^{1}and Nicholas T. Ouellette

^{1,a)}

### Abstract

Using a simple model, we study the transport dynamics of active, swimming particles advected in a two-dimensional chaotic flow field. We work with self-propelled, point-like particles that are either spherical or ellipsoidal. Swimming is modeled as a combination of a fixed intrinsic speed and stochastic terms in both the translational and rotational equations of motion. We show that the addition of motility to the particles causes them to feel the dynamical structure of the flow field in a different way from fluid particles, with macroscopic effects on the particle transport. At low swimming speeds, transport is suppressed due to trapping on transport barriers in the flow; we show that this effect is enhanced when stochastic terms are added to the swimming model or when the particles are elongated. At higher speeds, we find that elongated swimmers tend be attracted to the stable manifolds of hyperbolic fixed points, leading to increased transport relative to swimming spheres. Our results may have significant implications for models of real swimming organisms in finite-Reynolds-number flows.

I. INTRODUCTION

II. MODEL

A. Flow field

B. Swimmers

III. SPHERICAL PARTICLES

A. Deterministic swimming

B. Translational stochasticity

C. Rotational stochasticity

IV. ELLIPSOIDAL PARTICLES

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Diffusion
- 14.0
- Rheology and fluid dynamics
- 14.0
- Stokes flows
- 13.0
- Equations of motion
- 11.0
- Turbulent flows
- 9.0

## Figures

(a) Snapshot of one unit cell of the flow field at zero phase (that is, for *t* such that sin Ω*t* = 0). Arrows show the local fluid velocity, and the shading shows the vorticity. As time progresses, the flow field oscillates sinusoidally in the horizontal direction. (b) Poincare section (at zero phase) for a fluid element initially in the chaotic sea, for *B* = 0.12 and Ω = 6.28. Only one quarter of the unit cell is shown (corresponding to the lower-left vortex in (a)); the rest of the unit cell is related by symmetry. The central empty region is a period-1 elliptic island; the surrounding empty regions are a period-3 island chain. (c) Finite-time Lyapunov exponent (FTLE) field at zero phase. Again, only one quarter of the unit cell is shown.

(a) Snapshot of one unit cell of the flow field at zero phase (that is, for *t* such that sin Ω*t* = 0). Arrows show the local fluid velocity, and the shading shows the vorticity. As time progresses, the flow field oscillates sinusoidally in the horizontal direction. (b) Poincare section (at zero phase) for a fluid element initially in the chaotic sea, for *B* = 0.12 and Ω = 6.28. Only one quarter of the unit cell is shown (corresponding to the lower-left vortex in (a)); the rest of the unit cell is related by symmetry. The central empty region is a period-1 elliptic island; the surrounding empty regions are a period-3 island chain. (c) Finite-time Lyapunov exponent (FTLE) field at zero phase. Again, only one quarter of the unit cell is shown.

Chaotic diffusion coefficient *D* normalized by *D* _{0}, the diffusion coefficient for fluid elements, as a function of the swimming speed *v* _{ s }, for spherical swimmers with σ_{ s } = σ_{ r } = 0. Error bars are computed from the statistical fluctuations between many sets of simulations. Two distinct regions of suppressed long-time transport are seen, corresponding to trapping by the period-3 islands and by the period-1 islands.^{20}

Chaotic diffusion coefficient *D* normalized by *D* _{0}, the diffusion coefficient for fluid elements, as a function of the swimming speed *v* _{ s }, for spherical swimmers with σ_{ s } = σ_{ r } = 0. Error bars are computed from the statistical fluctuations between many sets of simulations. Two distinct regions of suppressed long-time transport are seen, corresponding to trapping by the period-3 islands and by the period-1 islands.^{20}

Chaotic diffusion coefficient *D* normalized by *D* _{0} as a function of σ_{ s }, for *v* _{ s } = 0, σ_{ r } = 0, and α = 0. When compared with Fig. 2, transport is more strongly suppressed, but the distinct signatures of each type of elliptic island are no longer present.

Chaotic diffusion coefficient *D* normalized by *D* _{0} as a function of σ_{ s }, for *v* _{ s } = 0, σ_{ r } = 0, and α = 0. When compared with Fig. 2, transport is more strongly suppressed, but the distinct signatures of each type of elliptic island are no longer present.

Average time for a swimmer to cross a cell boundary as a function of its spatial location for (a) *v* _{ s } = 0.004 and σ_{ s } = σ_{ r } = 0 and (b) σ_{ s } = 0.15 and *v* _{ s } = σ_{ r } = 0. Only one quarter of the flow domain is shown. The shade/color bar gives the cell-crossing time in flow cycles. For the deterministic swimmer in (a), trapping is strong in the period-3 islands and on a small ring just inside the period-1 island. The purely stochastic swimmer can wander into the core of the period-1 island, where trapping is strongest.

Average time for a swimmer to cross a cell boundary as a function of its spatial location for (a) *v* _{ s } = 0.004 and σ_{ s } = σ_{ r } = 0 and (b) σ_{ s } = 0.15 and *v* _{ s } = σ_{ r } = 0. Only one quarter of the flow domain is shown. The shade/color bar gives the cell-crossing time in flow cycles. For the deterministic swimmer in (a), trapping is strong in the period-3 islands and on a small ring just inside the period-1 island. The purely stochastic swimmer can wander into the core of the period-1 island, where trapping is strongest.

Chaotic diffusion coefficients *D* in the two-dimensional parameter space spanned by *v* _{ s } and σ_{ s } for σ_{ r } = 0. The shade/color bar shows *D* relative to *D* _{0}. The black line separates the regions of suppressed transport (*D*/*D* _{0} < 1) from those of enhanced transport (*D*/*D* _{0} > 1).

Chaotic diffusion coefficients *D* in the two-dimensional parameter space spanned by *v* _{ s } and σ_{ s } for σ_{ r } = 0. The shade/color bar shows *D* relative to *D* _{0}. The black line separates the regions of suppressed transport (*D*/*D* _{0} < 1) from those of enhanced transport (*D*/*D* _{0} > 1).

Chaotic diffusion coefficients *D* in the two-dimensional parameter space spanned by *v* _{ s } and σ_{ r } for σ_{ s } = 0. The shade/color bar shows *D* relative to *D* _{0}. The black line separates the regions of suppressed transport (*D*/*D* _{0} < 1) from those of enhanced transport (*D*/*D* _{0} > 1). Transport is suppressed in larger region of parameter space than it was for purely translational stochasticity case in Fig. 5.

Chaotic diffusion coefficients *D* in the two-dimensional parameter space spanned by *v* _{ s } and σ_{ r } for σ_{ s } = 0. The shade/color bar shows *D* relative to *D* _{0}. The black line separates the regions of suppressed transport (*D*/*D* _{0} < 1) from those of enhanced transport (*D*/*D* _{0} > 1). Transport is suppressed in larger region of parameter space than it was for purely translational stochasticity case in Fig. 5.

Spatially resolved maps of the average time to cross a cell boundary for *v* _{ s } = 0.004, and σ_{ r } = (a) 0.1, (b) 1.0, (c) 3.0, and (d) 6.0. σ_{ s } = 0 for all panels. The shade/color bar gives the cell-crossing times in flow cycles. The times are much longer than the comparable *v* _{ s } = 0.004, σ_{ r } = 0 case shown in Fig. 4(a).

Spatially resolved maps of the average time to cross a cell boundary for *v* _{ s } = 0.004, and σ_{ r } = (a) 0.1, (b) 1.0, (c) 3.0, and (d) 6.0. σ_{ s } = 0 for all panels. The shade/color bar gives the cell-crossing times in flow cycles. The times are much longer than the comparable *v* _{ s } = 0.004, σ_{ r } = 0 case shown in Fig. 4(a).

(a) Chaotic diffusion coefficient *D* normalized by *D* _{0} as a function of eccentricity α for deterministic swimmers with *v* _{ s } = 0.002. Transport is much more strongly suppressed for ellipsoids of intermediate eccentricities than it is for spheres. (b)–(d) Probability density functions of swimmer position for (b) α = 0, (c) α = 0.5, and (d) α = 1. The strong suppression of transport for ellipsoidal particles is due to the formation of attractors.

(a) Chaotic diffusion coefficient *D* normalized by *D* _{0} as a function of eccentricity α for deterministic swimmers with *v* _{ s } = 0.002. Transport is much more strongly suppressed for ellipsoids of intermediate eccentricities than it is for spheres. (b)–(d) Probability density functions of swimmer position for (b) α = 0, (c) α = 0.5, and (d) α = 1. The strong suppression of transport for ellipsoidal particles is due to the formation of attractors.

(a) Chaotic diffusion coefficient *D* normalized by *D* _{0} as a function of eccentricity α for deterministic swimmers with *v* _{ s } = 0.08. For this speed, transport of ellipsoidal particles is strongly enhanced relative to spheres. (b)–(d) Probability density functions of swimmer position for (b) α = 0, (c) α = 0.5, and (d) α = 1. The strong enhancement of transport is due to the clustering of ellipsoids on the stable manifolds of the hyperbolic fixed points.

(a) Chaotic diffusion coefficient *D* normalized by *D* _{0} as a function of eccentricity α for deterministic swimmers with *v* _{ s } = 0.08. For this speed, transport of ellipsoidal particles is strongly enhanced relative to spheres. (b)–(d) Probability density functions of swimmer position for (b) α = 0, (c) α = 0.5, and (d) α = 1. The strong enhancement of transport is due to the clustering of ellipsoids on the stable manifolds of the hyperbolic fixed points.

Mean nearest neighbor distance δ_{ NN } scaled by the value for passive particles δ_{0} as a function of eccentricity for (a) *v* _{ s } = 0.002 and (b) *v* _{ s } = 0.08. High aspect ratio swimmers tend to be closer to each than spherical swimmers are, leading to enhanced encounter rates.

Mean nearest neighbor distance δ_{ NN } scaled by the value for passive particles δ_{0} as a function of eccentricity for (a) *v* _{ s } = 0.002 and (b) *v* _{ s } = 0.08. High aspect ratio swimmers tend to be closer to each than spherical swimmers are, leading to enhanced encounter rates.

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