# Elasto-Inertial Particle Focusing in Microchannel with T-Shaped Cross-Section
## 3 Results and Discussion
### Simulation Results
As mentioned earlier, the horizontal axis in Figure 9 used an actual physical unit, which is better for describing the particle location with respect to the origin. Symbols for each case are denoted in the legend in Figure 9. Solid lines connecting through the data points were made by piecewise cubic interpolation to show a shape-preserving trend in F\({}_{\text{net}}\) along the x and y axis. Note that the current simulation could not resolve a valid F\({}_{\text{net}}\) after a certain point when the spherical particle resided in the proximity of the channel wall. It should also be noted that the negative value in F\({}_{\text{net}}\) when x (or y) \(>0\) represents a directional force toward the origin, while its positive value when x (or y) \(<0\) indicates a directional force toward the wall. Figure 8: Simulation results of the (**a**,**b**) axial velocity magnitude \(U\), normalized by the maximum axial velocity \(U_{\text{max}}\), for Type 2 and Type 3 channels, and (**c**,**d**) shear rate, \(\dot{\gamma}\) for each Type 2 and Type 3 channels at the flow rate of 10 \(\upmu\)L/min. For both the 5 and 13 \(\upmu\)m particles shown in Figure 9a,b, \(F_{x,net}\) is always negative and has a single zero crossing at the origin at a flow rate of 100 \(\upmu\)L/min for Type 2 (i.e., T\({}_{2}\)D5Q100, T\({}_{2}\)D13Q100, red symbols). This indicates that both 5 and 13 \(\upmu\)m particles suspended in a viscoelastic fluid are likely to drift toward the origin in a lateral direction when the fluid inertia is sufficient. This numerical observation agrees well with our experiments, showing a single train of particles for both sizes at the same flow rate, as shown in Figure 5c,d. However, at the lower flow rate (i.e., 1 and 10 \(\upmu\)L/min) in Figure 9a,b, a different pattern in \(F_{x,net}\) between two inertial particles can be observed. For example, \(F_{x,net}\) for the larger particle (i.e., 13 \(\upmu\)m) still shows a negative with a single equilibrium point at the origin despite some fluctuations in \(F_{x,net}\) at the lowest flow rate (i.e., 1 \(\upmu\)L/min). This means the lateral drift consistently occurred for large particles (13 \(\upmu\)m) independent of the flow rate within our measured range. For the case of small particles (5 \(\upmu\)m), on the other hand, \(F_{x,net}\) has multiples of zero crossings at the lower flow rates, as shown in Figure 9a, indicating wide dispersion of inertial particles along the lateral direction. These numerical results confirm a unique migrating behavior of the inertial particles depending on the particle size and the flow rate reported in our experiments (see Figure 5d). Similar drifting patterns of elasto-inertial particles can also be observed for the vertical direction in Figure 9c,d. For the large particle size (i.e., 13 \(\upmu\)m), the equilibrium point in \(F_{y,net}\) is made independent of the current flow rates and consistently resides at 10 \(\upmu\)m away from the vertical origin for all cases (see Figure 9d). The case of the small particle also shows a similar trend in \(F_{y,net}\) and an equilibrium point at the highest flow rate, as shown in Figure 9c. This observation in our numerical study explains the particle focusing when the effect of the elasto-inertial force is large enough due to either a high degree of the flow rate or a large particle surface area. Furthermore, the vertical equilibrium point observed herein is where the local minimum \(\dot{\gamma}\) and the local maximum \(U\) are positioned. In this study, our experimental result shows a different migrating pattern of particles depending on their size, particularly in the Type 3 channel, and thus has a potential implication for particle separation by leveraging the aspect ratio of the current T-shaped channel. Therefore, in this numerical study, it is of particular importance to confirm such migrating behavior can be observed in the Type 3 channel and explore their background mechanism compared to the one in the Type 2 channel.