Abstract
“Ghost Rods” are periodic structures in a two-dimensional flow that have an effect on material lines that is similar to real stirring rods. An example is a periodic island: material lines exterior to it must wrap around such an island, because determinism forbids them from crossing through it. Hence, islands act as topological obstacles to material lines, just like physical rods, and lower bounds on the rate of stretching of material lines can be deduced from the motion of islands and rods. Here, we show that unstable periodic orbits can also act as ghost rods, as long as material lines can “fold” around the orbit, which requires the orbit to be parabolic. We investigate the factors that determine the effective size of ghost rods, that is, the magnitude of their impact on material lines.
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Bibliography
B. J. Binder and S. M. Cox. A mixer design for the pigtail braid. Fluid Dyn. Res., 49:34–44, 2008.
P. L. Boyland, H. Aref, and M. A. Stremler. Topological fluid mechanics of stirring. J. Fluid Mech., 403:277–304, 2000.
P. L. Boyland, M. A. Stremler, and H. Aref. Topological fluid mechanics of point vortex motions. Physica D, 175:69–95, 2003.
M. D. Finn, S. M. Cox, and H. M. Byrne. Topological chaos in inviscid and viscous mixers. J. Fluid Mech., 493:345–361, 2003.
M. D. Finn, J.-L. Thiffeault, and E. Gouillart. Topological chaos in spatially periodic mixers. Physica D, 221:92–100, 2006.
M. Giona, S. Cerbelli, F. J. Muzzio, and A. Adrover. Non-uniform stationary measure properties of chaotic area-preserving dynamical systems. Physica A, 254:451–465, 1998.
E. Gouillart, J.-L. Thiffeault, and M. D. Finn. Topological Mixing with ghost rods. Phys. Rev. E, 73:036311 2006.
M. Liu and F. J. Muzzio. The curvature of material lines in chaotic cavity flows. Phys. Fluids, 8:75–83, 1996.
S. Newhouse and T. Pignataro, On the estimation of topological entropy. J. Stat. Phys., 72:1331, 1993.
R. T. Pierrehumbert. Tracer microstructure in the large-eddy dominated regime. Chaos Solitons Fractals, 4:1091–1110, 1994.
M. A. Stremler and J. Chen. Generating topological chaos in lid-driven cavity flow. Phys. Fluids, 19:103602, 2007.
J.-L. Thiffeault. Stretching and curvature along material lines in chaotic flows. Physica D, 198:169–181, 2004.
J.-L. Thiffeault. Measuring topological chaos. Phys. Rev. Lett., 94:084502, 2005.
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© 2009 CISM, Udine
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Thiffeault, JL., Gouillart, E., Finn, M.D. (2009). The Size of Ghost Rods. In: Cortelezzi, L., Mezić, I. (eds) Analysis and Control of Mixing with an Application to Micro and Macro Flow Processes. CISM International Centre for Mechanical Sciences, vol 510. Springer, Vienna. https://doi.org/10.1007/978-3-211-99346-0_10
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DOI: https://doi.org/10.1007/978-3-211-99346-0_10
Publisher Name: Springer, Vienna
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