## 1 Introduction  
Superparamagnetic particles with their unique magnetic properties (low coercivity, fast magnetic response, etc.) are widely used in various applications such as biomedicine, magnetic adsorbents, magnetic ore dressing, magnetic recording materials, etc. [1, 2, 3, 4]. Magnetic particles are also widely used in mechanical systems. For example, [PERSON] et al. reported that magnetic fluids can be used for heating and cooling mechanical systems [5]. [PERSON] and [PERSON] et al. investigated the sealing effect of magnetic fluids in rotating machinery [6, 7]. Due to their excellent physical properties, some magnetic nanoparticles have also been added to lubricants as nano-additives [8, 9, 10]. In particular, some magnetic particles need to be frequently manipulated by an external magnetic field to achieve the desired effect. For example, in the field of biomedicine, the magnetic drug needs to be assisted by an external magnetic field to reach known cancerous areas [11; 12]. Our previous work has also found that magnetically lubricated particles can exhibit superior performance with an external magnetic field [13]. If the time and content of the magnetic particles reaching the target area are known, then this will be very useful to maximize the effect of the particles and also to improve the utilization of the particles. At this point, establishing an accurate kinetic model of magnetic particles in liquid-phase fluids under magnetic field traction was necessary.  
The motion analysis of magnetic particles in fluids under the action of magnetic fields can generally be divided into two kinds, one is to consider the individual particles as the object of study. The forces acting on the particles are analyzed, respectively, mainly including the fluid traction, magnetic field forces, and gravity as the key influencing factors. The weak interactions between particles are usually ignored when the content is very small in fluid, and the [PERSON]'s second law formula is used to build the dynamic equations of the particles [14; 15; 16; 17; 18]. Another approach is to consider a homogeneous mixed phase containing magnetic nanoparticles, where the magnetic field force is considered as a bulk force acting on the mixed phase, and the N-S equation and the convective diffusion equation are solved to obtain the distribution of particle concentration in the mixed phase over time [19; 20; 21; 22]. Due to the magnetic saturation strength of magnetic particles, it is also of interest to accurately calculate the magnetic force on magnetic particles at different magnetic field strengths. In fact, the particle size distribution of magnetic nanoparticles in the fluid is not constant because the nanoparticles tend to agglomerate. Therefore, it is worth noting how changes in particle size and magnetic field strength affect the movement of magnetic particles, taking into consideration the magnetic saturation phenomenon.  
In this work, the true magnetization phase of a superparamagnetic particle in a magnetic field was described by a segmentation function, providing a new magnetization model for the calculation of magnetic forces in magnetic fields for different particles. Then, the motion characteristics of magnetic particles with different particle sizes under the different magnetic field strengths were analyzed, taking water as the study system. By establishing the mechanical model, the factors affecting the trajectory of the particles were analyzed, possibly providing significant guidance for realizing the precise targeting of magnetic nanoparticles under an external magnetic field manipulation.