## 2 Models and Analyses
### Static Models  
In order to analyze the force on the magnetic particles present in the suspension, the magnetic field distribution on the workpiece surface, especially the magnetic field leakage distribution above a crack, was first determined. This magnetic field leakage directly determines the final aggregation of the magnetic particles present in a suspension [18]. The magnetic dipole model was used to calculate the crack magnetic field leakage distribution. Figure 1 displays the magnetic field distribution at the position \(P\left(x,y\right)\) above a crack, \(d_{\mathrm{c}}\) is the depth and \(2\,w_{0}\) is the width of the crack. In order to make the dynamic analysis of magnetic particles simple and clear, the theoretical analysis analyzes the force on a single magnetic particle [19]. First, for the case of a single magnetic particle, the force and motion of the particle in the MPI system were analyzed. Assume that the particle is a small sphere with a density \(\rho\) and a diameter \(d_{m}\). The acceleration of gravity is \(g\), the density of the liquid is \(\rho_{w}\), and the suspension height from the workpiece surface is \(h_{0}\). As the size of each magnetic particle was small (3-8 \(\upmu\)m; much smaller than the overall size of the research system (cm range)), the volume effect of the particle was ignored. It was also assumed that the magnetic field strength was uniform at the center of the magnetic field of the sphere. Due to the one-to-one correspondence between the magnetic field strength and the x-coordinate, \(H\left(x,y\right)\) is abbreviated as \(H\left(x\right)\) in the subsequent analysis since the thickness of the magnetic particle suspension (\(y\)) is thin enough to allow an effect on H. The potential energy of the small particle in the magnetic field can be expressed as [20; 21]:  
\[U=-\int_{V}\frac{\mu_{0}\kappa H^{2}(x)}{2}dx \tag{1}\]  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]  
where \(H_{w}\) is the magnetic field strength, \(\mu_{0}\) is the vacuum permeability (\(4\pi\times 10^{-7}\mathrm{T}\cdot\mathrm{m}\,/\mathrm{A}\)), and \(\kappa\) is the magnetic susceptibility. The magnetic force acting on the small particles \(F_{m}\) was represented by a negative gradient of the potential energy [21]. \[F_{m}=-\
abla(U) \tag{3}\]  
The sphere was also subject to resistance according to the settlement dynamics:  
\[F_{d}=C\frac{\pi}{4}d_{m}^{2}\cdot\frac{\rho u^{2}}{2} \tag{4}\]  
Figure 1: Magnetic dipole model. where \(F_{d}\) is the drag force, \(C\) is the drag coefficient, and \(u\) is the speed of the particle. The volume of the spherical particles is \(V=\frac{\pi}{6}d_{m}^{3}\). Thus, in addition to the above forces, the magnetic particles were also subjected to gravity \(F_{g}\) and buoyancy \(F_{b}\) in the magnetic particle suspension carrier. They can be obtained by multiplying the product of the volume and the acceleration of gravity \(g\), respectively, with the density of the magnetic particle and the liquid suspension carrier, as:  
\[F_{g}=\frac{\pi}{6}d_{m}^{3}\rho g \tag{5}\]  
\[F_{b}=\frac{\pi}{6}d_{m}^{3}\rho_{w}g \tag{6}\]  
The fluid viscosity was also considered. Particles in the flow field were subjected to increased resistance due to Bassett forces \(F_{Ba}\)[22]. \[F_{Ba}=\frac{3}{2}d_{m}^{2}(\pi\rho_{w}\mu)^{0.5}\cdot\int_{0}^{t}(t-t^{\prime })^{-0.5}\frac{d(u)}{dt}\cdot dt^{\prime} \tag{7}\]  
where \(t\) is the current time with the unit of second (s) and \(t^{\prime}\) is an integral viable with the unit of second (s). After the particles settled on the workpiece surface, they were subjected to static friction \(f_{1}\).