Deformable microswimmer in a swirl: Capturing and scattering dynamics

Inspired by the classical Kepler and Rutherford problem, we investigate an analogous setup in the context of active microswimmers: the behavior of a deformable microswimmer in a swirl flow. First, we identify new steady bound states in the swirl flow and analyze their stability. Second, we study the dynamics of a self-propelled swimmer heading towards the vortex center, and we observe the subsequent capturing and scattering dynamics. We distinguish between two major types of swimmers, those that tend to elongate perpendicularly to the propulsion direction and those that pursue a parallel elongation. While the first ones can get caught by the swirl, the second ones were always observed to be scattered, which proposes a promising escape strategy. This offers a route to design artificial microswimmers that show the desired behavior in complicated flow fields. It should be straightforward to verify our results in a corresponding quasi-two-dimensional experiment using self-propelled droplets on water surfaces.

Figure 1

Variables characterizing the orientational degrees of freedom of an elliptically deformable active microswimmer when viewed from the laboratory frame (schematic).

Figure 2

Regions of stable circular motions for soft deformable particles in the perpendicular configuration. $r_0$ gives the radius of the circular trajectory. $\gamma >0$ measures the self-propulsion speed of active swimmers (diamonds), whereas $\gamma <0$ characterizes the friction for passive particles (plusses). We indicate by the vertical dashed line the value of the radius $r_{0,\min }$, see main text and Eq. (A14). The solid (blue) line gives the stability range of circular trajectories for deformable passive particles. This line can be calculated analytically, see Eqs. (A16) and (A17). Passive systems initialized in the “dip” region are horizontally expelled from that area as indicated by the arrows. Active swimmers perform different types of motion, depending on the initial conditions: an active circular motion (diamonds) or an escape (not indicated) for $0<\gamma <0.5$ (lower hatched area); an active circular motion (diamonds) or a lunar-type motion (not indicated) for $\gamma >0.5$ (upper hatched area). The system parameters are set to $\kappa =0.5$, $a_1=b_1=-1$, ${\nu }_1=1$, and $\mu =1$.

Figure 3

Example trajectories of (a) and (b) lunar-type motions and (c) a multicircular motion, each for $\gamma =1$ ($>{\gamma }_c$). Further parameter values are (a) $a_1=b_1=-1$ (perpendicular case) and $r_{\mathrm{init}}=10$, as well as (b) and (c) $a_1=b_1=+1$ (parallel case) together with (b) $r_{\mathrm{init}}=10$ and (c) $r_{\mathrm{init}}=0.5$. The remaining parameters are set to $\kappa =0.5$, ${\nu }_1=1$, and $\mu =1$. Different colors mark trajectory pieces of different intervals of swimming time. Black superimposed silhouettes indicate particle orientations and degrees of deformation.

Figure 4

Definition of the impact parameter $d_{\mathrm{imp}}$ depicted using example trajectories of incident swimmers (a) in a perpendicular configuration ($a_1=b_1=-1$) and (b) in a parallel configuration ($a_1=b_1=1$) for $\gamma =0.3$. The impact parameter $d_{\mathrm{imp}}$ measures the distance by which the swimmer would miss the swirl center if its trajectory were not affected by the swirl flow. Gray arrows on the axis $x=0$ indicate the direction of the swirl flow, pointing to the left for $y>0$ and to the right for $y<0$. The sign of $d_{\mathrm{imp}}$ is defined as positive when the swimmer initially heads towards the side $y>0$ of oppositely directed fluid flow; it is chosen negative when the swimmer initially heads towards the side $y<0$ of identically oriented fluid flow. For illustration, the scales of the $x$ and $y$ axes are chosen differently and distances in the $y$ direction are enlarged by a factor of 10.

Figure 5

Scattering dynamics for $a_1=b_1=-1$ (perpendicular case): (a) scattering angle ${\eta }_{\mathrm{scat}}$ as a function of the impact parameter $d_{\mathrm{imp}}$ for various propulsion strengths $\gamma$; [(b) and (c)] example trajectories for $\gamma =0.3$ and for different impact parameters $d_{\mathrm{imp}}$. Black superimposed silhouettes indicate the swimmer orientations and degrees of deformation. Arrows mark the direction of motion. Furthermore, the gray dotted lines illustrate the direction of the fluid flow, with the vortex center marked by the plus symbol. (c) There is a finite interval of impact parameters for which the swimmer cannot escape from the vortex any more but gets captured by the swirl. (Other parameter values are $\kappa =0.5$, ${\nu }_1=1$, and $\mu =1$.)

Figure 6

Dynamical phase diagram as a function of the impact parameter $d_{\mathrm{imp}}$ and the strength of active propulsion $\gamma$ for the perpendicular case ($a_1=b_1=-1$). Two dynamic events are distinguished: getting scattered by the swirl with a subsequent escape (area outside the lines) and getting captured by the vortex (area between the lines). Our results do not qualitatively depend on the initial distance of the swimmer from the vortex center as can be seen when comparing panels (a) and (b) with each other or when comparing the solid lines with the dashed lines within each panel. Reentrance of the dynamic events can be observed in both directions of the parameter space.

Figure 7

Scattering dynamics for $a_1=b_1=+1$ (parallel case): (a) scattering angle ${\eta }_{\mathrm{scat}}$ as a function of the impact parameter $d_{\mathrm{imp}}$ for various propulsion strengths $\gamma$; [(b) and (c)] example trajectories for $\gamma =0.3$ and for different impact parameters $d_{\mathrm{imp}}$. Black superimposed silhouettes indicate the swimmer orientations and degrees of deformation. Arrows mark the direction of motion. In addition, the gray dotted lines illustrate the direction of the fluid flow. No capturing events are observed in this case, but the orbit can come very close to the vortex center. (c) In a finite interval of intermediate impact parameters, the swimmer gets transiently caught and loops around the vortex center. As highlighted by the inset, these looping trajectories in close vicinity to the vortex center always show an identical sense of curvature prescribed by the rotational sense of the swirl flow. (Other parameter values are $\kappa =0.5$, ${\nu }_1=1$, and $\mu =1$.)

Figure 8

Effect of thermal noise on the scattering dynamics. We test the stability of the capturing event for a swimmer in the perpendicular configuration that is characterized by the same parameter values as in Fig. 5 ($\gamma =0.3$). (a) Probability for the swimmer to get captured by the swirl when heading from an initial distance $r_{\mathrm{init}}=1.5\times {10}^4$ towards the vortex center. The probability is plotted as a function of the impact parameter $d_{\mathrm{imp}}$ and the strength $\sigma$ of the thermal noise. For the color code see the scale bar on the right. (b) Probability $P_{\mathrm{ac}}$ of a captured swimmer to still remain on a captured active circular trajectory after $N_r$ circulations, plotted as a function of the stochastic noise strength $\sigma$.