Collisions of particles advected in random flows

TL;DR

This paper derives exact expressions for the steady-state collision rate of particles in rapidly fluctuating random flows, showing that previous models like Saffman-Turner only describe initial transients and providing more accurate estimates.

Contribution

It introduces a new theoretical framework for calculating steady-state collision rates in fluctuating flows, improving upon the Saffman-Turner model by accounting for flow fluctuations.

Findings

Saffman-Turner theory is only valid for initial transient behavior.

Derived exact expressions for steady-state collision rates.

Saffman-Turner estimate is an upper bound for incompressible flows.

Abstract

We consider collisions of particles advected in a fluid. As already pointed out by Smoluchowski [Z. f. physik. Chemie XCII, 129-168, (1917)], macroscopic motion of the fluid can significantly enhance the frequency of collisions between the suspended particles. This effect was invoked by Saffman and Turner [J. Fluid Mech. 1, 16-30, (1956)] to estimate collision rates of small water droplets in turbulent rain clouds, the macroscopic motion being caused by turbulence. Here we show that the Saffman-Turner theory is unsatisfactory because it describes an initial transient only. The reason for this failure is that the local flow in the vicinity of a particle is treated as if it were a steady hyperbolic flow, whereas in reality it must fluctuate. We derive exact expressions for the steady-state collision rate for particles suspended in rapidly fluctuating random flows and compute how this steady state is approached. For incompressible flows, the Saffman-Turner expression is an upper bound.