## 3 Results and Discussion
### Sedimentation of a Single Particle  
In this section, the behavior of a single settling solid spherical particle in the PAAm solution is presented, which was used to compare with that of two side-by-side particles in Section 3.3. In the previous work, we obtained the settling velocity of the corresponding particles in water [40]. The results show that the sedimentation velocity in the initial acceleration stage is basically proportional to the time for \(t<0.2\) s, regardless of the particle density. After this linear acceleration stage, all the particles in water achieved their terminal velocities at \(t\approx 0.4\) s.  
Figure 9 shows the settling behaviors of a single particle in the PAAm solution with different particle densities. It is clear that for all cases the particle motion in both \(x\)-\(z\) and \(y\)-\(z\) planes is almost straight and the transverse motion can be neglected. In other words, the particle settling path can be regarded as a vertical line in the PAAm solution. This phenomenon can be caused by the viscoelastic property of PAAm solution, the fluidization boundary between the settling particles and the viscoelastic fluid acts as a cylinder wall [24; 28]. Therefore, the friction generated by the fluidization boundary position in relation to the settling particle surface results in a sharp decline in the transverse velocity of the particle settling [41]. Thus, the transverse migration for a single particle can be neglected and the settling path can be regarded as a vertical line. In contrast, particles settling in water usually exhibit a three-dimensional unstable path [36; 40] due to the unsteady trajectory and the wake turning to three-dimensional and unsteady behind the settling particle. Figure 9: Settling behaviors of single particle. The time interval between adjacent particles is 19 ms. In addition, Figure 10 shows the variation of the particle velocity and the normalized trajectory in PAAm solution for different particles. The migration velocity is decomposed into the transverse velocities \(v_{x}\) in \(x\)-\(z\) plane, \(v_{y}\) in \(y\)-\(z\) plane and the vertical velocity \(u\). The transverse migration is normalized by the particle diameter and denoted as \(s^{*}\), which is also plotted as a function of time in Figure 10. The very small \(s^{*}\) quantitatively reflects that there is no obvious transverse motion of the settling particles. This can also be confirmed from the near-zero transverse velocities during the whole testing process. In our experiments, before reaching the terminal velocity, the particle would experience a short acceleration leading to an increase in the vertical velocity, while this acceleration hardly changes the transverse velocity in the viscoelastic fluid. In PAAm solution, although the difference in particle density leads to a difference in the terminal settling velocity, the particles almost achieved their terminal velocities at \(t\approx 0.4\) s. As shown in the Figure 11, 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. Based on this, we speculate that there is a critical value of \(Re^{*}\), i.e., another particle with a different density, and its terminal settling velocity in the PAAm solution is consistent with that in water. The reason why the terminal settling velocity of the particle (Si\({}_{3}\)N\({}_{4}\)) in the PAAm solution is lower in water but not the other two particles is the relatively weak shear thinning effect. The modified particle Reynolds number (321 (for stainless steel), 256 (for ZrO\({}_{2}\)), and 124 (for Si\({}_{3}\)N\({}_{4}\))) was mentioned for the settling particle in the PAAm solution.