Collective vibrations of a hydrodynamic active lattice

Area of Science:

• Hydrodynamics
• Nonlinear Dynamics
• Soft Matter Physics

Background:

• Quasi-one-dimensional lattices of self-propelled droplets exhibit collective instabilities.
• These instabilities manifest as out-of-phase oscillations and solitary-like waves.
• Hydrodynamic coupling is mediated by Faraday waves generated by external forcing of a fluid bath.

Purpose of the Study:

• To rationalize the form and onset of instability in droplet lattices.
• To model the droplet lattice as a memory-endowed system with spatially non-local coupling.
• To classify the observed instabilities and discover new ones.

Main Methods:

• Modeling the droplet lattice as a memory-endowed system.
• Applying spatially non-local coupling.
• Systematic weakly nonlinear analysis to classify Hopf bifurcations.
• Numerical simulations.

Main Results:

• Identified memory-driven instability as a function of droplet number.
• Determined precluded equispaced lattice configurations due to geometrical constraints.
• Classified instabilities as supercritical or subcritical Hopf bifurcations.
• Discovered a novel symmetry-breaking instability (oscillatory-rotary motion).

Conclusions:

• The memory-endowed, non-local coupling model successfully rationalizes experimental observations of droplet lattice instabilities.
• The study provides a theoretical framework for understanding this new class of dynamical oscillators.
• Further investigations into the nonlinear dynamics of these systems are warranted.