Gustavsson, K.; Mehlig, B. and Wilkinson, M.
Collisions of particles advected in random flows.
New Journal of Physics, 10,
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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.
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