## 3 Results and Discussion
### Spatial and Temporal Changes in the Effective Viscosity  
The fluctuation of effective viscosity in time and space due to particle behavior was found to be quite significant [12; 13]. In this section, the effects of the particle flow position and rotational motion on the spatial distribution of the relative viscosity are discussed, and the spatial and temporal changes in the viscosity are evaluated. Figure 9 shows distributions of local WSS on the upper and lower wall for a circular particle flowing at \(y/D=0.2\) with various confinements \(C_{3}=0.05,0.1,\) and \(0.2\). Figure 9b shows axial velocity distribution and particle position in connection with Figure 9a,c for \(C_{3}=0.2\). Note that local WSS can also be interpreted as the local relative viscosity from Equation (32). The lowest local WSS lies on the lower wall in the region near the particle (Figure 9c). This is already reported by previous studies [12; 14; 28]. The higher regions of WSS on the lower wall were observed near the edge of the particle. The peak in front of the particle is denoted by \(\tau_{1}\), and that behind the particle is denoted by \(\tau_{2}\) (Figure 9c). In contrast, on the upper wall, the local relative viscosity near the particle becomes higher (Figure 9a). These characteristics are qualitatively consistent with the results from [PERSON] et al. [28] who studied the inertial migration of a single circular particle flowing between the parallel plates using IBM-LBM. In order to capture the characteristics of the variation of the spatial WSS distribution with rotational motions of the particle, we focused on the difference of peaks \(\tau_{1}-\tau_{2}\). Figure 10 shows the time history of angular acceleration and difference of peak \(\tau_{1}-\tau_{2}\). The elliptical particle rotates, accelerating and deaccelerating during the cycle. When the peak is positive, \(\tau_{1}\) is higher than \(\tau_{2}\), and vice versa. When the angular acceleration of the particle equals \(0\), the particle is oriented in the flow direction around non-dimensional time \(t=4.6\) and \(t=7\) in Figure 10, and the particle is perpendicular to the flow direction around non-dimensional time \(t=6\) and \(t=8\) in Figure 10. In addition, the negative difference of peak is dominant, which indicates that \(\tau_{2}\) is higher than \(\tau_{1}\) when the particle flows at \(y/D=0.2\), and this depends on the particle-wall distance. Figure 8: Comparison of relative viscosity of different aspect ratios and confinement. Note that the vertical axis scale is different for each figure for clarity. Figure 9: Distributions of local relative viscosity determined by wall shear stress on the (**a**) upper and (**c**) lower wall for a circular particle flowing at \(y/D=0.2\) with various confinements \(C_{3}=0.05\) (blue dashed line), \(0.1\) (green dot-dashed line), and \(0.2\) (red solid line). The red shadow indicates the region where the particle exists for \(C_{3}=0.2\) as shown in (**b**) with axial velocity distribution. The forward and backward peaks of the local relative viscosity on the lower wall are denoted by \(\tau_{1}\) and \(\tau_{2}\), respectively. Since the rotational motion of the non-circular particle is unsteady, the effective viscosity dramatically changes not only in space but also in time. The spatial and temporal changes in viscosity were evaluated, and these effects were compared. Figure 11 shows the comparison of spatial and temporal standard deviation (SD) of effective viscosity. The temporal and spatial SD was calculated by  
\[\text{SD}_{\text{time}}=\sqrt{\frac{1}{n}\sum_{t=1}^{n}\Bigl{(}\left\langle\tau( t)\right\rangle-\overline{\left\langle\tau(t)\right\rangle}\Bigr{)}^{2}}, \tag{33}\]  
\[\text{SD}_{\text{space}}(t)=\sqrt{\frac{1}{2L}\sum_{i=1}^{2L}(\tau(t,\ i)- \left\langle\tau(t,\ i)\right\rangle)^{2}}, \tag{34}\]  
where \(\left\langle\tau\right\rangle\) symbolizes the spatial average for \(\tau\), and \(\overline{\tau}\) symbolizes time-average for \(\tau\).