Advective Collisions in Random Flows

Abstract: In nature, suspensions of small particles in fluids are common. An important example are rain droplets suspended in turbulent clouds. Such clouds can start to rain very quickly and the reason for this is still not fully explained, but it is believed that the turbulent motion in the cloud plays an important role. This thesis gives an introduction to the model we use to describe this system and some results coming from this model. In particular, we will consider collisions between very small droplets that are so light that their motion can be approximated by that of the air flow in the cloud. Collisions between small particles suspended in a fluid can occur due to a macroscopic motion of the fluid [ Z. f. physik. Chemie, XCII, 129-168, (1917)]. This fact was used by Saffman & Turner to estimate the frequency of collisions between the small droplets in clouds [ J. Fluid I., 1, 16-30, (1956)]. We show that this estimate is not adequate, because it only describes a short lived initial transient. We will also discuss different asymptotic behaviors of the model in one spatial dimension. The type of motion of droplets or other particles governed by the model changes as the dimensionless parameters change. It is possible to divide the space spanned by the dimensionless parameters into different regimes, where the particles within each regime exhibit the same kind of motion. Some of these regimes have been discussed in earlier work, whereas others are not as well explored.