# Sedimentation of Fractal Aggregates in Shear-Thinning Fluids
## 2 Mathematical Model and Numerical Method
### Governing Equations
Notice that, for the problem under investigation (settling dynamics of a particle in an unbounded fluid), the evolution of the aggregate center of volume does not affect the orientational dynamics. Therefore, Equation (10) can be removed from the set of equations to be solved. The governing equations can be made dimensionless by choosing appropriate characteristic quantities for length, time, and stress. As characteristic length, we choose the effective radius of the aggregate, defined as the radius of a sphere with the same volume \(V\) of the aggregate, \(R_{\mathrm{eff}}=(\frac{3V}{4\pi})^{1/3}\). The characteristic time is chosen as the inverse of a characteristic shear rate \((U/R_{\mathrm{eff}})^{-1}\), where \(U\) is the velocity along the direction of the applied force. The characteristic stress is \(m(U/R_{\mathrm{eff}})^{n}\). By using such characteristic quantities, we can recast the governing equations and boundary conditions in their dimensionless form. In these equations, only the flow index \(n\) appears as dimensionless parameter. Therefore, the investigated system is fully determined by specifying \(n\) and the geometry of the aggregate defined by the parameters in Equation (1). In this regard, the radius of the primary particles \(a\) is related to \(R_{\mathrm{eff}}\) through \(N_{\mathrm{P}}\). Moreover, the radius of gyration \(R_{\mathrm{g}}\) is determined once the other three parameters in Equation (1) are chosen. Therefore, the geometrical parameters that need to be specified are the number of particles \(N_{\mathrm{P}}\), the fractal dimension \(D_{t}\), and the fractal pre-factor \(k_{\mathrm{f}}\). To quantify the hydrodynamic resistance of the particle to the applied force, we introduce the drag correction coefficient [25]:  
\[X=\frac{F}{6\pi\,m\left(\frac{U}{2R_{\mathrm{eff}}}\right)^{n-1}U\,R_{\mathrm{ eff}}} \tag{12}\]  
defined as the applied force divided by the modified Stokes drag coefficient where the Newtonian viscosity is replaced by the power-law model. Of course, \(X=1\) for a sphere in a Newtonian fluid. The value of \(X\) depends on the orientation of the aggregate and, as such, changes in time since the particle varies its orientation while sedimenting. As it will be discussed below, the aggregate orientation can achieve various regimes, leading to different regime drag correction coefficients \(X_{\mathrm{R}}\). As \(X_{\mathrm{R}}\) is affected by the initial particle orientation, we average over \(C\) initial configurations:  
\[\langle X_{\mathrm{R}}\rangle=\frac{1}{C}\sum_{C}X_{\mathrm{R}} \tag{13}\]Finally, to make the results independent of the seed used in the algorithm to generate the aggregate, for each set of fractal parameters in Equation (1), the simulation is repeated with different seeds. The ensemble-average drag correction coefficient is computed as  
\[\langle X_{\mathrm{R}}\rangle_{\mathrm{m}}=\frac{1}{N_{\mathrm{seed}}}\sum_{N_{ \mathrm{seed}}}\left\langle X_{\mathrm{R}}\right\rangle \tag{14}\]  
with \(N_{\mathrm{seed}}\) the number of seeds. In this work, the fractal pre-factor is fixed to \(k_{\mathrm{f}}=1.3\), which is a value commonly used in the literature to describe realistic aggregate shapes [8]. The sedimentation dynamics is studied by varying the flow index, the fractal dimension, and the number of primary spheres forming the particle.