# Sedimentation of Fractal Aggregates in Shear-Thinning Fluids
###### Abstract
It can be readily seen that the hydrodynamic radius is inversely proportional to the aforementioned mean of the eigenvalues of the translational mobility tensor. The knowledge of the hydrodynamic radius for an aggregate with arbitrary shape is, then, sufficient to characterize its average settling velocity. The ratio of the hydrodynamic radius and the gyration radius has been found to be a function of the parameters of the fractal Equation (1) [13; 19; 20]. Specifically, such a ratio is an increasing function of \(D_{\rm f}\), assuming a value of \(\sim\)1 for \(D_{\rm f}=2\), up to a limiting value of about 1.29 for \(D_{\rm f}=3\)[13; 15; 20]. The number of primary particles strongly affects the ratio for low fractal dimensions (rod-like particles), whereas it has a weak influence for more spherical aggregates. Finally, increasing the fractal pre-factor moves the ratio to higher values without altering the dependence on \(D_{\rm f}\) and \(N_{\rm p}\)[13]. All the aforementioned studies consider a Newtonian suspending liquid. In addition, the developed methodologies used to compute the hydrodynamic drag are only applicable to Newtonian fluids. Active sludges, however, show a non-Newtonian rheology [21; 22], which has a strong influence on the particle settling dynamics. For a spherical particle in an unbounded shear-thinning inelastic fluid, modeled with the power-law constitutive equation, several works are available showing that the drag force deviates from the Stokes' law [23; 24; 25; 26]. A drag correction coefficient has been defined as the ratio between the applied force and the Stokes' drag law, where the viscosity is replaced by the power-law constitutive equation with a characteristic shear rate given by the terminal settling velocity divided by the particle diameter. The coefficient is 1 for a Newtonian fluid and increases as the flow index decreases (i.e., fluid shear-thinning increases). For non-spherical particles, [PERSON] et al. [27] carried out finite element simulations to study the flow of a power-law fluid over prolate and oblate spheroidal particles aligned with the flow direction. In this particular orientation, the dependence of the total drag coefficient on the flow index was found to be qualitatively similar to that observed for spherical particles. As the aspect ratio of the prolate spheroid increases, the drag becomes relatively insensitive to the degree of shear-thinning. Oblate spheroids with high aspect ratio experience a lower drag as compared to spheres. In summary, many works exist dealing with drag calculation of particles with non-spherical shape in Newtonian liquids. Concerning power-law fluids, the only results available are for spheres [25] and spheroids [27], the latter only considering particles oriented with a principal axis along the force direction. To the best of our knowledge, a study on the combined effect of non-Newtonian rheology and complex particle shape on the hydrodynamic drag is missing. In this work, we investigate the hydrodynamic drag experienced by fractal aggregates suspended in a non-Newtonian fluid by numerical simulations. We assume that the aggregates are sufficiently large to neglect Brownian motion and that their concentration is low enough (less than 5% in volume) to avoid hydrodynamic interactions. This allows us to consider a single-particle problem. The suspending fluid is assumed to be inelastic and shear-thinning, and is modeled by the power-law constitutive equation. A map of particle velocities is precomputed by running finite element simulations for orientations of the applied force uniformly distributed over the unit sphere. Such velocities are then interpolated and used to reconstruct the aggregate dynamics by integrating the evolution equation of the particle position and orientation. The drag correction coefficient at long times is averaged over several initial orientations and particle shapes with the same fractal parameters.