# Sedimentation of Fractal Aggregates in Shear-Thinning Fluids
## 3 Results  
We investigate the aggregate dynamics and the resulting drag correction coefficient by varying the fractal dimension, the flow index, and the number of primary particles forming the aggregate. The values selected for the three parameters are \(D_{\mathrm{f}}=[1.5,2.0,2.5]\), \(n=[1.0,0.8,0.6]\), and \(N_{\mathrm{P}}=[20,50,100]\). As discussed in Section 2.2, for each set of these parameters, we first run single-step simulations for different orientations of the applied force uniformly distributed over the unit sphere. Figure 3a-c reports the drag correction coefficient \(X\) as a function of the polar and azimuthal spherical coordinates \((0\leq\theta\leq\pi,-\pi\leq\phi\leq\pi)\), identifying the orientation of the applied force for \(n=1\) (i.e., the Newtonian case), \(n=0.8\), and \(n=0.6\), respectively. The aggregate is the one shown in Figure 1a, i.e., with \(N_{\mathrm{P}}=20\) and \(D_{\mathrm{f}}=1.5\). It can be readily observed that (i) the drag correction coefficient depends on the orientation of the force; (ii) the distributions are symmetric since \(X\) is the same for a specific force orientation \((\theta,\phi)\) and its opposite \((\pi-\theta,\phi\pm\pi)\); (iii) in agreement with the spherical particle case [25], the drag correction coefficient increases as the flow index decreases (see the bar legends on the right of the panels); and (iv) the distributions are not affected by the flow index (for instance, the maxima and minima are observed for the same orientations of the force). Previous results have evidenced a trend between the drag force experienced by a fractal aggregate and its area projected along the direction of the applied force [14], although this geometrical quantity is not sufficient to accurately predict the drag. The dimensionless area of the aggregate projected along the force direction is reported in Figure 3d. Specifically, we take the directions identified by the 162 vertices of the unity sphere and, for each of them, we compute the area of the aggregate projected on a plane orthogonal to that direction (identified by the spherical coordinates \(\theta\) and \(\phi\)). The comparison with panels (a)-(c) shows some similarities between the distributions, e.g., the position of the maxima and minima is approximately the same, although a strict correlation is not observed. Figure 2: Drag correction coefficient \(X\) as a function of the flow index \(n\) for a spherical particle in an unbounded power-law fluid. The black circles are the simulation results by [PERSON] and [PERSON] [25], and the triangles are obtained by our simulations for different mesh resolutions and size of the external domain. The same quantities are reported in Figure 4 for the more sphere-like aggregate in Figure 1b (\(N_{\rm p}=20\), \(D_{\rm f}=2.5\)). As for the previous case, similar distributions are observed as the flow index is varied, with higher values of \(X\) for more shear-thinning fluids. The projected area also shows a trend similar to the drag correction coefficient. Due to the higher value of the fractal dimension leading to an aggregate with a more spherical shape, the range of variation of both the projected area and the drag correction coefficient is narrower than in the case at \(D_{\rm f}=1.5\), i.e., the influence of the force orientation is weaker. Figure 3: (**a**–**c**) Drag correction coefficient for the aggregate shown in Figure 1a (\(N_{\rm p}=20\), \(D_{\rm f}=1.5\)) as a function of the force direction identified by the spherical coordinates \((\theta,\phi)\) for the Newtonian fluid (**a**), the power-law fluid with \(n=0.8\) (**b**), and \(n=0.6\) (**c**). (**d**) Dimensionless area projected along the direction of the applied force for the same aggregate as in panels (**a**–**c**). The symbols denote the direction of the force attained at long times for the Newtonian fluid (circle), power-law fluid with \(n=0.8\) (square) and \(n=0.6\) (triangle).