## 2 Numerical Methods  
### Computational Models  
We carried out two-dimensional numerical simulations of pressure-driven plane Poiseuille flow including a single circular or elliptical particle. Figure 1 shows the simulation model. The particle was initially located at the lower side and constrained in the \(y\)-directional motion (details in Section 2.5). The computational domain was set to width \(D=400\)\(\upmu\)m and length \(L=4D=1600\)\(\upmu\)m with a periodic boundary condition with pressure difference in the flow direction [12] to reduce the computational costs. In Section 3.1, the effects of periodic length and grid resolution are assessed. The computational domain was filled with an orthogonal grid of equal spacing \(\Delta x=\Delta y=0.5\)\(\mu\)m. The Reynolds number was set to \(Re=32\) for all simulations to consider inertial effects.  
The suspended particle was assumed to be rigid, neutrally buoyant, and non-Brownian. Aspect ratio was set to \(AR=1\), \(2\), and \(4\). Confinement is usually defined as the ratio of particle size to the channel width; however, when the particle is non-spherical, it is difficult to define it uniquely. Therefore, in this study, it was defined using three different characteristic lengths as follows: \(C_{1}=2b/D\), \(C_{2}=2a/D\), and \(C_{3}=2r/D\), where \(a\) is semi-major axis, \(b\) is semi-minor axis, and \(r\) is equivalent radius of the circular particle with the same area as shown in Figure 2. Case 1 is for comparison of the same minor diameter. Case 2 is for comparison of the same major diameter. Case 3 is for comparison of the same area for different aspect ratios. The parameters of confinement were set to \(C_{3}=0.05\), \(0.1\), and \(0.2\), which corresponds to the area fraction \(\phi=0.05\%\), \(0.20\%\), \(0.79\%\).  
In the studies on the inertial migration using numerical simulation, the comparison in case 2 (same major diameter) is often adopted [5; 16]. [PERSON] et al. [4] adopted the comparison similar to case 1 (the confinement increases as increasing aspect ratio). However, comparison in case 3 (with the same area fraction (particle concentration)) should be considered for rheological evaluation because the effective viscosity is predominantly dependent on particle concentration. [PERSON] et al. [17] adopted case 3 to compare the different geometrical shapes of particles in settling or flowing particle simulation. However, they investigated only trajectories of particles, and equilibrium positions of the inertial migration were not  
Figure 1: Simulation model including a single particle in pressure-driven plane Poiseuille flow. The center of the circular or elliptical particle was initially located at \(y=y_{0}\) with particle angle to the flow direction \(\theta_{\text{p}}=0\). The computational domain was \(L\times D\), and periodic boundary condition with pressure difference was applied in the flow direction.  
Figure 2: Schematic view of elliptical particles for comparison of three different definitions of confinement. The parameter of confinement \(C\) was chosen by the researchers, i.e., semi-major axis \(a\), semi-minor axis \(b\), and equivalent radius \(r\) of a circular particle with the same area.  
discussed. From the above background, these three definitions of confinement effects are discussed in this study.