# Research on the Inertial Migration Characteristics of Bi-Disperse Particles in Channel Flow
## 1 Introduction  
The precise focusing process of particles is a critical step of separation and screening techniques [1; 2; 3] and is widely used in the fields of biochemical engineering and medical areas. In recent decades, microfluidic technology has been widely used, owing to its unique advantages in many fields and rapid development, among which inertial microfluidic technology is one of the typical representatives [4]. Inertial microfluidic, as a passive method, has attracted wide attention for its advantages of simple operation, simple structure, being harmless to cell viability, and high-throughput. The suspended particles naturally gathered together and migrated to the equilibrium position, under specific flow fields and geometric conditions, confirming, by [PERSON] and [PERSON] [5] with extensive experiments, that the particles gathered into a concentric ring on the cross section of the channel and continued to move in the flow direction. This focusing was thought to be a balance of shear-induced, wall-induced, and rotation-induced lift forces, and the underlying mechanisms have been extensively investigated [6]. Remarkably, [PERSON] and [PERSON] [5] proposed that the particles not only migrated by inertia to the equilibrium position of the cross section, but also ordered longitudinally along the main flow direction to form particle trains that may result from particle-particle interactions. [PERSON] et al. [7] observed that, after particles randomly enter the pipe, the particles would be arranged along the flow direction to form a multi-particle train structure, defined as the particle train [7; 8] when three or more particles are aligned with a regular inter-particle spacing. [PERSON] et al. [8] found that increasing the Reynolds number was beneficial to the formation of particle trains, but it reduced the number of particles in the particle trains. The distance between adjacent particles decreases with the increase in Reynolds number. [PERSON] et al. [9] found in numerical simulations that, during inertial migration, owing to the force between particles and the fluid, with the effect of inertia, the particles moved to the equilibrium position one by one, forming a line of evenly spaced particle structures. However, the particle spacing increases slowly with the increase in the inertial migrationlength; the particle inertial migration length decreases with the decrease in the power-law exponent, and the shear thinning effect will increase the formation of ordered particle trains; the particle spacing decreases with the increase in Reynolds number and blocking ratio. [PERSON] et al. (2010) showed that, for single-line particle pairs, the inter-particle spacing increased to a larger value for further downstream, while two lines of 12 particles would self-organize the staggered particle trains, which were dependent on the Reynolds number and blockage ratio. However, the orderly arrangement technology of particle trains is difficult to apply to cell populations because of size dispersion and deformability of cells ([PERSON] et al., 2010). The particle size not only affects the equilibrium position of the particles ([PERSON] et al., 2010), but also affects the migration speed of the particles ([PERSON] et al., 2010) and the arrangement of the flow direction ([PERSON] et al., 2010); therefore, label-free enrichment or separation of particles in polydisperse suspensions can be achieved. However, most of these works only focus on the separation efficiency, and there is little research on the physical mechanism of particle inertial migration by the size dispersion of particles. Up to now, a large number of theoretical, numerical, and experimental studies have focused on exploring the physical mechanism of particle migration in a closed environment. Only a few works have studied the effect of the interaction between particles of different sizes in multi-scale situations. The effect of particle multiscale on inertial behavior is still unclear and needs further study.