# Elasto-Inertial Particle Focusing in Microchannel with T-Shaped Cross-Section
## 3 Results and Discussion
### Simulation Results  
In order to first assess the flow pattern without the inertial particle on the channel cross-plane, Figure 8a,b show contour maps of the axial velocity (\(U\)) at the flow rate of 10 \(\upmu\)L/min as a representative case for Type 2 and Type 3, respectively. The axial velocity was normalized by its maximum (\(U_{max}\)), while we used an actual physical unit for both the horizontal and vertical axis. Figure 8 shows that the aspect ratio of the channel significantly affects the mean flow structure on the cross-plane. For example, the horizontal location of \(U_{max}\) for both Type 2 and Type 3 is the same, residing on the symmetric plane (i.e., \(x=0\)). However, the vertical location of \(U_{max}\) is different depending on the aspect ratio of the channel cross-section. For Type 2, the local maximum in \(U\) occurs below the junction (i.e., \(y=10\)\(\upmu\)m), whereas Type 3 has a local maximum above the junction on the vertical axis (i.e., \(y=25\)\(\upmu\)m). This is because the shallow bottom channel for Type 3 induces a higher-pressure zone that moves the location of \(U_{max}\) further away from the bottom wall. Furthermore, the presence of the higher-pressure zone for Type 3 affects the mean flow topology, as shown in Figure 8a,b. Since H/W = 1 for Type 2, most of the fluid travels through the bottom channel, while the fluid in Type 3 mainly goes through the upper channel. Figure 6: Fluorescence intensity under viscoelastic fluid (top view) for 5 \(\upmu\)m and 13 \(\upmu\)m particles in (a) square channel, (b) Type 1 channel, (c) Type 2 channel and (d) Type 3 channel. Figure 7: Positions of 5 \(\upmu\)m and 13 \(\upmu\)m particles distributed within the channel for the flow rate of 100 \(\upmu\)L/min. (a) Type 1 channel, (b) Type 2 channel and (c) Type 3 channel. The contour maps of the shear rate for both channels are presented in Figure 8c,d. As reported in the previous study [23], the shear rate as a differential of the fluid velocity plays a crucial role in particle migration in the viscoelastic regime. This is because the shear rate is closely correlated with the competing mechanism of the inertial lift force between the shear gradient and wall interaction. Furthermore, previous studies [7,23] have shown a preferential migrating pattern of the inertial particles toward the region where the shear rate is zero, particularly when the second normal stress difference in a viscoelastic fluid is negligible. With these notions, Figure 8c,d give a qualitative picture of the migration pattern of the elasto-inertial particles for the cross-section shapes considered in this study. It is clear that each wall created a relatively higher shear rate, whereas the center and each corner of the channel cross-section had the lowest shear rate for both Type 2 and Type 3, as observed in the previous studies [3,11,20]. Most importantly, the protruding sharp edges at the junction caused the highest shear rate for both channels. The effect of these sharp edges became more significant when they were coupled with the bottom wall effect as the aspect ratio increased (see Figure 8d). For example, the topology of the low shear rate (i.e., \(\dot{\gamma}<1000\) s\({}^{-1}\)) seems bottleneck-like for Type 3. Based on these observations, it can be expected that the migrating pattern of the elasto-inertial particles passing through the Type 3 channel would be different to those in the Type 2 channel. Figure 9 presents the variation in x- and y-directional net elasto-inertial force (F\({}_{\text{net}}\)) for 5 and 13 \(\upmu\)m particles in the Type 2 channel along the aforementioned lines, as illustrated in Figure 2. The normalization was made for F\({}_{\text{net}}\) by \(\rho_{\text{f}}\)U\({}_{\text{max}}\)\({}^{2}\) d\({}^{4}\)/D\({}_{\text{h}}\)\({}^{2}\), where \(\rho_{\text{f}}\) and U\({}_{\text{max}}\) denote the fluid density and the maximum velocity of the channel.