# Sedimentation of Fractal Aggregates in Shear-Thinning Fluids
## 3 Results
The data presented so far refer to the instantaneous drag correction factor, i.e., the one obtained by solving the fluid dynamics equations for a fixed orientation of the force (or, equivalently, of the aggregate for a fixed force). The applied force, however, generates a rotation of the aggregate (and, of course, a translation) leading to a change of the orientation and, in turn, of the drag correction coefficient. The knowledge of the orientational dynamics of the aggregate is then crucial to determine the time evolution of the drag correction coefficient and the regime attained by the particle. By using the procedure described in the previous section, we compute the orientational dynamics of the applied force for different initial orientations. Figure 5 shows the orbits for the aggregates reported in Figure 1a (top row) and Figure 1b (bottom row), and for the Newtonian (left column) and power-law fluid with \(n=0.6\) (right column). Twelve initial orientations uniformly distributed over the unit sphere are considered (blue circles). For these sets of parameters, the orbits converge towards a unique equilibrium point (green circle) regardless of the initial orientation. In the fixed reference frame, this means that the aggregate achieves a stable orientation. Specifically, our simulations show that, once the regime is achieved, the particle still rotates around the applied force, although with a very small rotation rate (the resulting linear velocity, obtained as the angular velocity around the applied force times the effective radius, is 2-3 orders of magnitude smaller than the sedimentation velocity). It is important to point out, however, that this rotation does not influence the drag as any configuration  
Figure 4: (**a–c**) Drag correction coefficient for the aggregate shown in Figure 1b (\(N_{\text{p}}=20\), \(D_{\text{f}}=2.5\)) 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). around the force is equivalent (i.e., the force direction is a symmetry axis). The orientations of the force corresponding to the equilibrium points (green circles in Figure 5) are shown as symbols in the previous Figures 3 and 4. It can be readily observed that shear-thinning slightly affects the equilibrium orientation only for rod-like particles, whereas it has no influence for higher values of the fractal dimension (the symbols in Figure 4 overlap). Moreover, in both cases, the equilibrium orientation does not correspond to any special value of the projected area (for instance the minimum). Therefore, this quantity is not representative of the final orientation achieved by the aggregate and, as such, it is not helpful to estimate the drag correction factor at long times. On the contrary, the detailed microstructure of the aggregate needs to be considered to correctly predict the sedimentation dynamics. To further investigate on the effect of aggregate morphology, we have repeated the calculations by varying the seed of the random number generator. We recall that, although the fractal parameters in Equation (1) are fixed, the morphologies obtained by varying the seed are different. In the leftmost panels of Figure 6, the regime drag correction coefficient is shown as a function of the seed for \(N_{\text{p}}=20\) and for different values of the fractal dimension and the flow index.