Finite-time Singularity

water drop

A drop of water in the process of breaking apart. At the point of breakup, the fission event looks like a cone pointing into an approximately spherical drop. The cascade of structure that is produced in this process is of uncommon beauty. As the drop falls, a long neck, connecting two masses of fluid, stretches out and then breaks. What is the shape of the drop at the instant of breaking apart?

Something dire must happen to the mathematical description of the liquid at that point since the drop undergoes a topological transition where it starts out as a single, connected fluid and ends up in two or more separate pieces. This is an example of a finite-time singularity since the drop breakup occurs in a short time after the drop becomes unstable and starts to fall. At the transition, a singularity occurs since the radius of the neck holding the drop to the nozzle becomes vanishingly thin. As its radius goes to zero, the curvature diverges and the surface tension forces become infinite.

How can such dramatic dynamics occur in something which had such smooth and innocuous initial conditions and forcing terms? Using photographic techniques, we have been studying transitions such as these to understand how the non-linearities in the governing equations (in this case the Navier-Stokes equations) can be tamed and understood. Singularities of this kind occur in many areas of physics from stellar structure to turbulence to bacterial colony growth. This drop breakup problem is one of the simplest places to start an experiment which directly probes the singularity itself. We have uncovered a variety of different singularities - some of which surprisingly retain a memory of their initial conditions throughout the entire breakup process.

These and related phenomena are currently being studied experimentally by the research group of Professor Sidney Nagel and theoretically by the group of Professor Wendy Zhang. The particular image shown here is taken from X. D. Shi, M. P. Brenner, and S. R. Nagel, Science, volume 265, p. 219, 1994.

> Professor Sidney Nagel
> Professor Wendy Zhang