# Sedimentation of Fractal Aggregates in Shear-Thinning Fluids
## 2 Mathematical Model and Numerical Method
### Governing Equations
\begin{table}
\begin{tabular}{c c} \hline \(a\) & Radius of primary particles \\ \(C\) & Number of initial aggregate configurations \\ \(D\) & Rate-of-deformation tensor \\ \(D_{\rm f}\) & Fractal dimension \\ \(\mathbf{f}\) & Applied force in the body reference frame \\ \(\mathbf{f}_{0}\) & Initial applied force in the body reference frame \\ \(\mathbf{F}\) & Applied force in the fixed reference frame \\ \(\mathbf{I}\) & 3x3 unity tensor \\ \(k_{\rm f}\) & Fractal pre-factor \\ \(m\) & Consistency index \\ \(n\) & Flow index \\ \(\mathbf{n}\) & Unit vector normal to the particle surface \\ \(N_{\rm elem}\) & Number of elements of the tetrahedral mesh \\ \(N_{\rm p}\) & Number of primary particles composing the aggregate \\ \(N_{\rm seed}\) & Number of seeds \\ \(p\) & Fluid pressure \\ \(R_{\rm eff}\) & Radius of a sphere with the same volume of the aggregate \\ \(R_{\rm g}\) & Radius of gyration \\ \(R_{\rm out}\) & Radius of the external sphere \\ \(S\) & Particle surface \\ \(t\) & Time \\ \(\mathbf{T}\) & Torque on the particle \\ \(\mathbf{u}\) & Fluid velocity \\ \(U\) & Aggregate velocity along the direction of the applied force \\ \(\mathbf{U}_{\rm p}\) & Particle translational velocity \\ \(V\) & Volume of the aggregate \\ \(\mathbf{x}\) & Point of particle boundary \\ \(\mathbf{x}_{\rm p}\) & Particle center of volume \\ \(\mathbf{x}_{\rm p,0}\) & Initial particle position \\ \(X\) & Drag correction coefficient \\ \(\langle X_{\rm R}\rangle\) & Regime drag correction coefficient \\ \(\langle X_{\rm R}\rangle_{\rm m}\) & Ensemble-average drag correction coefficient \\ \hline \(\dot{\gamma}\) & Effective deformation rate \\ \(\Delta x\) & Size of the elements on the aggregate \\ \(\Delta x_{\rm out}\) & Size of the elements on the external surface \\ \(\eta\) & Fluid viscosity \\ \(\theta\) & Polar spherical coordinate \\ \(\theta_{\rm p}\) & Particle rotation angle \\ \(\theta_{\rm p,0}\) & Initial particle rotation angle \\ \(\sigma\) & Fluid stress tensor \\ \(\phi\) & Azimuthal spherical coordinate \\ \(\omega_{\rm p}\) & Particle angular velocity \\ \hline \end{tabular}
\end{table} 
Table 1: List of symbols. Figure 1: Examples of aggregate shapes obtained from the particle–cluster method for \(N_{\rm p}=20\), \(k_{\rm f}=1.3\), and \(D_{\rm f}=1.5\) (**a**), and \(D_{\rm f}=2.5\) (**b**). To avoid numerical issues due to the tangent point, the centers of the spheres in contact are connected with a set of cylinders with radius \(0.732a\). In panels (**c**,**d**), the final geometry of the aggregates and the surface mesh are shown. The computational domain and the mesh on the external spherical surface is displayed in panel (**e**). Finally, we need to specify the hydrodynamic total force and torque acting on the aggregate. Under the assumption of inertialess particle, the following equations hold,  
\[\mathbf{F}=\int_{S}\mathbf{\sigma}\cdot\mathbf{n}\,dS \tag{8}\] \[\mathbf{T}=\int_{S}(\mathbf{x}-\mathbf{x}_{\mathrm{P}})\times(\mathbf{\sigma}\cdot \mathbf{n})\,dS=\mathbf{0} \tag{9}\]  
where \(\mathbf{T}\) is the total torque on the particle surface \(S\) and \(\mathbf{n}\) is the unit vector normal to the particle surface pointing from the fluid to the boundary. Notice that the aggregate is torque-free, whereas the only external force is the applied force \(\mathbf{F}\). The solution of the governing equations gives the fluid velocity and pressure fields, along with the particle translational and angular velocities. The translational and orientational dynamics can be computed by integrating the following equations,  
\[\frac{d\mathbf{x}_{\mathrm{P}}}{dt} =\mathbf{U}_{\mathrm{P}} \tag{10}\] \[\frac{d\mathbf{\theta}_{\mathrm{P}}}{dt} =\mathbf{\omega}_{\mathrm{P}} \tag{11}\]  
with initial conditions \(\mathbf{x}_{\mathrm{P}}|_{t=0}=\mathbf{x}_{\mathrm{P},0}\) and \(\mathbf{\theta}_{\mathrm{P}}|_{t=0}=\mathbf{\theta}_{\mathrm{P},0}\).