# A Non-inertial Model for Particle Aggregation Under Turbulence

## TL;DR

This paper derives a formula for the mean collision rate of particles aggregating under turbulent conditions using a non-inertial model.

## Contribution

The novelty lies in deriving a general formula for collision rates in turbulence and showing its connection to the Saffman–Turner formula.

## Key findings

- A formula for the mean collision rate R is derived for particle aggregation under turbulence.
- The derived formula reduces to the Saffman–Turner formula under specific turbulence assumptions.
- The model uses a Gaussian white noise approximation with a defined correlation time.

## 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 \documentclass[12pt]{minimal}
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\begin{document}$$\tau _{\eta }$$\end{document}τη, up to approximations the formula reads \documentclass[12pt]{minimal}
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\begin{document}$$R\sim \tau _{\eta }\left\langle \left| \Delta _{a}u\right| ^{2}\right\rangle a\cdot n^{2}$$\end{document}R∼τηΔau2a·n2 where n is the particle number per unit of volume and \documentclass[12pt]{minimal}
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\begin{document}$$\left\langle \left| \Delta _{a}u\right| ^{2}\right\rangle $$\end{document}Δau2 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 \documentclass[12pt]{minimal}
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\begin{document}$$\tau _{\eta }\sim \left( \frac{\nu }{\varepsilon }\right) ^{1/2}$$\end{document}τη∼νε1/2 and we assume that a is in the dissipative range where \documentclass[12pt]{minimal}
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\begin{document}$$\left\langle \left| \Delta _{a}u\right| ^{2}\right\rangle \sim \left( \frac{\varepsilon }{\nu }\right) a^{2}$$\end{document}Δau2∼ενa2, we get Saffman–Turner formula for the collision rate R.