Collective vibrations of a hydrodynamic active lattice

Abstract

Recent experiments show that quasi-one-dimensional lattices of self-propelled droplets exhibit collective instabilities in the form of out-of-phase oscillations and solitary-like waves. This hydrodynamic lattice is driven by the external forcing of a vertically vibrating fluid bath, which invokes a field of subcritical Faraday waves on the bath surface, mediating the spatio-temporal droplet coupling. By modelling the droplet lattice as a memory-endowed system with spatially non-local coupling, we herein rationalize the form and onset of instability in this new class of dynamical oscillator. We identify the memory-driven instability of the lattice as a function of the number of droplets, and determine equispaced lattice configurations precluded by geometrical constraints. Each memory-driven instability is then classified as either a super- or subcritical Hopf bifurcation via a systematic weakly nonlinear analysis, rationalizing experimental observations. We further discover a previously unreported symmetry-breaking instability, manifest as an oscillatory–rotary motion of the lattice. Numerical simulations support our findings and prompt further investigations of this nonlinear dynamical system.

Free rings of bouncing droplets: stability and dynamics

“…Strings of walking droplets confined by a submerged annular channel form a coherent wavefield that allows them to walk faster than an individual droplet (Filoux, Hubert & Vandewalle 2015). A droplet chain that spans the entire circumference of an annular channel is found to destabilize through either out-of-phase oscillations or propagating solitary-like waves as the vibrational acceleration is increased (Thomson, Couchman & Bush 2020 a ; Thomson, Durey & Rosales 2020 b ), phenomena similar to those predicted by theoretical studies of one-dimensional driven dissipative lattices (Ebeling et al. 2000).…”

“…Our study thus highlights the importance of accounting for modulations in the impact phase when modelling the interactions of multiple droplets. We note that Thomson et al (2020b) have recently developed a weakly nonlinear analysis to study the collective vibrations in a one-dimensional lattice of bouncing droplets. A similar analysis could be developed to capture the dynamics of rings beyond the initial instability threshold, such as the amplitude of the azimuthal oscillations in figure 3(c) or the azimuthal travelling wave in figure 4.…”

Collective vibrations of confined levitating droplets

“…We now summarize a mathematical model and accompanying stability analysis, presented in [40], aimed at rationalizing qualitatively the different bifurcations that can arise as the number of droplets N varies. The model presented in [40] considers N equispaced droplets of equal mass which bounce in periodic synchrony, confined to a circle of constant radius R. The arc-length position of each drop x n evolves according to the stroboscopic model derived by Oza et al [52], which describes the time-averaged motion of the droplet in terms of a balance between inertia, drag, and the propulsive wave force enacted on each droplet as it lands on the sloping crest of its local wave field. As shown in [40], the radially symmetric, N-droplet analog of the stroboscopic model, valid below and near to the point of instability, in dimensionless form reads as…”

“…The model presented in [40] considers N equispaced droplets of equal mass which bounce in periodic synchrony, confined to a circle of constant radius R. The arc-length position of each drop x n evolves according to the stroboscopic model derived by Oza et al [52], which describes the time-averaged motion of the droplet in terms of a balance between inertia, drag, and the propulsive wave force enacted on each droplet as it lands on the sloping crest of its local wave field. As shown in [40], the radially symmetric, N-droplet analog of the stroboscopic model, valid below and near to the point of instability, in dimensionless form reads as…”

“…We demonstrate that this driven-dissipative oscillator exhibits two distinctive emergent features, specifically out-of-phase oscillations and solitary wave propagation. Using the theoretical model and stability analysis presented in [40], we rationalize A shallow layer of depth h = 0.56 mm acts as a damper so that wave motion is largely confined to the channel where the fluid depth H = 5.12 mm. The entire assembly is vibrated vertically with maximum acceleration γ and frequency f .…”