## 3 Simulation Results  
### Magnetic Field Strength and Magnetic Volume Force Distribution  
Figure 3 shows the distribution of magnetic field strength in the fluid domain when the permanent magnets have different remanences, 0, 0.5, 1, and 1.5 T, respectively. The critical magnetic field strength calculated from the above is 58.26 kA/m, and the corresponding isoline is shown in the red curve in Figure 3. The fluid domain could be then divided into two regions by the isoline, i.e., the non-magnetic saturation region above the isoline and the magnetic saturation region below the isoline. Obviously, the greater the remanence, the greater the magnetic saturation zone, which also indicates that it is necessary to consider the existence of a magnetic saturation phenomenon when the magnetic field is strong enough.  
To compare the magnetic forces calculated using a different magnetization model, Figure 4 shows the distribution of the magnetic volume force distribution in the fluid domain. Here, I represents the conventional magnetization model, i.e., Equation (7), and II represents the magnetization model proposed in this paper, i.e., Equation (8). The black boundary line in the figure was the isoline of the magnetic field strength, with 58.26 kA/m. It can be seen that there is a clear difference in the magnitude of the magnetic volume force calculated using the two magnetization models in the region below the black critical isoline. In order to quantify the difference, a vertical path was selected, as shown by the black vertical line in Figure 5, and the magnetic volume forces calculated using the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Vacuum** \\ **Permeability** \\ **Air** \\ \end{tabular} & \begin{tabular}{c} **Relative** \\ **Remaneability of** \\ **Air** \\ \end{tabular} & \begin{tabular}{c} **Remanence** \\ **Fluid Density** \\ **Viscosity** \\ \end{tabular} & \
\begin{tabular}{c} **Fluid Dynamic** \\ **Viscosity** \\ \end{tabular} &
\begin{tabular}{c} **Particle Density** \\ **Particle Density** \\ \end{tabular} \\ \hline \(4\pi\times 10^{-7}\) H/m & 1 & 0.5 T, 1 T, 1.5 T & 1000 kg/m\({}^{3}\) & 8.94 \(\times 10^{-4}\) Pa-s & 5180 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The physical parameters required for the simulation.  
Figure 3: Magnetic field strength distribution in the fluid domain for different remanences.  
two magnetization models were compared along this path. The calculated results are shown in Figure 5b,d,f, corresponding to Br = 0.5, 1, and 1.5 T, respectively. It was found that the magnetic volume forces calculated by the two models were the same in the unsaturated region, while the deviation occurred when magnetic field strength was higher than the critical strength. Obviously, the higher the Br value, the earlier and larger the deviation was, as shown in Figure 5b,d,f. When the magnetic field strength reached 175 kA/m, the magnetic forces calculated by the two models were up to four times different in value. It can be seen that when the magnetic field strength did not reach the magnetic saturation value of the magnetic particles, the traditional model was still applicable. However, when the magnetic field strength exceeded the critical strength \(H_{0}\), it was necessary to consider the influence of the magnetic saturation of the magnetic particles in order to accurately calculate the magnetic volume force.  
Figure 4: Magnetic volume force distribution in the fluid domain for different remanent magnetic strengths. I represents the magnetic force calculated according to the conventional magnetization model, i.e., Equation (7); II represents new magnetization model, i.e., Equation (8).  
Figure 5: Quantitative comparison of the magnetic volume forces calculated by different magnetization models under the action of different remanences. (**a**,**b**) Br = 0.5 T; (**c**,**d**) Br = 1 T; (**e**,**f**) Br = 1.5 T.