Abstract
The advection of tracer particles in hydrodynamical flows represents one of the successful applications of chaos theory. The basic observation is that molecular diffusion is negligible on the typical time scale of the flow. As a result, in the absence of any diffusion-enhancing mechanism such as hydrodynamical turbulence, advection dominates. Indeed, the main physical mechanism for fluid stirring is advection, whose efficiency can be enhanced greatly by chaotic dynamics. The spreading of pollutants on large scales is also dominated by advection. Potential applications of chaotic advection range from laboratory investigations of fluid dynamics to the study of large-scale environmental flows. From the point of view of dynamical systems, an appealing feature of the passive advection problem is that its phase space coincides with the physical space of the fluid, rendering possible direct experimental observation and characterization of fractal structures associated with chaotic dynamics.
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Notes
- 1.
In particular, if the length and velocity scales are L and U, respectively, the typical hydrodynamical time scale is L ∕ U and the diffusive time scale is L 2 ∕ D diff, where D diff is the diffusion coefficient, whose typical value for water and most tracer substances is of order \(1{0}^{-8}\ {\mbox{ m}}^{2}/\mbox{ s}\). Suppose L = 1 m and \(U = 0.1\ \mbox{ m}/\mbox{ s}\). The hydrodynamical and the diffusive time scales are thus ten seconds and one thousand days, respectively, rendering physically irrelevant any diffusive processes in the flow.
- 2.
Note that Fig. 10.25 represents a plot of initial conditions. For both ideal (Hamiltonian) and inertial particles, those with long lifetimes belong to the stable foliation of the nonattracting chaotic set. It is known that for a general Hamiltonian system, under weak dissipation, the stable foliations are converted into the basin boundaries between the coexisting attractors [534], which are fractals.
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Lai, YC., Tél, T. (2011). Chaotic Advection in Fluid Flows. In: Transient Chaos. Applied Mathematical Sciences, vol 173. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6987-3_10
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DOI: https://doi.org/10.1007/978-1-4419-6987-3_10
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