Dynamics of sessile drops: symmetry classes and a minimal model

We consider two approaches to the question of how liquid drops vibrate on a solid support. First, a spectral approach: Motions of a drop on a vertically oscillated support exhibit vibrational modes.  These are solutions of a “droplet Schrödinger equation” which is reduced to an eigenvalue problem for mode shapes (characterized by azimuthal and polar wavenumbers) and scaled frequencies. The solutions exhibit spectral splitting and mixing when symmetry is broken. We propose a method of identifying observed motions of drops via a neural network.  Triangle drop approach: It is also advantageous for application purposes to simulate large populations of droplets that are translating and deforming dynamically; this motivates a minimal model of a moving droplet. We consider a 2D triangular drop whose motions are governed by pressure and surface tension under Newton’s law. The resulting four-dimensional dynamical system includes fixed points, periodic, quasi-periodic and chaotic trajectories. We suggest a connection to real-world drops via the Steiner circumellipse.