# Elasto-Inertial Particle Focusing in Microchannel with T-Shaped Cross-Section
## 3 Results and Discussion  
The elasto-inertial particle focusing in viscoelastic non-Newtonian fluid flowing through a straight microchannel is a result of competition between the inertial lift force and elastic force [6]. It has also been clarified that the channel cross-section plays an essential role in particle focusing. When flowing through a straight microchannel, initially random suspensions of particles reach certain equilibrium positions.  
The Reynolds number (Re) is defined as \(\text{Re}=\rho\ U_{max}\ D_{h}/\mu\) (\(\rho\) is the fluid density, \(U_{max}\) is the characteristic velocity, \(D_{h}\) is the hydraulic diameter, \(\mu\) is the mean viscosity), which is used to quantify the relative importance of the inertial and viscous effect for a given flow condition. A dimensionless Weissenberg number (Wi) is defined as \(\text{Wi}=\lambda\ \dot{\gamma}=2\ \lambda\ U_{max}/D_{h}\) (\(\lambda\) is the relaxation time of the polymer solution, \(\dot{\gamma}\) is the shear rate), which is used to quantify the viscoelastic effects of a fluid and describes the ratio of elastic and viscous forces. To compare the relative importance of the elastic effect to the inertial effect, the fluid elasticity (EI) is defined as the ratio of the Weissenberg number to the Reynolds number,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **Cross-Section** & \(d\) & **Flow Rate** & \(\mathbf{U}_{max}\) & **Re** & \(\text{Re}_{p}\) \\ \hline  & & (\(\mu\)m) & (\(\mu\)L/min) & (m/s) & & \\ \hline T\({}_{2}\)D5Q1 & Type 2 & 5 & 1 & 0.0037 & 0.29 & \(1.84\times 10^{-2}\) \\ T\({}_{2}\)D5Q10 & Type 2 & 5 & 10 & 0.037 & 2.90 & \(1.84\times 10^{-1}\) \\ T\({}_{2}\)D5Q100 & Type 2 & 5 & 100 & 0.37 & 29.0 & \(1.84\times 10^{0}\) \\ \hline T\({}_{2}\)D13Q1 & Type 2 & 13 & 1 & 0.0037 & 0.29 & \(4.79\times 10^{-2}\) \\ T\({}_{2}\)D13Q10 & Type 2 & 13 & 10 & 0.037 & 2.90 & \(4.79\times 10^{-1}\) \\ T\({}_{2}\)D13Q100 & Type 2 & 13 & 100 & 0.37 & 29.0 & \(4.79\times 10^{0}\) \\ \hline T\({}_{3}\)D5Q1 & Type 3 & 5 & 1 & 0.0038 & 0.24 & \(1.84\times 10^{-2}\) \\ T\({}_{3}\)D5Q10 & Type 3 & 5 & 100 & 0.038 & 2.42 & \(1.84\times 10^{-1}\) \\ T\({}_{3}\)D5Q100 & Type 3 & 5 & 100 & 0.38 & 24.2 & \(1.84\times 10^{0}\) \\ T\({}_{3}\)D5Q150 & Type 3 & 5 & 150 & 0.59 & 37.6 & \(2.94\times 10^{0}\) \\ \hline T\({}_{3}\)D13Q1 & Type 3 & 13 & 1 & 0.0038 & 0.24 & \(1.84\times 10^{-2}\) \\ T\({}_{3}\)D13Q10 & Type 3 & 13 & 100 & 0.38 & 24.2 & \(1.84\times 10^{-1}\) \\ T\({}_{3}\)D13Q150 & Type 3 & 13 & 150 & 0.59 & 37.6 & \(2.94\times 10^{0}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Flow and particle parameters in the numerical simulation.  
i.e., El = Wi/Re = 2 \(\lambda\)\(\mu\)/(\(\rho\)\(D_{h}\)\({}^{2}\)) [19], which is a dimensionless number and can determine the number of particle focusing points under a non-Newtonian viscoelastic fluid.  
Figure 3 shows the SEM (scanning electron microscope) images of three kinds of microchannels with different T-shaped cross-sections. They show that the two PDMS molds were perfectly aligned and bonded thanks to their geometrical shapes, as shown in Figure 1. The detailed dimensions of the fabricated microchannels are listed in Table 1.