## 4 Discussion  
A force analysis of a single magnetic particle in a fluid was carried out to understand the effect of particle diameter and remanence of a magnet on the trajectory of the magnetic particles, as shown in Figure 8. It can be found that the motion of the particle along the X direction is mainly determined by the magnetic field force \(\mathbf{F_{Mx}}\), while the motion along the Y direction is determined by both gravity and the magnetic field force \(\mathbf{F_{My}}\). Since both the magnetic field force and gravity are bulk forces, magnetic particles with different diameters move along the same trajectory, and the speeds of different diameter particles are different due to \(\mathbf{F_{M}}\) and \(\mathbf{G}\) and are proportional to volume. The larger the particle size, the greater the gravitational and magnetic forces, and the faster the motion.  
However, the strength of the magnetic field in the fluid domain changed as the remanence varied, which in turn changed the magnitude of \(\mathbf{F_{Mx}}\) and \(\mathbf{F_{My}}\). As the gravity in the Y direction was constant, the forces on the particles along the X and Y directions no longer changed proportionally. This could be the main reason as to why the trajectory of the particles changed when a different remanence was adopted, as shown in Figure 7. However, it is worth noting that, as shown in Figure 7, when the releasing height was 5 mm, the trajectory of the particles was same for all remanences. Based on this phenomena, we further focused on the ratio of the Y-directional force, the gravity \(\mathbf{G}\) to \(\mathbf{F_{My}}\), as shown in Figure 9. It was clear that the closer to the bottom of the fluid domain, the smaller the value of \(\mathbf{G}/\mathbf{F_{My}}\). It can be seen that the ratio of the \(\mathbf{G}/\mathbf{F_{My}}\) was only about 5% for three different remanences at 5 mm from the bottom. Combining the trajectory of the particles shown in Figure 7a, when the proportion of gravity in the Y-directional force on the particle was so small, about 5%, that it could be ignored, the motion direction of particles was mainly determined by the magnetic field force. Because \(\mathbf{F_{Mx}}\) and \(\mathbf{F_{My}}\) varied proportionally as the remanences changed, this may be the reason as to why the trajectory of particles remained the same when different remanences were applied at a releasing height of 5 mm.  
Figure 7: The effect of remanence on the trajectory of particles at different releasing positions. (**a**) Releasing position 5 mm from the bottom; (**b**) releasing position 10 mm from the bottom; (**c**) releasing position 15 mm from the bottom. The diameter of particle is 1 \(\upmu\)m.  
Considering the limited experimental conditions, it is difficult to observe the motion of individual nanoparticles under a magnetic field. So, we designed a simple experiment to observe the motion of magnetic particles in a liquid-phase fluid under a magnetic field. The experimental details were as follows. A transparent cuvette containing aqueous solution was placed on a cylindrical permanent magnet with a remanence of approximately 1.2 T. The mixture containing the magnetic particles was then injected into the aqueous solution using a pipette, and the trajectory of the magnetic particles was recorded in real time using a high definition camera. The trajectory of the particles in the absence of a magnet was  
Figure 8: Force analysis of magnetic particles in fluid domain.  
Figure 9: The distribution of the value of \(G/F_{My}\) in the fluid domain under the action of different remanences.  
also observed as a comparison. The observed images are shown in Figure 10. Magnetic particles disorderly fell under gravity in the absence of a magnetic field. In the presence of a magnetic field, the magnetic particles showed a lateral shift toward the center of the cuvette as they fell; this trend is similar to the simulated results in Figures 6 and 7.