Periodic Orbits of Active Particles Induced by Hydrodynamic Monopoles

Terrestrial experiments on active particles, such as Volvox, involve gravitational forces, torques and accompanying monopolar fluid flows. Taking these into account, we analyze the dynamics of a pair of self-propelling, self-spinning active particles between widely separated parallel planes. Neglecting flow reflected by the planes, the dynamics of orientation and horizontal separation is symplectic, with a Hamiltonian exactly determining limit cycle oscillations. Near the bottom plane, gravitational torque damps and reflected flow excites this oscillator, sustaining a second limit cycle that can be perturbatively related to the first. Our work provides a theory for dancing Volvox and highlights the importance of monopolar flow in active matter.

Figure 1

Fixed point and limit cycles of the five-dimensional dynamics system of Eqs. (2) and (3). First column: coordinate system used to describe a pair of particles between two parallel plane surfaces. Second column: streamlines of the monopolar flow (red curved arrows show flow-induced rotations of the particles) superimposed on a pseudo color map of the flow speed. The plots correspond to the following three cases: top row: near the top surface, middle row: away from the surfaces, and bottom row: near the bottom surface. Third column: stroboscopic images of the two-particle dynamics in the three configurations. The dynamical system admits a fixed point at the top surface, while limit cycles are formed away from the surfaces and near the bottom surface. The last two columns contain the plot of relative orientation $\theta$ and average height $h$ as a function of $x$ for the three cases. The color bar indicates time in the final three columns and $L$ is the separation of the planes. See movie 1 of the Supplemental Material [27].

Figure 2

Exact solution of the Hamiltonian limit cycle. (a) level sets of the Hamiltonian in which the reflection symmetries $x\to -x$ and $\theta \to -\theta$ are clearly visible, (b) variation of the period of large oscillations $T_E$ as a function of their amplitude scaled by the period of small oscillations. Using experimental values [11], we obtain $T_0\sim 8\mathrm{s}$ giving $T_E\sim 12$ s for $x_m=z=3$. This agrees well with the experimentally measured time period of the “minuet” bound state [11], (c) sedimentation speed as a function of the oscillation amplitude, and (d) orbit on a constant “energy” manifold in $x$, $\theta$, and $h$ with the level sets of $H$ shown on a cross section. Exact analytical results are compared with numerical simulations of the full [Eq. (1)] and reduced [Eqs. (2b), (3a), (3c)] equations in (b) and (c) with $T_R,1/h=0$.

Figure 3

Perturbative solution of the two-body system near the bottom plane, (a) a stable limit cycle is formed in the presence of the bottom plane at height $h^{\star }$ and “energy” $E^{\star }$, (b) an overlay of the numeric and analytic contours with $E=2.2$. The color map shows instantaneous “power” input into the numerical limit cycle from perturbative effects which produce small deformations to the shape of the orbit, (c) the period averaged levitation height from the bottom plane $h^{\star }/b$ against the maximum amplitude $x_m$ for the limit cycle as ${\omega }_R$ is varied. Excellent quantitative agreement between the analytic solution in Eqs. (6) and (7) and the reduced numerics justifies, a posteriori, averaging over the Hamiltonian orbit. Qualitative agreement between analytics and the full equations of motion shows the monotonic increase of $h^{\star }$ with $E^{\star }$ survives when the complete dynamics is considered. A fixed value of $z/b=2$ was used for analytic and reduced numerical computations while a more realistic short-ranged repulsive harmonic potential was implemented in the full numerics, resulting in a nonzero $\dot{z}$. (d) the spikes in $\dot{z}/b$ are caused by steric repulsion induced by the short-ranged potential. Thus, over an oscillation, the value of $z$ changes, leading to quantitative disagreement as seen in (c).