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Nonlinear dynamics refers to phenomena governed by nonlinear differential equations, often fluids. It has important interfaces with soft matter and biological physics. Fluid dynamics has important connections to astrophysics, geophysics, and engineering. The projects to be described involved a team approach: a postdoctoral researcher, one or more undergraduates, and me.
Here are some of the research questions we examined over the last few years: Do small particles accurately follow fluid flows? How do elongated particles orient themselves in a fluid? Do converging flows exhibit spontaneous swirl? When is the flow of a fluid containing particles reversible? What unique flow properties are manifested by polymeric fluids? How do swimming cells interact? I'll say just a little about each of these projects, and then comment on the role of research in Physics with students at Haverford.
Undergraduate Peter O'Malley looked at the question of whether small particles accurately follow fluid flows, working with postdoctoral fellow Nick Ouellette, now at Yale. Particles certainly do not follow the fluid if their density is different and the particles are accelerated. But what happens if there is no density difference? The question is important because particle tracking is the primary method by which flow phenomena are studied. Peter used electromagnetic forcing to drive a chaotic flow, and tracked particles of different diameter: 80 micron particles, which followed these slow flows well, to determine the velocity field, and 1-2 mm particles of the same density to look for deviations. Very significant velocity differences between particles and the local fluid elements were detected for the larger particles, and the results were published in Physical Review Letters.1 Peter entered a graduate program in physics after his time at Haverford.
Subsequently, undergraduate Monica Kishore, working with Nick Ouellette and postdoc Jeffrey Guasto, investigated the question of how elongated particles orient themselves in the same type of fluid flow. Monica made excellent progress on this problem, and the eventual result, finished after her departure for graduate study in medical physics, was that particle alignment can be explained using what we call stretching fields. These fields, which can be computed from measured velocity fields, give the local strength of the stretching of fluid elements. This work has been submitted to Physics of Fluids, in collaboration with Greg Voth's group, which studied the same problem in parallel with our work at Haverford.2 Greg had earlier pioneered the process of measuring stretching fields3 while a postdoctoral fellow at Haverford, before joining the Wesleyan University faculty.
Undergraduate Michael Jablin looked at the question of whether spontaneous swirl exists in a converging fluid flow, as had been claimed in published work. He designed and built an apparatus to test this hypothesis, and became an expert in particle tracking to look for a small azimuthal (non-radial) velocity in converging flows. His measurements were quite sensitive, but there was no convincing evidence for the claimed effect. While this outcome was disappointing to us, the work resulted in excellent training for Michael, who obtained a job after graduation providing user support at the Los Alamos SPEAR neutron reflectometer facility. His work there led to diverse publications and eventual graduate study. So experiments don't always have to discover or characterize a new phenomenon to produce a useful educational outcome.
Undergraduate Andrew Ross worked on the reversibility of low Reynolds number flow containing particles. We knew from earlier work that such flows can be irreversible, as a result of chaotic interactions between particles.4 Andrew, working with postdoctoral fellow Jeffrey Guasto, looked at channel flows, where the fluid is sheared non-uniformly. This work, completed after Andrew went on to work with a colleague on quantum computing, showed that channel flows can produce irreversibility everywhere, even in places where the shear is small. The results were published in Physical Review E.5
Students James Diorio and Charles Thomas, working with postdoctoral associate Paulo Arratia (now at University of Pennsylvania), studied instabilities in polymeric solutions using microfluidic flows. They detected two new instabilities which occur at low Reynolds number, where Newtonian fluids would flow without instability. The resulting paper6 stimulated quite a bit of theoretical work and garnered 25 citations. Charles is now a graduate student at Penn in nonlinear physics, and James got a Ph.D. at University of Maryland in Mechanical Engineering.
Recently, my group has been working on the fluid flows induced by swimming algal cells only 10 microns across, which use twin flagella moving in a breaststroke pattern to propel themselves.7 Algal cells account for a significant contribution to the world's oxygen production, and their flagella are similar to those found in some cells in the human body. Current undergraduates Andrew Sturner and Ivy Tao have been working to understand the interactions between these swimming cells. Are they mainly hydrodynamic (where each swimmer's induced velocity field advects the other cells)? Or do the cells sense each other and respond?
What happened to the postdoctoral scholars who worked with the undergraduates on these projects? they have faculty appointments elsewhere, along with independent research funding, and continue to work with students.
Scholarly investigations first became a requirement for the undergraduate degree at Haverford in 1920. I understand that Reed College was also an early adopter of this approach, and would be curious to hear of others. Undergraduate research mentoring is also built into the teaching responsibilities of faculty members. It takes a considerable time investment to make research experiences available to students, and suitably designed advanced laboratory courses can prepare them effectively (for example, see ours at http://www.haverford.edu/physics-astro/course_materials/phys326/phys326.html ). Summer research opportunities are also critical for students. This past summer, 22 students did research at Haverford with physics faculty members, and 7 did so elsewhere, altogether at least 76% of our junior and senior majors. The fields represented were diverse, including quantum gravity, biological physics, nanoscale (condensed matter) physics, and near field cosmology, in addition to the work on nonlinear/fluid dynamics described in this summary. Active engagement in research has been rewarding for our students, many of whom have won awards (Goldwater, Churchill, Fullbright, NSF, Apker, etc.) At least four former research students later won NSF Career Awards when they became faculty members. Our website shows the diverse careers of Haverford physics graduates, many of whom chose directions outside of scientific research. However, we believe (and they indicate) that their lives and careers have been significantly enriched by their research experiences as undergraduates.
The work described in this summary was presented at the Gordon Research Conference on Physics Research and Education in June 2010, and was supported by NSF-DMR-0803153.
1. N. T. Ouellette, P. J. J. O'MALLEY and J. P. Gollub, Phys. Rev. Lett. 101, 174504 (2008).
2. S. Parsa, J. S. Guasto, M. KISHORE, N. T. Ouellette and J. P. Gollub (submitted, 2010).
3. G. A. Voth, G. Haller and J. P. Gollub, Phys. Rev. Lett. 88, 25401 (2002).
4. D. J. Pine, J. P. Gollub, J. F. Brady and A. M. Leshansky, Nature 438, 997-1000 (2005).
5. J. S. Guasto, A. S. ROSS and J. P. Gollub, Phys. Rev. E 81, 061401 (2010).
6. P. E. Arratia, C. C. THOMAS, J. DIORIO and J. P. Gollub, Phys. Rev. Lett. 96, 144502 (2006).
7. J. S. Guasto, K. Johnson and J. P. Gollub (Phys. Rev. Lett. in press).
Jerry Gollub is a professor of physics at Haverford College, where he has worked with undergraduates in research for 40 years. He was the first recipient of the APS Award for Research in an Undergraduate Institution, and is a member of the National Academy of Sciences.