## 4 Results and Discussion
### Relative Viscosity  
To examine the rheology of suspensions in a two-dimensional parallel-plate flow, evaluations were performed based on relative viscosity. Figure 8 shows the average of 24 analyses of the time history of relative viscosity. The blue, black, and red colors correspond to power-law indices \(n=0.6,\ 1.0,\ 1.4\), respectively, while the error bars indicate the standard deviations. The black dotted line represents the results from [PERSON]'s viscosity formula for this concentration.  
The relative viscosity shows similar trends across all power-law indices, and its values are lower than those predicted by [PERSON]'s viscosity formula. It increases during the non-dimensional time \(T^{*}=\) 20-30, and decreases thereafter. During \(T^{*}=\) 20-30, the calculated condition is impact departure, leading to an increase in relative viscosity owing to the rise in shear rate. The reasons for the decrease in relative viscosity after \(T^{*}=\) 40 are discussed below.  
According to previous studies, [PERSON] et al. (2012) proposed that macroscopic relative viscosity could be expressed as the sum of individual particle contributions, which depend on their relative positions in the channel. When particles flow near the wall, the relative viscosity increases, even under the same concentration conditions. This phenomenon was  
Figure 8: Relative viscosity changes over time for different power-law indices (\(n=\) 0.6, 1.0, 1.4). Black dotted line represents [PERSON]’s viscosity formula.  
Figure 7: Normalized PDD for different power-law indices. (a) \(n=0.6\), (b) \(n=1.0\), and (c) \(n=1.4\).  
also noted by [PERSON] et al. [13], who indicated that relative viscosity could be estimated by summing the contributions of each particle. As observed in Figures 7 and 8, it can be inferred that the relative viscosity decreases over time as particles migrate toward the center. This microscopic inertial migration of particles in the suspension results in the observed changes in macroscopic relative viscosity.  
Here, we present the results for the power-law index \(n=1.4\), regarding the time history of relative viscosity. A fluid characterized by a power-law index \(n=1.4\) demonstrates shear-thickening behavior. When the solvent of the suspension is a Newtonian fluid, the viscosity can be predicted by the [PERSON] viscosity formula, but when the solvent is shear-thickening, the suspension's overall viscosity will also be influenced by shear-thickening effects. Therefore, one would expect the relative viscosity of the suspension to exceed that predicted by [PERSON]'s viscosity formula. However, the findings show a deviation from this expectation, with the relative viscosity being lower than [PERSON]'s prediction. This discrepancy may be attributed to the inertial migration of particles, the presence of a non-Newtonian solvent, and particle-particle interactions, none of which are accounted for in [PERSON]'s viscosity formula. Among these factors, the loss of particles near the walls owing to inertial migration is believed to considerably influence the reduction in relative viscosity. In other words, it appears that the microstructure has a more substantial effect on the flow characteristics of the suspension than the properties of the solvent. Therefore, monitoring changes in the microstructure using PDD is valuable for predicting the flow characteristics of the suspension.