Sketch of the vortex pair (A1, A2), together with the two mirror vortices .
Sketch of the stretching flow induced by the wall in the reference frame translating with I. This flow stretches and compresses the segment A 1 A 2 twice during each complete rotation of the vortex pair.
Streamlines of the leading-order flow in the rotating frame, in the vicinity of the vortices (inner asymptotic field).
(Color online) Poincaré sections of 20 particles, together with the theoretical boundary of the inner stochastic zone (black thick line). Graph (a): ɛ = 0.05 and graph (b): ɛ = 0.2.
Sketch of the separatrix map used to calculate the boundary of the inner stochastic layer.
(Color online) Thickness of the inner stochastic zone at point C of Σ2, versus ɛ = d 0/L 0, obtained from the separatrix map (Eq. (26), thick solid line) and estimated from numerical computations: Poincaré sections (empty circles), particle cloud using the asymptotic velocity in the rotating frame (squares), and particle cloud using the exact potential velocity in the laboratory frame (filled circles). The dashed line is the theoretical thickness of the outer stochastic zone near the separatrix S 1 S 2 discussed in Sec. IV.
(Color online) Particle clouds at t = 50π, computed in the laboratory frame by using the exact potential velocity field, for L 0 = 10d 0 (ɛ = 0.1) (a), and L 0 = 5d 0 (ɛ = 0.2) (b). Solid lines are streamlines.
Graph (a): streamlines in the frame translating with the vortices, when ɛ = 0.3. Both the real (y > 0) and mirror (y < 0) vortices are shown; the wall corresponds to y = 0. Solid lines: exact four-vortex potential flow (1). Dashed lines: outer asymptotic streamfunction (29). Graph (b): leading-order flow (ɛ = 0) in the same frame. S 1 and S 2 are saddle points located at . The thick line is the homoclinic cycle S 1 S 2. (Lengths have been rescaled by L 0, stars have been removed.)
The role of the inner and outer separatrices in chaotic mixing. The four graphs show particle clouds in the laboratory frame at t = 100π, obtained by using the exact four-vortex potential solution. Particles marked with black dots have been injected near the upper vortices at t = 0 (in box A of Fig. 8(b)). Those marked with empty circles have been injected outside the cycle S 1 S 2, ahead of vortices (in box B of Fig. 8(b)). Graph (a): . Graph (b): . Graph (c): . Graph (d): . Solid lines are streamlines.
(Color online) Graph (a): linearized flow in the vicinity of stagnation point A. Graph (b): comparison between the theoretical period of trajectories near the separatrix (Eq. (23), dashed line) and the numerical one (dots).
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