^{1}

### Abstract

The purpose of this work is to analyze the flow due to a potential point vortex pair in the vicinity of a symmetry line (or “wall”), in order to understand why the presence of the wall, even far from the vortices, accelerates fluid mixing around the vortex pair. An asymptotic analysis, in the limit of large distances to the wall, allows to approximate the wall effect as a constant translation of the vortex pair parallel to the wall, plus a straining flow which induces a natural blinking vortex mechanism with period half the rotation period. A Melnikov analysis of lagrangian particles, in the frame translating and rotating with the vortices, shows that a homoclinic bifurcation indeed occurs, so that the various separatrices located near the vortex pair (and rotating with it) do not survive when a wall is present. The thickness of the resulting inner stochastic layer is estimated by using the separatrix map method and is shown to scale like the inverse of the squared distance to the wall. This estimation provides a lower-bound to the numerical thickness measured from either Poincaré sections or simulations of lagrangian particles transported by the exact potential velocity field in the laboratory frame. In addition, it is shown that the outer homoclinic cycle, separating the vortices from the external (open) flow, is also perturbed from inside by the rotation of the vortex pair. As a consequence, a stochastic layer is shown to exist also in the vicinity of this cycle, allowing fluid exchange between the vortices and the outer flow. However, the thickness of this outer stochastic zone is observed to be much smaller than the one of the inner stochastic zone near vortices, as soon as the distance to the wall is large enough.

The author would like to thank A. Motter for fruitful discussions at Northwestern University, and T. Nizkaya for her comments.

I. INTRODUCTION

II. INNER ASYMPTOTIC VELOCITY FIELD NEAR VORTICES

A. Approximate vortex motion

B. Flow in the rotating frame

III. WALL-INDUCED CHAOTIC ADVECTION NEAR VORTICES

A. Splitting of the inner homoclinic cycle

B. Characterization of the inner stochastic zone

C. Comparison with transport by the exact potential velocity field

IV. MIXING WITH THE OUTER OPEN FLOW

A. Asymptotic analysis of the outer homoclinic cycle

B. Numerical verification

V. CONCLUSION

### Key Topics

- Rotating flows
- 124.0
- Flow instabilities
- 6.0
- Rheology and fluid dynamics
- 6.0
- Vortex dynamics
- 6.0
- Mirrors
- 4.0

## Figures

Sketch of the vortex pair (A1, A2), together with the two mirror vortices .

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.

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).

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.

(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.

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 Σ_{2}, 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) Thickness of the inner stochastic zone at point *C* of Σ_{2}, 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* = 50*π*, computed in the laboratory frame by using the exact potential velocity field, for *L* _{0} = 10*d* _{0} (*ɛ* = 0.1) (a), and *L* _{0} = 5*d* _{0} (*ɛ* = 0.2) (b). Solid lines are streamlines.

(Color online) Particle clouds at *t* = 50*π*, computed in the laboratory frame by using the exact potential velocity field, for *L* _{0} = 10*d* _{0} (*ɛ* = 0.1) (a), and *L* _{0} = 5*d* _{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.)

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* = 100*π*, 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.

The role of the inner and outer separatrices in chaotic mixing. The four graphs show particle clouds in the laboratory frame at *t* = 100*π*, 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).

(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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