Walking droplets & Galloping bubbles
Pedro J. Sáenz
Assistant Professor, Department of Mathematics
University of North Carolina at Chapel Hill, USA
Physical Mathematics Laboratory

www.pml.unc.edu
In the first part of this talk, we present a classical wave-particle analog of Anderson localization
with walking droplets, which self-propel across the surface of a vibrating fluid bath through a resonantinteraction with its own guiding wave field. These walking droplets, or ‘walkers’, push the boundaries ofclassical mechanics to include behaviors previously thought to be exclusive to the quantum realm. By
investigating the erratic motion of walkers over submerged random topographies, we demonstrate theemergence of localized statistics analogous to those of quantum particles. Consideration of an ensembleof walker trajectories reveals a suppression of diffusion when the guiding wave field extends over thedisordered topography. The emergent statistics are rationalized mechanistically by virtue of the wavemediated resonant coupling between the droplet and topography, which generates an attractive wave
potential about the localization region. This hydrodynamic quantum analog demonstrates how a classicalparticle may localize like a wave, and so suggests new directions for future research.
The second part of this talk is dedicated to introducing a new symmetry breaking that prompts
bubbles to ‘gallop’ along horizontal surfaces inside a vertically-vibrated fluid chamber, self-propelled
by a resonant interaction between their shape oscillation modes. These active bubbles exhibit distinct
trajectory regimes, including rectilinear, orbital, and run-and-tumble motions, which can be dynamicallytuned via external forcing. Through periodic body deformations, galloping bubbles swim by leveraging
inertial forces rather than vortex shedding, enabling them to maneuver even when viscous traction is notviable. The galloping symmetry breaking provides a robust self-propulsion mechanism, arising in bubbles
whether separated from the wall by a liquid film or directly attached to it. The bubble’s spectrum anda perturbative stability analysis show that the galloping instability results from the coupling between
symmetric and asymmetric shape modes. Through proof-of-concept demonstrations, we showcase thetechnological potential of galloping locomotion for applications involving bubble generation and removal,transport and sorting, navigating complex fluid networks, and surface cleaning. The rich dynamics ofgalloping bubbles suggest opportunities in heat transfer, microfluidics, soft robotics, and active matter.

Fig. 1: A walking droplet and a galloping bubble.

Biosketch – Pedro is an Assistant Professor and the director of the Physical Mathematics Laboratory(www.pml.unc.edu) in the Department of Mathematics at UNC. From 2015 to 2019, he was an Instructor
in Applied Mathematics at MIT. Pedro received his Ph.D. from the University of Edinburgh in 2014
and was a post-doctoral fellow at Imperial College London in 2015. He was awarded an Alfred P. Sloan
Research Fellowship in 2023 and an NSF CAREER award in 2021. Pedro has also received multiple scientific visualization awards, including the American Physical Society Gallery Fluid of Motion award in 2017and 2022. His research blends experiments, numerical simulations, and theory to address fundamental
problems that find motivation in physics and engineering.