## 1 Introduction  
Particle sedimentation can be found in a wide range of natural and industrial processes, such as river flows, blood flows, microfluidic chips and fluidization etc. [1, 2, 3, 4, 5]. The settling particle is strongly affected by the rheological properties of surrounding fluid as well as the physical properties of the particle. Knowledge on particle behaviors in Newtonian liquids is relatively extensive. [PERSON] [6] summarized the forces that act on spherical particles, which usually include steady drag, history force, added mass effects and lift force. In addition, it is possible to predict interactions between dispersed particles moving in Newtonian fluids [7]. [PERSON] proposed that when the particle Reynolds number is large (\(750<Re<2\times 10^{5}\)), the viscous resistance of the particle is negligible compared with the inertial force [5]. Considering the relationship between the Reynolds number (\(Re\)) and the mass ratio (\(m\)\"), three typical sedimentation trajectories of a single particle settling in glycerin-water mixture were found: vertical, oblique and zigzag [8], where (\(m\)\") is the relative density of the particle compared to the fluid, a map of regimes of the particle trajectories is presented for \(Re=100\)-\(12000\) and \(m^{*}=0\)-\(1.5\). Five types of trajectories of a free-settling particle in Newtonian fluids have been found: steady and oblique, oblique and oscillating regime with low frequency, oblique and oscillation regime with high frequency, zigzagging periodic regime and three-dimensional chaotic regime [9]. The settling velocities of dual particles would be larger than a single particle at small Reynolds numbers [10]. It has been widely confirmed that the drag coefficient of a particle settling in Newtonian fluids only depends on \(Re\). In the case of two particles, based on a numerical study using the three-dimensional lattice Boltzmann method, [PERSON] and [PERSON] [11] identified six sedimentation patterns for two side-by-side particles of different densities settling in a Newtonian fluid. They also simulated two particles of different densities released in tandem [12]. As for the theoretical analysis, the asymptotic forms of the far-field and near-field for the particle velocities were given by [PERSON] and [PERSON] [13]. In this work, the two particles were unequal in size and density, and both the inertia of the particles and the fluid can be neglected, as opposed to with the viscous forces. Since many processes involving particle settling occur in non-Newtonian fluids [14; 15], such as hydraulic fracture for oil exploration [16], sedimentation velocity analytical ultracentrifugation in biopharmaceuticals [17], the behavior of particles in the human respiratory tract [18] and the transport of the particles in microfluidic chips [19], understanding particle settling behavior in non-Newtonian fluids is of significant importance to the above applications. However, this is much more complex for particles moving in non-Newtonian fluids. Specifically, the viscosity of the fluid around the particle changes with the variation of the distribution of shear stress caused by the settling particle, which finally results in a change in the drag coefficient [7]. Thus, the conclusions drawn from the investigation of Newtonian fluid are not appropriate for non-Newtonian fluids. Various experimental and numerical studies have been carried out on the sedimentation of particles in different non-Newtonian fluids, such as power-law fluids [20], Carreau fluids [21], and Boger fluids [22; 23]. Particles can behave in completely different ways in non-Newtonian fluids due to the coupling effects of viscoelasticity and shear-dependent viscosity. For example, obvious negative wakes would occur when particle settling in different viscoelastic fluids. The increase in the elasticity would cause the negative wakes to move closer to the particle [24]. The pioneering studies on the two particles settling in viscoelastic fluids were performed by [PERSON] et al. [25], [PERSON] et al.