The Stability of a Hydrodynamic Bravais Lattice
Name
symmetry-14-01524-v3.pdf
Size
1.24 MB
Format
Adobe PDF
Checksum (MD5)
51a656edbef1b81bbd336bbd4a8b0aca
Author(s)
Couchman, Miles M. P.
Evans, Davis J.
Bush, John W. M.
Date Issued
July 26, 2022
Publisher
Multidisciplinary Digital Publishing Institute
Citation
Symmetry 14 (8): 1524 (2022)
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.3390/sym14081524
