Particles in Flows

Collision, Fragmentation, Coagulation in a Turbulent Environment

As an example, let us start with a model for formation of raindrops. The process of rain starts with moisture condensing on very small particles of dust which goes by the name cloud condensation nuclei (CCN). The very small droplets of water thus formed grows further by condensation of moisture on their surfaces. If these drops could grow only by condensation, then raindrops would form agonisingly slow. How could they grow faster ? One possible way could be if the small droplets could collide and consequently merge to form bigger dropelts. This possibility was addressed by Saffman and Turner, (Journal of Fluid Mechanics, 1, page 16, 1956). They noted that two small droplets fall with the same terminal speed in a quiescent environment, hence their relative velocities are close to zero. Consequently, the only way two droplets can collide is if atmospheric turbulent flows make them collide.

In turbulent flows, the difference of velocity between two points

$$ \delta u(\ell) \sim \ell $$

for very small \(\ell\) (this essentially assumes that the velocity at small scales is Taylor-expandable). Hence, if the small water droplets move with the same velocity as the local fluid then they would not collide either, because as they come close and close together their relative velocities goes to zero.

Of course, droplets do not move with the local fluid velocity. Because of the drag of the fluid on them, the simplest equation describing the motion of droplets is given by

$$ \begin{eqnarray} \dot{\bf x} &=& {\bf v} \\ \dot{\bf v} &=& \frac{1}{\tau}[{\bf u} - {\bf v}] \end{eqnarray} $$

Here \(\tau = 6\pi\mu a/m\), with \(a\) the radius and \(m\) the mass of a spherical particle, and \({\bf u}\) is the fluid velocity at the location of the particle. Water droplets of a range sizes satisfy these equations of motion to a good approximation. They are called heavy inertial particles. In a recent breakthrough (independently by three different groups of scientists), it has been argued that the relative velocities between such inertial particles can be independent of their separation, at small separation. The turbulent flow determines the relative velocities of collisions between the particles. This is a problem in fluid mechanics. But once they collide are they going to merge or break up into even smaller droplets? This is a problem of material science. We aim to bring these two fields together.