More Like this

1. Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs

   https://doi.org/10.1017/jfm.2019.293  

   Couchman, Miles M.; Turton, Sam E.; Bush, John W. (July 2019, Journal of Fluid Mechanics)

   We present the results of an integrated experimental and theoretical investigation of the vertical motion of millimetric droplets bouncing on a vibrating fluid bath. We characterize experimentally the dependence of the phase of impact and contact force between a drop and the bath on the drop’s size and the bath’s vibrational acceleration. This characterization guides the development of a new theoretical model for the coupling between a drop’s vertical and horizontal motion. Our model allows us to relax the assumption of constant impact phase made in models based on the time-averaged trajectory equation of Moláček and Bush ( J. Fluid Mech. , vol. 727, 2013b, pp. 612–647) and obtain a robust horizontal trajectory equation for a bouncing drop that accounts for modulations in the drop’s vertical dynamics as may arise when it interacts with boundaries or other drops. We demonstrate that such modulations have a critical influence on the stability and dynamics of interacting droplet pairs. As the bath’s vibrational acceleration is increased progressively, initially stationary pairs destabilize into a variety of dynamical states including rectilinear oscillations, circular orbits and side-by-side promenading motion. The theoretical predictions of our variable-impact-phase model rationalize our observations and underscore the critical importance of accounting for variability in the vertical motion when modelling droplet–droplet interactions. 

   more » « less
2. Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent statistics

   https://doi.org/10.1103/PhysRevE.108.034213  

   Kutz, J Nathan; Nachbin, André; Baddoo, Peter J; Bush, John_W M (September 2023, Physical Review E)

   We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath’s vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data. Thus, DMD reduces the complex nonlinear interactions between pilot waves and droplet to a low-dimensional linear superposition of DMD modes characterizing the wave field. In particular, it provides a low-dimensional characterization of the bifurcation structure of the pilot-wave physics, wherein the excitation and recruitment of additional modes in the linear superposition models the bifurcation sequence. This DMD characterization yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics. 

   more » « less
3. The Stability of a Hydrodynamic Bravais Lattice

   https://doi.org/10.3390/sym14081524  

   Couchman, Miles M.; Evans, Davis J.; Bush, John W. (August 2022, Symmetry)

   We present the results of a theoretical investigation of the stability and collective vibrations of a two-dimensional hydrodynamic lattice comprised of millimetric droplets bouncing on the surface of a vibrating liquid bath. We derive the linearized equations of motion describing the dynamics of a generic Bravais lattice, as encompasses all possible tilings of parallelograms in an infinite plane-filling array. Focusing on square and triangular lattice geometries, we demonstrate that for relatively low driving accelerations of the bath, only a subset of inter-drop spacings exist for which stable lattices may be achieved. The range of stable spacings is prescribed by the structure of the underlying wavefield. As the driving acceleration is increased progressively, the initially stationary lattices destabilize into coherent oscillatory motion. Our analysis yields both the instability threshold and the wavevector and polarization of the most unstable vibrational mode. The non-Markovian nature of the droplet dynamics renders the stability analysis of the hydrodynamic lattice more rich and subtle than that of its solid state counterpart. 

   more » « less
4. Gradient Droplet Arrays by Acceleration‐Mode Dip‐Coating

   https://doi.org/10.1002/admi.202200667  

   Mandsberg, Nikolaj K.; Shneidman, Anna V.; Jensen, Kaare H.; Taboryski, Rafael; Nielsen, Line H.; Aizenberg, Joanna; Boisen, Anja (August 2022, Advanced Materials Interfaces)

   Abstract Droplet microarray technology is of great interest in biology and chemistry as it allows for significant reactant savings and massive parallelization of experiments. Upon scaling down the footprint of each droplet in an array, it becomes increasingly challenging to produce the array drop‐by‐drop. Therefore, techniques for parallelized droplet production are developed, e.g., dip‐coating of biphilic substrates. However, it is in general difficult to tailor the characteristics of individual droplets, such as size and content, without updating the substrate. Here, the method of dip‐coating of uniformly patterned biphilic substrates in so‐called “acceleration‐mode” to produce droplet arrays featuring gradients in droplet height for fixed droplet footprint is developed. The results herein present this method applied to produce drops with base diameters varying over orders of magnitude, from as high as 6 mm to as small as 50 µm; importantly, the experimentally measured power‐law‐dependency of volume on capillary‐number matches analytical theory for droplet formation on heterogenous substrates though the precise quantitative values likely differ due to 2D substrate patterning. Gradient characteristics, including average droplet volume, steepness of the gradient, and its monotonicity, can all be tuned by changing the dip‐coating parameters, thus providing a robust method for high‐throughput screening applications and experiments. 

   more » « less
5. Speed and Acceleration of Droplets Generated by Breaking Wind‐Forced Waves

   https://doi.org/10.1029/2022GL098426  

   Erinin, M. A.; Néel, B.; Ruth, D. J.; Mazzatenta, M.; Jaquette, R. D.; Veron, F.; Deike, L. (July 2022, Geophysical Research Letters)

   Abstract Laboratory measurements of droplet size, velocity, and accelerations generated by mechanically and wind‐forced water breaking waves are reported. The wind free stream velocity is up to 12 m/s, leading to wave slopes from 0.15 to 0.35 at a fetch of 23 m. The ratio of wind free stream and wave phase speed ranges from 5.9 to 11.1, depending on the mechanical wave frequency. The droplet size distribution in all configurations can be represented by two power laws,N(d) ∝ d⁻¹for drops from 30 to 600 μm andN(d) ∝ d⁻⁴above 600 μm. The horizontal and vertical droplet velocities appear correlated, with drops with slower horizontal speed more likely to move upward. The velocity and acceleration distributions are found to be asymmetric, with the velocity probability density functions (PDFs) being described by a normal‐inverse‐Gaussian distribution. The horizontal acceleration PDF are found to follow a shape close to the one predicted for small particles in homogeneous and isotropic turbulence, while the vertical distribution follows an asymmetric normal shape, showing that both acceleration components are controlled by different physical processes. 

   more » « less