Aggregation and fragmentation dynamics in random flows: From tracers to inertial aggregates

We investigate the aggregation and fragmentation dynamics of tracers and inertial aggregates in random flows leading to steady-state size distributions. Our objective is to elucidate the impact of changes in aggregation rates due to differences in advection dynamics, especially with respect to the influence of inertial effects. This aggregation process is, at the same time, balanced by fragmentation triggered by local hydrodynamic stress. Our study employs an individual-particle-based model, tracking the position, velocity, and size of each aggregate. We compare the steady-state size distribution formed by tracers and inertial aggregates, characterized by different sizes and densities. On the one hand, we show that the size distributions change their shape with changes in the dilution rate of the suspension. On the other hand, we obtain that the size distributions formed with different binding strengths between monomers can be rescaled to a single form with the use of a characteristic size for both dense inertial particles and tracer monomers. Nevertheless, this last scaling relation also fails if the size distribution contains aggregates that behave as tracer-like and inertial-like, which results in a crossover between different scalings.

Figure 1

Nonhomogeneous spatial distribution of inertial particles, with $\text{St}=0.5$ for a flow field with $\text{Ku}=1$.

Figure 2

(a) Evolution of the size distribution in time. (b) Size distribution of aggregates in the steady state; the dashed gray line represents the mean aggregate size ${〈\alpha 〉}_e$. $N$ is the total number of aggregates in the system ($N={\sum }_{\alpha }{N}_{\alpha }$).

Figure 3

(a) PDF of the shear forces for a random flow, obtained from space sampling of $\mathbf{u}(\mathbf{X},t)$, with $\text{Ku}=1$ (${\tau }_f=1$), and (b) the corresponding maximum aggregate's size allowed by these forces; the parameters used are $\gamma =10$ and $\kappa =3$.

Figure 4

Schematic representation of the changes in critical shear attributed to an aggregate, which changes due to aggregation and fragmentation events (red), in comparison to the shear force of the flow at the aggregate's position (black).

Figure 5

(a) Steady-state size distribution of noninertial (tracer) aggregates obtained by changing the number of monomers in the flow (inset: the same in log-lin representation); (b) number of aggregation (or fragmentation) events per aggregate in one correlation time of the flow; (c) average aggregate size as a function of the total number of monomers in the suspension. Tracer monomers characterized by a size $r_0=5\times {10}^{-4}{\lambda }_f$ and binding strength $\gamma =10$ (solid lines) and $\gamma =4$ [gray dashed lines in (b) and (c)].

Figure 6

(a) Size distributions of aggregates initialized with different numbers of monomers $M$, for monomers with $r_p=5\times {10}^{-4}{\lambda }_f, \beta =0.1$, which corresponds to $\text{St}=0.8$ and binding strength $\gamma =4$. The size distribution represented with red stars corresponds to the limit distribution, described in Sec. 2c. (b) Number of aggregation (or fragmentation) events per aggregate in one correlation time of the flow; (c) average aggregate size as a function of the total number of monomers in the suspension.

Figure 7

Steady-state size distributions of tracer aggregates for different binding strengths $\gamma$. Tracer monomers are characterized by a size $r_p=5\times {10}^{-4}{\lambda }_f$ in a suspension with $M={10}^6$. (a) The size distribution in its original form. (b) Rescaled size distribution according to Eq. (3).

Figure 8

Steady-state size distributions $N_{\alpha }$ changing $\gamma$, binding strength among monomers. The data are scaled using the average aggregate size in the distribution ${〈\alpha 〉}_e$, according to Eq. (3). The simulations contains inertial aggregates ($\text{St}=0.83, \beta =0.1$) in suspension with $M={10}^6$ monomers.

Figure 9

(a) Time evolution of the average aggregate size in the suspension. (b) Size distribution of aggregates in the steady state. Each ensemble contained ${10}^6$ monomers, and the binding strength between monomers was $\gamma =4$.

Figure 10

Average aggregate size in the steady state ${〈\alpha 〉}_e$, and equivalently its time average ${〈\alpha 〉}_t$, for inertial ($\text{St}=0.83, \beta =0.1$), small inertial ($\text{St}=0.08, \beta =0.1$), weakly inertial ($\text{St}=0.08, \beta =0.99$), and tracer aggregates changing the binding strength $\gamma$; the simulation was initialized with $M={10}^6$. The inset shows ${〈\alpha 〉}_t$ for a system with ${10}^6$ (black dots) and another with ${10}^5$ (dashed line) monomers, characterized by $\beta =0.1$.

Figure 11

Steady-state size distribution for weakly inertial monomers ($\text{St}=0.08, \beta =0.99$). The simulation contains ${10}^6$ monomers. (a) The size distribution in its original form. (b) Rescaled size distribution according to Eq. (3).