# Elasto-Inertial Particle Focusing in Microchannel with T-Shaped Cross-Section
## 2 Materials and Methods
### Simulation for Viscoelastic Flow  
The current simulation aimed to explore the influence of T-shaped cross-sections with varying aspect ratios (i.e., H/W) on elasto-inertial particle migration, mainly when the particle was suspended in a viscoelastic fluid. This numerical study was designed to corroborate the results of the experimental study and thus provide an underlying mechanism for particle migration. To this end, we used a finite-volume-based flow solver (ANSYS Fluent) to assess the equilibrium position of the inertial particles, where all the forces acting upon the particles balance each other in the microchannel. In this numerical study, we utilized two types of channels (i.e., Type 2 and Type 3), with an aspect ratio (i.e., H/W see Figure 2) of 1 and 1.4, respectively. Each microchannel had an axial length of 1 mm to ensure a fully developed condition at the downstream measurement location. The physical dimensions of each channel cross-section are shown in Table 1, and they were kept identical to those used in the experimental works for an appropriate comparison. The \(x\) origin resided on the symmetric plane for both cross-sections. In contrast, the \(y\) origin was at the half-height of the bottom channel, corresponding to 25 \(\upmu\)m and 15 \(\upmu\)m away from the bottom wall for Type 2 and Type 3, respectively. Note that the Type 2 channel was considered as a baseline for comparison since it had an aspect ratio of unity. Due to the larger aspect ratio, the Type 3 channel was characterized by a shallow channel depth (i.e., 30 \(\upmu\)m) and a long neck (i.e., 70 \(\upmu\)m). Thus, a coupled effect between the bottom wall and the sharp edge at the junction could be promoted as compared to the case of the Type 2 channel. Furthermore, it should be noted that the vertical length for both Type 2 and Type 3 channels was slightly shorter than those used in the present experiment. We intentionally used this shorter length to keep the H/W as 1 and 1.4 for each microchannel. Nonetheless, the difference in \(D_{h}\) between the numerical and experimental studies was less than 5% for all cases and was thus negligible. In this numerical study, uniform velocity and pressure were applied for each inlet and outlet. Four selective flow rates (i.e., 1, 10, 100 and 150 \(\upmu\)L/min) were used, corresponding to the centerline velocity, \(U_{max}\), in a range of 0.0037-0.59 m/s. Flow parameters and the resulting channel Reynolds number (Re = \(\rho\)\(U_{max}\)\(D_{h}\)/\(\mu\), where \(\rho\) is the fluid density, \(U_{max}\) is the characteristic velocity, \(D_{h}\) is the hydraulic diameter of the current microchannels, \(\mu\) is the mean viscosity, respectively) are summarized in Table 2. Two particle diameters (i.e., \(d\) = 5 and 13 \(\upmu\)m) were considered similar to the experiments. The particle Reynolds number (Re\({}_{p}\) = Re (\(d/D_{h}\))\({}^{2}\), where \(d\) is the particle diameter) was thus in a range of O(10\({}^{-2}\))-O(10\({}^{0}\)). The exact value of the particle parameters in this numerical study are also reported in Table 2.  
Figure 2: Cross-sectional shape and dimensions of (**a**) Type 2 and (**b**) Type 3 microchannel considered in the numerical simulation. x and y on a cross-plane represent a horizontal and a vertical axis in the Cartesian coordinate system considered in the numerical simulation. A single, rigid, spherical particle was set at a fixed location on the cross-plane for both the Type 2 and Type 3 channels. Then, a net lift force acting on the particles was computed, varying with the particle location along the horizontal and vertical axis as depicted in Figure 2. Since the viscoelastic effect on the inertial particle was of particular interest in this study, we simulated the viscoelastic fluid by leveraging the user-defined function (UDF) provided by the software package. This UDF allowed us to model the viscosity of the current working fluid (i.e., PEO500 solution) depending on a range of shear rates.