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 $$ , up to approximations the formula reads $$ where n is the particle number per unit of volume and $$ is the square-average of the increment of random velocity field u between points at distance a, the particle radius. If we choose the Kolmogorov time scale $$ and we assume that a is in the dissipative range where $$ , we get Saffman-Turner formula for the collision rate R.