Gilbert Oppy is an undergraduate student at Monash University, Melbourne. He is about to undertake the 4th year of a double degree in Engineering and Science, specialising in electrical engineering and applied mathematics (and physics) respectively. He has also just completed a diploma in philosophy at Monash, which he worked on in the previous two summers, along the way receiving the Robert Pargetter Prize for First Year Philosophy. Gilbert has undertaken a wide range of different units in mathematics, currently needing to complete one more maths unit in 2015 to finish of a double major (which is 12 semester-long units in total). Broad areas of mathematics that he now has experience with include complex analysis, differential equations (ordinary and partial), integral transforms, probability theory and fluid dynamics. Gilbert is in particular interested in applying skills from these areas to model and investigate real world phenomenon; in a recently completed mathematical modelling unit, he helped code a Matlab program for simulating and modelling traffic flow along a road with signal-lights at either end. Seeing mathematical theory work in practice such as was the case here is one of the more rewarding things a mathematician can experience. With this in mind, Gilbert has chosen to investigate bouncing fluid droplets, aiming to further enhance his knowledge in the field of fluid dynamics (a topic he particularly enjoyed studying) while simultaneously hoping to observe some maths in action.
Discrete dynamics of a bouncing ball
Usually when a droplet of fluid impacts a bath of the same fluid, it coalesces. However, if the bath is vibrated vertically with sufficient vigour, then the droplet does not have time to coalesce before the bath is whisked out from under it. The droplet can then bounce indefinitely on the bath’s surface. These bounces also generate waves on the bath surface, which in turn can influence the bouncing droplet’s motion. In some cases, the droplet can start to “walk” on the bath’s surface guided by this “pilot” wave. This association of a wave with a particle allows macroscopicre production of surprisingly many quantum phenomena. An example is single-electron double-slit interference fringes: In the macroscopic analogue, the droplet goes through one slit while its wave goes through both. The waves emanating from the two slits interfere with one another and guide the droplet to a preferred set of paths. Recently, a very complete and accurate model has been developed for the motion of a single bouncing droplet. It contains only one parameter, which in principle can be found by understanding the interaction between the droplet, the bath and the intervening air cushion during a bounce. However, the theoretical value is approximately twice the experimentally observed one and it is thought that neglecting rotation of the droplet is responsible. This project would use a scaling argument to develop a model for the thinning air cushion in two dimensions with the aim of calculating the torque on the droplet during the bounce.
The project is also looking at the simplified, analogue problem of a bouncing ball on a seesaw as a dynamical system and investigating the transition to chaos