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Alignment rule and geometric confinement lead to stability of a vortex in active flow

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Abstract

Vortices are hallmarks of a wide range of nonequilibrium phenomena in fluids at multiple length scales. In this work, we numerically study the whirling motion of self-propelled soft point particles confined in circular domain, and aim at addressing the stability issue of the coherent vortex structure. By the combination of dynamical and statistical analysis at the individual particle level, we reveal the persistence of the whirling motion resulting from the subtle competition of activity and geometric confinement. In the stable whirling motion, the scenario of the coexistence of the irregular microscopic motions of individual particles and the regular global whirling motion is fundamentally different from the motion of a vortex in passive fluid. Possible orientational order coexisting with the whirling are further explored. This work shows the stability mechanism of vortical dynamics in active media under the alignment rule in confined space and may have implications in creating and harnessing macroscale coherent dynamical states by tuning the confining geometry.

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Data Availability Statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants No. BC4190050).

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  1. School of Physics and Astronomy, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yi-Heng Zhang & Zhenwei Yao

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ZY proposed the research project. YZ and ZY designed the research. YZ performed the research.

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Correspondence to Yi-Heng Zhang.

Appendix A Simplification of the order parameter

Appendix A Simplification of the order parameter

Since the repulsive force is reciprocal and adds up to zero, we take time average of Eq. (1a) and have

$$\begin{aligned} P&=\frac{1}{Nv_0}\left<\left|\sum _i^N\frac{\mathrm {d}\mathbf {r}_i(t)}{\mathrm {d}t}\right|\right>_t\nonumber \\&=\frac{1}{Nv_0}\left<\sqrt{\left( \sum _i^N\mathbf {v}_i\cdot \mathbf {\hat{e}}_\varphi \right) ^2 + \left( \sum _i^N\mathbf {v}_i\cdot \mathbf {\hat{e}}_\rho \right) ^2}\right>_t\ . \end{aligned}$$
(A1)

We separate \(\sum _i^N\mathbf {v}_i(t)\) at any given time by its time average \(\overline{\mathbf {V}}\) and the variation \(\delta \mathbf {V}(t)\)

$$\begin{aligned} \sum _i^N\mathbf {v}_i(t)&=\overline{\mathbf {V}}+\delta \mathbf {V}(t)\nonumber \\&=\left( \overline{V}_\varphi +\delta V_\varphi (t)\right) \mathbf {\hat{e}}_\varphi +\delta V_\rho (t)\mathbf {\hat{e}}_\rho \ . \end{aligned}$$
(A2)

The last equal sign is due to the conservation of N, and there is no net flow along the radial direction. By expanding Eq. (A1) with respect to the variation, we have

(A3)

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Zhang, YH., Yao, Z. Alignment rule and geometric confinement lead to stability of a vortex in active flow. Eur. Phys. J. E 46, 4 (2023). https://doi.org/10.1140/epje/s10189-023-00260-3

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