The goal of this research is to characterize and predict the formation of nonlinear singularities in a broad class of physical phenomena. This knowledge applies to diverse systems: turbulence, breaking surface waves, and buckling of solids. All of these systems share a common backbone: they are governed by nonlinear partial differential equations.

This work would occur in the laboratory of Prof. D. Lathrop. Prof. Lathrop has had success in undergraduate mentoring, including 11 students, one of whom published results in a Nature [9] paper on the formation of fluid singularities. One goal of this plan is to increase the number of undergraduate projects resulting in publications. One way of fostering this goal is the practice of having undergraduate students responsible for their own research project (and not just performing busy work for other students' work).

The undergraduate projects include three proposed experiments exploring local singularities. The first project involves self-focusing surface waves, while the next two involve breakup and topology change (specifically, pinch-off of pendant drops and the generation of foam from breaking wave states).

• Two-dimensional focusing of surface waves. Most of our progress in understanding self-focusing surface waves and jets has been for a particular symmetry: axisymmetric. We have tantalizing hints that there are focusing phenomena of importance for other symmetries for free surfaces. The spherically symmetric case is clear -- cavitation collapse is known to produce extreme pressures, damage, and, in the case of sono-luminescence, light. Another case which forms one proposed undergraduate project is that of translational symmetry. In that case, the fluid motion is confined to a plane, and the collapse causes a jet which is sheet-like, similar to the familiar situation of ocean waves which break on cliffs and give a vertical sheet of water spray and foam. These waves, breaking on sea walls, are known to cause significant damage during storms.

• Pinch-off of pendant drops. The pinch-off of a pendant drop has been studied extensively both theoretically and experimentally in work centered around the University of Chicago MRSEC. Recent molecular modeling and associated theoretical studies have indicated that the behavior of the collapsing neck dramatically changes as the neck becomes as small as 100nm. A challenging project would be to determine the best way to measure the properties of the neck just as the droplet pinches off.

• Foam generation by breaking waves. Another project involves characterizing the generation of foam from breaking wave states. Past research in Prof. Lathrop's laboratory has characterized the threshold for the production of spray and air bubble entrainment in vertically oscillated waves. With higher forcing levels, these turbulent wave states have been observed to further transition into a foamy state, with a statistically steady foam layer evolving on the surface. The project would entail measuring and characterizing this foam.

Additional information about the experimental studies of nonlinear singularities in fluids and solids can be found at http://complex.umd.edu and by contacting Professor Lathrop at (301) 405-1594 or lathrop@umd.edu.