## 4 Conclusions  
The contribution of a single circular or elliptical particle motion, especially the particle-wall distance and rotational motion to the effective viscosity was studied using the regularized lattice Boltzmann method coupled with the virtual flux method. First, the effects of elliptical particle shape on the equilibrium position of inertial migration were summarized using three definitions of confinement. It was found that the effects of aspect ratio on the equilibrium position depend on the confinement definition. The effects of particle shape on the viscosity were also studied, and the contribution of particle shape to the effective viscosity was found to be enhanced when the particle flows near the wall. In terms of temporal and spatial variation of the relative viscosity, spatial variation is larger than temporal variation regardless of the aspect ratio and particle-wall distance.  
Conceptualization, M.K. and T.F.; methodology, M.K. and T.F.; software, M.K., T.F, and K.M.; validation, M.K.; investigation, M.K.; resources, T.F. and K.M.; writing--original draft preparation, M.K.; writing--review and editing, M.K., T.F., and K.M.; visualization, M.K.; supervision, T.F. and K.M.; project administration, T.F. and K.M.; funding acquisition, T.F. All authors have read and agreed to the published version of the manuscript.  
This research was funded in part by JSPS KAKENHI Grant Number JP20K04266.  
Data Availability Statement: Data sharing not applicable.  
The authors declare no conflict of interest.  
Figure 11: Comparison of temporal (circle) and spatial (triangle) standard deviation of effective viscosity for \(AR=2\) with confinement \(C_{3}=0.05\) (filled in blue) and \(0.1\) (red).  
\begin{tabular}{l l} \(c\) & lattice speed \\ \(D\) & channel width (characteristic length) \\ \(D_{p}\) & diameter of sphere \\ \(e_{\alpha}\) & discrete velocity vector \\ \(f_{\alpha}\) & distribution function \\ \(f_{\alpha}^{\rm eq}\) & equilibrium distribution function \\ \(f_{\alpha}^{\rm neq}\) & non-equilibrium distribution function \\ \(f_{\alpha}^{*}\) & virtual distribution function \\ \(f_{\alpha}^{\rm eq*}\) & virtual equilibrium distribution function \\ \(F_{\rm P}\) & total force vector acting on the particle \\ \(I\) & moment of inertia \\ \(L\) & channel length \\ \(m\) & mass of the suspended particle \\ \(n\) & normal unit vector \\ \(p_{0}\) & reference pressure \\ \(p_{\rm P}\) & pressure on the solid particle surface \\ \(\Delta p_{0}\) & pressure drop for particle-free flow \\ \(\Delta p\) & pressure drop \\ \(p_{\alpha}\) & pressure distribution function \\ \(Re\) & Reynolds number \\ \(Re_{\rm P}\) & particle Reynolds number \\ \(r\) & radius of circular particle \\ \(t\) & time \\ \(T_{\rm P}\) & torque for particle \\ \(T_{\rm w}\) & stress tensor \\ \(U\) & characteristic velocity \\ \(u\) & fluid velocity vector \\ \(\Delta x\), \(\Delta y\) & grid resolution \\ \(x_{\rm P}\) & position vector for particle \\ \(y_{\rm eq}\) & equilibrium position of inertial migration \\ \(y_{0}\) & initial position of a particle in y-coordinate \\ \(w_{\alpha}\) & weight coefficients \\ \(e\) & ellipticity \\ \(\eta_{\rm eff}\) & effective viscosity \\ \(\eta_{0}\) & viscosity of solvent \\ \([\eta]\) & intrinsic viscosity \\ \(\theta_{\rm p}\) & particle angle \\ \(\Pi_{ij}\) & non-equilibrium stress tensor \\ \(\rho\) & fluid density \\ \(\tau\) & single relaxation time \\ \(\tau_{\rm w}\) & shear stress tensor \\ \(\tau_{0}\) & wall shear stress for particle-free flow \\ \(\phi\) & area fraction \\ \end{tabular}  
\begin{tabular}{l l} eq & equilibrium \\ neq & non-equilibrium \\ eff & effective \\ \end{tabular}