## 4 Conclusions  
In this work, we studied the trajectories and velocities of two side-by-side particles of different density settling in a non-Newtonian fluid with shear thinning viscoelastic properties, as well as that for the corresponding single settling particle.  
In contrast to from the three-dimensional unstable path of a single particle settling in water, the particle settling path can be regarded as a vertical line in the PAAm solution. The terminal settling velocity of the particle (Si\({}_{3}\)N\({}_{4}\)) in PAAm solution is lower than that in water. Apart from this, the terminal velocities of other particles (stainless steel and ZrO\({}_{2}\)) are higher than that in water, and the deviation of terminal velocity from that of the particle settling in water increases with the particle density. The reason may be that there is a critical value of \(Re^{\star}\), i.e., another particle with a different density, and its terminal settling velocity in the PAAm solution is consistent with that in water.  
In addition to this, an experimental study on the effect of the initial separation distance on settling side-by-side particles motion at different density ratios was conducted in PAAm solution. The results show that the wake of the heavy particle, in a leading position,  
Figure 14: The velocities and the trajectories of the two side-by-side settling particles at different separation distance \(L^{\star}\); Left: the light particle; Right: the heavy particle.  
can affect the light particle settling into a non-linear path. For the heavy particle, the particle-particle interaction can also cause the transverse migration of the heavy one, and the smaller the density ratio, the greater the transverse migration. For the light particle, three basic path types were found: (i) the light particle is first attracted and then repelled by the wake of the heavy one; (ii) the light particle approaches and then largely traces within the path of the heavy one in the limited field of view; (iii) the light particle is first slightly shifted away from its original position and then returns to this initial position. In these, the light particle can be attracted or rebounded by the CRV behind the heavy particle. Meanwhile, we conclude that the \(u\) of light particle became higher than the terminal velocity of a single particle, due to the existence of a corridor with a reduced viscosity and the negative wake generated by the viscoelastic property. As explained above, particle-particle attraction is mainly caused by the effects of the shear-thinning viscoelasticity properties of the PAAm solution. As discussed throughout this study, the trajectory and velocity of particle sedimentation in PAAm solution are affected by the density difference and initial spacing of the two particles and fluid properties. These findings will make contributions to the future studies and applications in this area.  
Conceptualization, C.T., F.B.; methodology, S.Y., M.D., R.X. and C.T.; formal analysis, X.G. ([PERSON]) and S.Y.; investigation, S.Y., M.D. and X.G. ([PERSON]), and R.X.; writing--original draft preparation, S.Y., C.T., X.G. ([PERSON]), and M.D.; and writing--review and editing, C.T., and F.B.; All authors have read and agreed to the published version of the manuscript. This research received no external funding. This work was supported by the National Natural Science Foundation of China (Grants No. 11632016, No. 11972336, No. 11872217, and No. 11972334). The authors declare no conflict of interest.