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Chaotic Mixing


Stretching Field: lines of large past (red) and future (blue) stretching. Click to enlarge
Jerry Gollub of Haverford College is understandably quite famous for his pioneering work in measuring the onset of turbulence. With his conceptually clean (yet technically difficult because of their precision) experiments, he and his colleagues and students have produced a wide range of experimental and theoretical results that demonstrate the role of chaotic dynamics in fluid dynamics.

Gollub is at it again, this time with colleague Paulo Arratia of U. Penn. Gollub and Arratia designed a clever experiment in which they were able to observe the mixing of two fluids in a regime known as "chaotic advection," which is distinctly different from turbulence. (See the review article Mixing, Chaotic Advection, and Turbulence by J.M. Ottino for a good description of these different fluid regimes.

As described in the Feb., 2006 issue of Physics Today (and soon to be published in Phys. Rev. Lett.), Gollub and Arratia were able to measure the stretching field of their fluid. This field is the "local distortion of an infinitesimally small fluid element." This field, in turn, can be used to calculate the Lyapunov exponent for the fluid under different mixing conditions. (The Lyapunov exponent is a well-established measure of the tendency for the phase trajectories of chaotic systems to move apart.) Remarkably, Gollub and Arratia found that they could model the amount of chemical product formed from their mixing reactants as a function of Lyapunov exponent only for a large range of mixing conditions. This result is important because it demonstrates yet again one of the hallmarks of chaotic systems - universality, which is Feigenbaum’s contribution to chaos theory (and which the Gollub/Swinney rotating cylinder experiments helped establish as experimental fact.)

To view more ongoing Gollub projects (as well as interesting applets showing chaotic mixing), visit the Nonlinear Physics and Fluid Dynamics Lab of Haverford College.