## 1 Introduction  
Suspensions are prevalent in daily life and serve various purposes in food, medicine, and industry. The rheology of suspensions exhibits complex properties influenced by several factors, including particle size, shape, the spatial arrangement of the dispersed particles, the fluid forces acting on the particles, the interactions between the particles and their container, and the interactions between the particles themselves [1, 2, 3]. As a result, many aspects remain unsolved. A thorough understanding of suspension rheology, particularly viscosity, is crucial, because it can be easily modified to meet various requirements [4]. According to [PERSON]'s viscosity formula [5], the effective viscosity \(\eta_{\text{eff}}\) of a suspension can be determined based on the concentration of suspended particles. \[\eta_{\text{eff}}=\eta_{0}(1+[\eta]\phi). \tag{1}\]where \(\eta_{0}\) represents the viscosity of a particle-free fluid, \([\eta]\) denotes the intrinsic viscosity, and \(\phi\) signifies the area fraction in 2D systems. According to [PERSON]'s viscosity formula, intrinsic viscosity \([\eta]\) is well established as \([\eta]=2\) for 2D systems and \([\eta]=2.5\) for 3D systems [6]. Consequently, the viscosity can be determined based on the concentration. However, [PERSON]'s viscosity formula is only valid when particles are uniformly dispersed, small in size, present in low concentrations, and when the solvent is a Newtonian fluid. Therefore, [PERSON]'s viscosity equation is applicable when particles are isolated, but it cannot be applied when particle interactions or non-uniform particle dispersion occur or the solvent is non-Newtonian, and the behavior of relative viscosity under these conditions remains unclear. It has been reported that changes in the microstructure of a suspension, defined by the relative positioning of dispersed particles, considerably influence its rheology [7; 8]. One factor that can change this microstructure is the inertial migration of the particles. A notable example of this inertial migration is the Segre-Silberberg effect [9]. This phenomenon occurs in inertial flows in a channel, causing neutrally buoyant particles to concentrate at specific equilibrium positions in the radial direction. The equilibrium position occurs where the lift force, created by the velocity difference along the length of the particle, balances with the repulsive force from the channel wall [10]. [PERSON] et al. showed that the equilibrium position resulting from the Segre-Silberberg effect shifted toward the center of the channel as the confinement increased [11]. Here, confinement refers to the ratio of the particle diameter to the channel width. Furthermore, regarding the relationship between the suspension rheology and microstructure, it has been reported that the relative viscosity of suspensions containing a single particle remains the same when the vertical position of the particle and the level of confinement are equal [12]. It was found that as the particle approached the wall and the confinement increased, the relative viscosity increased. Building on this understanding, [PERSON] et al. explored the relationship between relative viscosity and the spatial arrangement of a single particle in two-dimensional parallel-plate flow. They proposed that the macroscopic viscosity of a low-concentration suspension can be estimated by summing the contributions to the relative viscosity resulting from the confinement and spatial arrangement of each particle [13]. Thus, the microstructure undergoes changes due to solvent effects and inertial migration, which in turn influence the rheology of the suspension. However, the abovementioned studies were performed on suspensions with a Newtonian solvent, and few numerical studies have focused on suspensions with a non-Newtonian solvent. In a previous study of suspensions with a non-Newtonian solvent, [PERSON] et al.