A non-inertial model for particle aggregation under turbulence

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 $\ddot{I}{„}_{\hat{I}}\cdot$, up to approximations the formula reads $R\sim \ddot{I}{„}_{\hat{I}}\cdot ⟨{\left|\hat{I}{”}_{a}u\right|}^2⟩a\cdot n^2$ where $n$ is the particle number per unit of volume and $⟨{\left|\hat{I}{”}_{a}u\right|}^2⟩$ 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 $\ddot{I}{„}_{\hat{I}}\cdot \sim {\left(\frac{\hat{I}½}{\epsilon }\right)}^{1/2}$ and we assume that $a$ is in the dissipative range where $⟨{\left|\hat{I}{”}_{a}u\right|}^2⟩\sim \left(\frac{\epsilon }{\hat{I}}½\right)a^2$, we get Saffman-Turner formula for the collision rate $R$.

45 pages