In the last decade a significant advance has been made in understanding the behaviour of small heavy particles advected by turbulent flows. There are two principal motivations to study such problem. One is to understand rain formation and the second is to understand the formation of planets in a circumstellar disk. In both of these cases the crucial question is whether small objects can collide (frequently enough, in the right range of relative speeds) to eventually merge and form bigger objects -- raindrop in one case and planetesimals in the other.
Most of the work has concentrated on study of small spherical objects. The behaviour of non-spherical but still small heavy objects can potentially be quite different from spherical ones. Note further that although raindrops are to a good approximation spherical, the astrophysical dust is not. Our aim in this project is to study clustering and possible collisions between non-spherical
The simplest model of a non-spherical object would be an
ellipsoid. Let us start by considering an ellipsoid with three
principal axis. In what follows, we shall call an ellipsoid "a
grain". We shall start by making the following
* The grains are passive objects, they do not modify the
* The grains are small and heavy such that the drag on them
due to the flow, which would generate both a drag force and a drag torque (which is absent for spherical particles) can be obtained by solving the Stokes equation.
* We ignore mutual interaction between two grains. This is
a reasonable approximation for a low number density of grains.
* The flow is not obtained by solving the Navier-Stokes
equation but is a time-independent solution of the Euler equation which has chaotic behaviour, in particular, the ABC
We would then have to solve the following numerical problem. The instantaneous velocity and angular velocity of a grain and the local velocity and its gradients (strain and vorticity) sets the
drag for and drag torque. These are given the by the Jeffery's
equation. With the force and torque thus determined, we need to solve the equation of motion of a rigid body. This last step is best done by using the quaternions to describe the rigid body
The aim of the project is to calculate the following quantities:
* How do the grains cluster ? This can be quantified by
calculating the radial distribution function of their center of
* Are the orientations of the grains correlated ?
* More generally, we can calculate the joint probability
density function of their relative position, relative angles and
relative velocities. From this we can calculate the average number of collisions per-unit-time, per-unit-volume (in a large volume) of between two grains.