## 1 Introduction  
Suspension flows are ubiquitous in industrial process such as filtration, cosmetics, and food processing and also in nature such as mud and blood flow. Rheological properties of a suspension flow partly depend on its microstructure, which refers to spatial arrangement and orientation of suspended particles [1]. Rheological properties of suspension play important roles in terms of product quality; therefore, it is important to reveal the relationship between macroscopic rheological properties of suspensions and their microstructures to handle suspension rheology effectively. In a channel flow, suspended particles migrate transversal to the streamlines and converge at the equilibrium position due to inertial effects. This phenomenon is known as the \"tubular pinch effect\" or \"[PERSON] effect\". [PERSON] and [PERSON] [2] found that rigid spheres in a Poiseuille flow through a tube accumulate at about 0.6 tube radii from the axis. After these findings, many attempts have been made to clarify the mechanism of inertial migration and to extend this phenomenon to broad applications. We have observed that the microstructure of a suspension changes in time mainly due to inertial effects of the suspended particles [3]. Such inertial effects might be significant in practical cases, such as industrial flow or blood flow; therefore, it is important to consider inertial migration. Strictly speaking, suspended particles are commonly not always spherical in nature, i.e., non-spherical shapes should be considered. [PERSON] et al. [4] investigated the motion of an elliptical particle in plane Poiseuille flow. The numerical results showed that the equilibrium position of the particle with higher aspect ratio is closer to the center of the channel. [PERSON] et al. [5] also numerically investigated the inertial migration of the elliptical particle, and they showed that the equilibrium position of the elliptical particle changes with aspect ratio. However, in these previous studies, confinement, which is the ratio of particle size to the channel diameter (or width), or particle concentration was not kept constant with changing aspect ratio. It is important to take into account not only aspect ratio but also confinement when suspension rheology is considered because effective viscosity strongly depends on confinement as well as particle concentration. Effective viscosity of a dilute suspension including spheres was firstly studied and summarized by [PERSON] [6]. According to his theory, the effective viscosity can be expressed as  
\[\eta_{\text{eff}}=\eta_{0}(1+[\eta]\phi), \tag{1}\]  
where \(\eta_{\text{eff}}\) is the effective viscosity, \(\eta_{0}\) is the viscosity of the solvent, \([\eta]\) is the intrinsic viscosity, and \(\phi\) is solid volume fraction in three-dimensional conditions or area fraction in two-dimensional conditions. Intrinsic viscosity \([\eta]\) is also called shape index because it depends on particle shape; for example, \([\eta]=2\)[7] for a rigid circular particle and \([\eta]=2.3\)[8] for an elliptical porous particle for Darcy number \(Da=10^{-12}\) with aspect ratio \(AR=2\) in two-dimensional conditions. [PERSON] [9] extended [PERSON]'s work to the case of ellipsoidal particles. He investigated the motion of a single ellipsoid in a shear flow and found that increase of viscosity can be represented by modifying the intrinsic viscosity. [PERSON] et al. [10] numerically demonstrated that the intrinsic viscosity of spheroids depends on rotational mode such as tumbling or log rolling mode. They showed that the intrinsic viscosity agrees well with [PERSON]'s analytical solution for various aspect ratios. [PERSON] et al. [11] studied the effects of confinement on the effective viscosity of two-dimensional suspensions. They found that intrinsic viscosity is also a function of confinement. As particle size becomes larger, intrinsic viscosity also increases.