Aggregation of inertial particles in random flows

We consider the trajectories of particles suspended in a randomly moving fluid. If the Lyapunov exponent of these trajectories is negative, the paths of these particles coalesce, so that particles aggregate. Here we give a detailed account of a method [B. Mehlig and M. Wilkinson, Phys. Rev. Lett. 92, 250602 (2004)] for calculating this exponent: it is expressed as the expectation value of a random variable evolving under a stochastic differential equation. We analyze this equation in detail in the limit where the correlation time of the velocity field of the fluid is very short, such that the stochastic differential equation is a Langevin equation. We derive an asymptotic perturbation expansion of the Lyapunov exponent for particles suspended in three-dimensional flows in terms of a dimensionless measure of the inertia of the particles, $ϵ$, and a measure of the relative intensities of potential and solenoidal components of the velocity field, $\Gamma$. We determine the phase diagram in the $ϵ\text{−}\Gamma$ plane.

Figure 1

(Color online) Illustrating the aggregation of noninteracting particles in a random three-dimensional flow, the motion is defined by Eqs. (1, 2, 3), using a simplified form explained in Sec. 3, see Eqs. (41, 42). Panel (a) shows the initial configuration at $t=0$, panels (b) and (c) show configurations after a long time for two different values of $\gamma$. In the case shown in the center panel, trajectories coalesce, until eventually all particles follow the same trajectory. The particles in the right panel exhibit density fluctuations, but the trajectories do not coalesce.

Figure 2

(Color online) (a) Lyapunov exponent as a function of $ϵ$, for a small value of $\chi$, results from Eqs. (38, 39, 40) are shown as solid lines, those from direct simulations as symbols, $\Gamma =1∕3$ (엯), $\Gamma =1$ (◻), and $\Gamma =2$ (◇). The simulations were performed with a simplified version of the dynamics using Eqs. (41, 42). (b) Lyapunov exponent ${\lambda }_1∕\left(ϵ\gamma \right)$ versus $ϵ$ for $\Gamma =0.45$. Shown are results from simulations (엯) as well as from the asymptotic series (68) summed to orders $l_{\max }=1$,…,6: up to ${ϵ}^{*}\left(l_{\max }\right)$ full lines and from ${ϵ}^{*}\left(l_{\max }\right)$ dashed lines. The simulations were performed by replacing the components of ${\mathbf{f}}_n\left(\mathbf{r}\right)$ in (41) by random numbers with the appropriate distributions. (c) Phase diagram in the $ϵ\text{−}\Gamma$ plane. Shown are results from numerical simulations as described above, 엯, summation of the asymptotic series (68) summed to $l_{\max }^{*}\left(ϵ\right)$, [(blue) line, showing discontinuities where $l_{\max }^{*}$ changes and plotted for $ϵ<0.25$], as well as results from Langevin simulations [smooth (red) line].