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arXiv · 0801.2927

Collisions of particles advected in random flows

K. Gustavsson·B. Mehlig·M. Wilkinson
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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.

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K. Gustavsson, B. Mehlig, M. Wilkinson. 2008-01-18. Collisions of particles advected in random flows. https://doi.org/10.1088/1367-2630%2F10%2F7%2F075014

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