A non-inertial model for particle aggregation under turbulence
Franco Flandoli, Ruojun Huang
TL;DR
The paper develops a non-inertial model for particle aggregation in turbulence and proves a rigorous mean-field limit: the empirical particle distribution converges to a density $f(t,x)$ solving $\partial_t f = \tfrac{1}{2}(\sigma^2+\sigma_0^2)\Delta f - \overline{\mathcal{R}} f^2$, where the effective collision rate $\overline{\mathcal{R}}$ is determined by a cell problem and, in the high-collision limit, by a capacity $Cap(K_{\theta},A)$. The reduction from an inertial framework to the non-inertial model is accomplished via two approximations, yielding a closed stochastic transport with a quadratic coagulation term and a finite mean free path regime $Na=\rho_0$ that supports a non-mean-field scaling. A central methodological contribution is the Itô-Tanaka approach, which constructs a regular cell problem $u^{\varepsilon}$ and analyzes the nonlinear interaction through a two-scale, probabilistic homogenization, establishing convergence to the limiting PDE with a precise expression for $\overline{\mathcal{R}}$. The work also clarifies how Kolmogorov-scale physics enters the macroscopic collision law, reproducing the Saffman-Turner formula in the dissipative range and connecting coalescence efficiency to potential-theoretic capacity in high-coupling regimes. Overall, the results bridge microscopic stochastic dynamics and macroscopic coagulation, offering a quantitative framework for predicting collision rates in turbulent suspensions.
Abstract
We consider an abstract non-inertial model of aggregation under the influence of a Gaussian white noise with prescribed space-covariance, and prove a formula for the mean collision rate $R$, per unit of time and volume. Specializing the abstract theory to a non-inertial model obtained by an inertial one, with physical constants, in the limit of infinitesimal relaxation time of the particles, and the white noise obtained as an approximation of a Gaussian noise with correlation time $τ_η$, up to approximations the formula reads $R\simτ_η\left\langle \left\vert Δ_{a}u\right\vert ^{2}\right\rangle a\cdot n^{2}$ where $n$ is the particle number per unit of volume and $\left\langle \left\vert Δ_{a}u\right\vert ^{2}\right\rangle $ is the square-average of the increment of random velocity field $u$ between points at distance $a$, the particle radius. If we choose the Kolmogorov time scale $τ_η\sim\left( \frac{ν}{\varepsilon}\right) ^{1/2}$ and we assume that $a$ is in the dissipative range where $\left\langle \left\vert Δ_{a}u\right\vert ^{2}\right\rangle \sim\left( \frac{\varepsilon}ν\right) a^{2}$, we get Saffman-Turner formula for the collision rate $R$.