Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2008-01-18
New. J. Phys. 10 (special issue: Focus on Cloud Physics), 075014 (2008)
Nonlinear Sciences
Chaotic Dynamics
24 pages, 3 figures
Scientific paper
10.1088/1367-2630/10/7/075014
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.
Gustavsson Kristian
Mehlig Bernhard
Wilkinson Mark
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