Collisions of particles advected in random flows

We consider collisions of particles advected in a fluid. As already pointed out by Smoluchowski (1917 Z. Phys. Chem. 92 129-68), macroscopic motion of the fluid can significantly enhance the frequency of collisions between the suspended particles. This effect was invoked by Saffman and Turner (1956 J. Fluid Mech. 1 16-30) 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