Vortex-sink flow

Figure 2.13 Escape times as a function of initial location, showing singularities on a fractal set for an ensemble of particles released on a line segment in the vortex-sink flow (from Karolyi and Tel (1997)).

Figure 2.15 Instantaneous view of the chaotic saddle (a), stable (b) and unstable (c) manifolds for the advection dynamics in the blinking vortex-sink flow. The vortex-sinks are at (x,y) = ( — 1,0) and (0,1) (from Karolyi and Tel (1997)).

A simple model that illustrates the chaotic scattering process and fractal manifolds in open flows is the blinking vortex-sink flow (Karolyi and Tel, 1997) that consists of two point vortex-sinks, with their centers separated by distance L, in an unbounded two-dimensional domain. Each vortex-sink is opened alternately for a half period T/2 and, while active, it generates a flow given by the streamfunction... 

Figure 7.4 Snapshots of the spatial distribution for the autocatalytic model (7.1) in the open blinking vortex-sink flow at time intervals equal to the flow period for a supercritical Damkohler number, Da = 7.0. Note, that after a transient time a time-periodic asymptotic state is reached where the autocatalytic growth, localized on the fractal unstable manifold, is balanced by the loss of product due to the outflow from the mixing region, in this case through the point sinks.

Figure 7.10 Total average concentration Ctotai in the stationary state vs Da obtained numerically for the bistable model in the open vortex-sink flow. Note the discontinuous jump to Ctotai = 0 at Da = Dac R 24.2, that is characteristic to the bistable dynamics.

Figure 7.11 Total concentration (C ) in the stationary state vs Da for the FitzHugh-Nagumo dynamics in the open blinking vortex-sink flow showing two discontinuous transitions.

This separation zone is characterized by a flat air vortex prevailing in a cylindrical chamber with tangential inlet and central outlet (Fig. lc). In this vortex air rotates and flows radially towards the chamber centre. The radial air movement (radial sink flow type) serves as the particles separation track. 

In the blinking vortex-sink chaotic open flow system we have again two qualitatively different regimes, depending on Da, with a transition at the critical value Dac 2.3. As before, for small Da the initial perturbation decays fast towards C = 0. This is, however, not followed by an homogeneous transition to the C = 1 state from the (7 0 unstable configuration. The perturbation is now completely... 

Figure 7.5 Total amount of reaction product (C) vs Da for the autocatalytic reaction advected in the alternating vortex-sink open flow system in the stationary state at a given fixed phase of the periodic flow. Note that a transition takes place at Da 2.2, below which the escape dominates and the autocatalytic component is washed out from the system. Above the critical value a non-zero steady state exists as a result of a dynamical equilibrium between the reaction and the outflow.

The flow shown in Fig. 10.11 is called a free vortex, it is not common alone in natur. However, the combination of a free vortex with the sink shown in Fig. 10.6, obtained by adding the potential functions, produces... 

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