## 3 Results and Discussion  
### Periodic Length and Grid Resolution  
The effects of periodic length, which refers to the length \(L\) of the parallel plates model, on the equilibrium position of inertial migration were assessed, and a grid independence study was conducted. First of all, the effects of periodic length were investigated. This length should be sufficient to neglect the effect of the boundary condition. The lift coefficient profiles were compared to determine the periodic length. Figure 4 shows the profiles of lift coefficient of a single circular particle for four different periodic lengths \(L/D=1\), 2, 4, and 6. On the lower wall side (\(y/D=0.2\)), the lift force exerted on the particle was positive, which indicates that particle in this region is lifted up toward the center. On the other hand, when the lift force is negative, particle is lifted down from center region to the wall side. The profiles for \(L/D=4\) and 6 were almost the same. Table 1 shows equilibrium positions obtained from the lift coefficient profiles. The periodic length \(L/D=4\) was thus adopted in the simulation conditions. Figure 4: Lift coefficient profiles of a single circular particle with \(C_{3}=0.05\) for four different periodic lengths \(L/D=1\), 2, 4, and 6 (the number of cells for \(D\) was 600). Secondly, a grid independence study was performed. Figure 5 shows profiles of lift coefficient with three different grid resolutions. When the particle flowed near the lower wall, the lift coefficient became slightly larger with increasing grid resolutions as illustrated in Figure 5. The equilibrium position, where the lift force is zero, moved slightly toward the center for higher resolution. Table 2 shows equilibrium positions obtained from the lift coefficient profiles. The results show that the number of cells for channel width is required to be at least 600. Thus, the number of cells for channel width \(D\) was set to 800 in all simulations to take non-spherical particle shapes into account. At least 20 cells were used for the grid resolution of the minor axis of \(AR=4\) in case 3 for \(C_{3}=0.05\). Effects of Aspect Ratio of Elliptical Particle on the Equilibrium Position Due to Inertial Migration  
The effects of aspect ratio on the lift coefficient profiles and the equilibrium positions of the particles with the same area fraction (in case 3) are discussed. Figure 6 shows profiles of lift coefficient of an elliptical particle for different aspect ratios with various confinements. Note that the vertical axis scale depends on each figure for clarity. The lift \begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Cells for \(D\) [Cells]** & **Number of Cells for \(2r\) [Cells]** & \(y_{\text{eq}}/D\) [-] \\ \hline
400 & 20 & 0.255 \\
600 & 30 & 0.258 \\
800 & 40 & 0.258 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Normalized equilibrium position \(y_{\text{eq}}/D\) of a single circular particle with \(C_{3}=0.05\) for different grid resolutions. The parameters \(D\) and \(2r\) denote the number of grids for channel width and particle diameter, respectively. Figure 5: Lift coefficient profiles of a single circular particle with \(C_{3}=0.05\) for different grid resolutions: 400, 600, and 800 cells for \(D\) in the condition of \(L/D=4\). coefficient profiles were similar between \(AR\) = 1 and \(AR\) = 2. On the contrary, those for \(AR\) = 4 were different from those for \(AR\) =1 and 2. When the particle flowed on the lower wall side, the lift coefficient became larger with increasing aspect ratio, i.e., the repulsive force from the wall became larger with increasing aspect ratio. Table 3 shows equilibrium positions for different aspect ratios with various confinements. The equilibrium position shifted toward the center with the increase of aspect ratio, and it was enhanced with higher confinement (larger particle). The effects of aspect ratio on the equilibrium positions with the different confinement definitions are discussed.