A non-inertial model for particle aggregation under turbulence

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 𝜏𝜂 , up to approximations the formula reads 𝑅 ∼ 𝜏𝜂 (Delta u ^2) 𝑎 ⋅ 𝑛2 where 𝑛 is the particle number per unit of volume and (Delta u ^2) 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 and we assume that 𝑎 is in the dissipative range, we get Saffman-Turner formula for the collision rate 𝑅.