# Sedimentation of Fractal Aggregates in Shear-Thinning Fluids
## 3 Results
If the force reaches an  
Figure 5: Orbits described by the orientation of the applied force for 12 initial orientations (blue circles) uniformly distributed over the unit sphere for: the aggregate shown in Figure 1a (\(N_{\text{p}}=20\), \(D_{\text{f}}=1.5\)) in a Newtonian (**a**) or power-law fluid with \(n=0.6\) (**b**), the aggregate shown in Figure 1b (\(N_{\text{p}}=20\), \(D_{\text{f}}=2.5\)) in a Newtonian (**c**) or power-law fluid with \(n=0.6\) (**d**). The equilibrium points are denoted by green circles. equilibrium point, regardless of the initial orientation like the orbits shown in Figure 5, \(X_{\rm R}\) is taken as the steady-state value. These points are represented as solid circles in Figure 6. The data show that, for \(N_{\rm P}=20\), the specific morphology (seed) has a relatively weak effect on \(X_{\rm R}\), with a maximum relative deviations of 7% from the average value. Furthermore, in all the investigated cases, a single equilibrium orientation is achieved, except in one case (\(D_{\rm f}=2.5\), seed \(=1\), and \(n=0.6\)) that will be discussed later. A relevant quantity for the sedimentation process is the time \(t_{\rm R}\) needed to achieve the final regime. For instance, with reference to Figure 5, this is the time needed to travel along the orbits from the initial orientation to the green circle. Of course, the time strongly depends on the initial orientation. Thus, we compute the orbits followed by the aggregate with orientation starting from the 162 vertices of the spherical triangle mesh discussed in the previous section and, for each orbit, we estimate the time needed to achieve the regime within a certain tolerance. In case a single steady-state regime exists, we calculate the time the force requires to align with the equilibrium orientation within an angle tolerance of \(5^{\circ}\). The results for \(N_{\rm P}=20\) and different values of the seed, fractal dimension, and flow index are shown as box plots in Figure 7. The lower and higher limits of each box plot represent the first and third quartile over the different initial orientations, whereas the black dash is the median. In general, the (dimensionless) time decreases as the flow index decreases, whereas it is  
Figure 6: Regime drag correction coefficient \(X_{\rm R}\) and its average over the initial orientations \(\langle X_{\rm R}\rangle\) for different number of primary particles (columns), fractal dimension (rows), random seed for the aggregate generation (bands), and flow index (orange \(n=1\), red \(n=0.8\), and green \(n=0.6\)). Solid circles and open squares denote steady-state and periodic regimes. The black dashes represent \(\langle X_{\rm R}\rangle\). rather unaffected by the fractal dimension, ranging between 10 and 100 for almost all the examined cases. There are, however, some exceptions leading to remarkably longer times. To investigate these particular cases more in detail, we show in Figure 8 the orbits described by the orientation of the force for (i) \(n=1\), \(D_{\rm f}=1.5\), seed \(=9\) (Figure 8a corresponding to the highest box plot in the top panel of Figure 7); (ii) \(n=0.8\), \(D_{\rm f}=2.5\), seed \(=1\) (Figure 8b corresponding to the leftmost red box plot in the bottom panel of Figure 7); and (iii) \(n=0.6\), \(D_{\rm f}=2.5\), seed \(=1\) (Figure 8c corresponding to the leftmost green box plot in the bottom panel of Figure 7). In the first case, we still observe a dynamics similar to what reported in Figure 5 with all the orbits converging to a single equilibrium point. However, at variance with the previous cases where the orbits independently moved towards the equilibrium point, now each trajectory converges first towards a common orbit and then, very slowly, to the equilibrium orientation, resulting in a drastic increase of the time needed to reach the steady-state regime. A similar dynamics is also observed for the same seed and for \(n=0.8\) (red box plot) and \(n=0.6\) (green box plot), as well as for \(D_{\rm f}=2.5\) and seed \(=3\).