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 \(R\sim \tau _{\eta }\left\langle \left| \Delta _{a}u\right| ^{2}\right\rangle a\cdot n^{2}\) where n is the particle number per unit of volume and \(\left\langle \left| \Delta _{a}u\right| ^{2}\right\rangle \) 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 \(\left\langle \left| \Delta _{a}u\right| ^{2}\right\rangle \sim \left( \frac{\varepsilon }{\nu }\right) a^{2}\), we get Saffman–Turner formula for the collision rate R.