Examples of Volume-Preserving Great Circle Flows of S3

This summer Herman Gluck and Weiqing Gu proved the last step in a process that took conformal maps between two complex spaces and related them to Volume Preserving Great Circle Fibrations of S3. These fibrations, which are non-intersecting flows, break down under certain conditions. We obtained the fibrations by applying the process to different conformal maps then calculated the angles where they intersect. This paper centers around the developments in the method for converting the conformal maps and finding the critical angles. Finally, the examples are included in their various stages of completeness

Dynamics of Perfectly Wetting Drops under Gravity

We study the dynamics of small droplets of polydimethylsiloxane (PDMS) silicone oil on a vertical, perfectly-wetting, silicon wafer. Interference videomicroscopy allows us to capture the dynamics of these droplets. We use droplets with a volumes typically ranging from 100 to 500 nanolitres (viscosities from 10 to 1000 centistokes) to understand long time derivations from classical solutions. Past researchers used one dimensional theory to understand the typical $t^{1/3}$ scaling for the position of the tip of the droplet in time $t$. We observe this regime in experiment for intermediate times and discover a two-dimensional, similarity solution of the shape of the droplet. However, at long times our droplets start to move more slowly down the plane than the $t^{1/3}$ scaling suggests and we observe deviations in droplet shape from the similarity solution. We match experimental data with simulations to show these deviations are consistent with retarded van der Waals forcing which should become significant at the small heights observed