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 $\tau_{\eta}$, up to approximations the formula reads $R\sim\tau_{\eta}\left\langle\left\vert \Delta_{a}u\right\vert ^{2}\right\rangle a\cdot n^{2}$ where $n$ is the particle number per unit of volume and $\left\langle \left\vert \Delta _{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 $\tau_{\eta}\sim\left( \frac{\nu}{\varepsilon}\right) ^{1/2}$ and we assume that $a$ is in the dissipative range where $\left\langle \left\vert \Delta_{a}u\right\vert{2}\right\rangle \sim\left( \frac{\varepsilon}{\nu}\right) a^{2}$, we get Saffman-Turner formula for the collision rate $R$.