Inertial Self-Caging: Dynamics of Macroscopic Swimmers at Moderate Reynolds Number Sustaining Chemical Wake Resonance

Abstract

Self-propelled phoretic swimmers are generally studied in the laminar flow regime, where their low speed renders inertial effects negligible and trajectories highly predictable. This research tackles the challenge of propulsion in the inertial regime, at moderate Reynolds numbers (100 < Re < 200), where fluid dynamics becomes non-linear. By using a chemically driven macroscopic hydrogel this work demonstrates, through experiments and modeling, the existence of stable resonant states under confined geometry: as the swimmer circles, it interacts with its own lasting chemical wake. This chemical self-feedback creates a complex, stable motion characterized by both universal exponential speed decay with superimposed significant periodic speed oscillations. Furthermore, a critical threshold speed is identified, where the system abruptly transitions from the resonant oscillatory regime to a stochastic stop & go behavior. These findings provide a fundamental understanding of how chemical fields, hydrodynamic inertia, and confinement couple to determine the motion properties of high-speed active matter.

Figures (12)

• Figure 1: Typical dynamics of a swimmer motion. (a-top) Profile of the speed of the swimmer. Initially the speed follows an exponential oscillating decay with superimposed significant oscillations. After a certain (and variable) time the swimmer comes to an abrupt stop and then it enters into a stochastic, stop & go, regime. (a-bottom) The rainbow line reports the cosine of the angle of the swimmer position with respect to $\hat{u_{x}}$ centered at the pool center. During the initial phase of the motion the swimmer performs closed trajectory following the edge of the container and they are characterized by a chirped period of the oscillations. (b) Typical trajectory of a swimmer: (Top) During the exponential decaying phase the inertial movement propels the swimmer along the pool wall edges. (Bottom) When the speed reaches a critical threshold value the motion stops abruptly and a wandering trajectory develops. The rainbow-coded color corresponds to the one of time traces in panels (a) and (b).
• Figure 2: Analysis of both temporal and spatial cross correlation between the speed and position of the swimmer. (a) Time correlation of the oscillations of the speed (detrended by subtracting the exponential decay, blue line) and cosine of the angular position of the swimmer (orange line) are overlapped and show a clear correlation. The inset shows the cross correlation of these signals; the two small red dots mark the minima of the cross correlation that correspond to $\pm \pi$ shift from the average period of the oscillations. (b) Spatial correlation between max and min of the speed. Symbols refer to same quantities of panel (a). Generally max and min tends to space apart but there is a significant noise. The three gray lines map three adjacent couples of max and adjacent minima. The inset shows the dispersion of the separations between adjacent max and min: the horizontal line marks the average value ($80 \pm 10\ mm$) and the shaded region is $\pm 1 \sigma$.
• Figure 3: (a-top) Spectrogram of the speed oscillations for a swimmer moving in a large pool. The trajectory that creates the spectrogram is shonw in (b). (a-bottom) Spectrogram for a swimmer confined inside a circular pool. The trajectory for this particular experiment is shown in (c). No pool outline was added to either (b) or (c) panel since the large size of the square pool would prevent a proper visualization of the details of the trajectory. In the large square pool, the swimmer explores an area of about $50 \times 25\ cm$, while the round small pool is slightly larger than the average circular loop (see the top panel of Fig.\ref{['fig:speed_vs_time']}(b)).
• Figure 4: Interpolated decay of the speed for experiments in which oscillations last for at least few hundreds of seconds (some tens of loops). The lines color refer to specific W/G ratio as indicated in the legend. Generally, decays stop abruptly at speed threshold values of about $10 \pm 2.5\ mm/s$.
• Figure 5: (a) Typical dynamic of the speed of the swimmer in the stop & go regime. Orange dots indicate speed peaks and red dots the adjacent local minima. Gray area are the widths used to integrate the peak area. Note that the time scale for the stop & go regime starts at breakpoint. (b) Mean lag-1 correlation for fundamental parameters of motion averaged over the entire set of experiments for W/G=4/1; PH: Peak Height, IPIs: Inter Peaks Intervals, Space: space traveled during the event (in mm). The colors of the map report the values of the correlations, while the numbers indicates the CVs: (IPIs, PH) and (Space, IPIs) are vaguely correlated, (Space, PH) are uncorrelated.