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 $τ_η$, 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 $R$.