathbf{f}_{b}\cdot\mathbf{v}dV+ \int_{V}S_{u}dV \tag{4}\] \[\frac{d}{dt}\int_{V}\rho_{s}C_{ps}TdV=-\oint_{A}\mathbf{q}\cdot d \mathbf{a}+\int_{V}S_{u}dV \tag{5}\]  
where \(\rho_{s}\) is the density of solid, \(\mathbf{q}\) is the heat flux vector, \(E\) and \(H\) are the total energy and enthalpy per unit mass of fluid, and \(C_{ps}\) is the specific heat of solid. \(S_{u}\) represents the energy source terms. This energy source term includes the heat source from the coil and the heat flux from radiative heat transfer. The radiation heat flux is obtained by solving the radiative transfer equation, which can be written in the format of Equation (6):  
\[\frac{dI\Big{(}\mathbf{r},\mathbf{s}\Big{)}}{ds}=-\rho_{r}(\mathbf{r})I\Big{(} \mathbf{r},\mathbf{s}\Big{)}+k_{a}(\mathbf{r})I_{b}(\mathbf{r})+\frac{k_{s}( \mathbf{r})}{4\pi}\int_{4\pi}I\Big{(}\mathbf{r},\mathbf{s}^{\prime\prime} \Big{)}\phi\Big{(}\mathbf{r},\mathbf{s}^{\prime\prime},\mathbf{s}\Big{)}d \Omega^{\prime} \tag{6}\]