1 Department of Applied Mathematics and Computer Science, Technical University of Denmark2 Dynamical Systems, Department of Applied Mathematics and Computer Science, Technical University of Denmark
The first part of this thesis deals with a droplet on a hydrophobic surface. We first present a basic introduction to fluid dynamics, including a description of relevant dimensionless numbers and a derivation of the Young-Laplace equation. An analytic approach to describing the oscillations of a droplet is then given, after which we set up a 2D computational Finite-Element Method (FEM) model for a neutrally buoyant drop immersed in another liquid. The model is validated by considering the volume loss over time. Subsequent to an introduction to the physics of wetting, the developed FEM model is then extended to include drop-surface interactions, and we describe a) the initial descent of a droplet down an inclined hydrophobic substrate, and b) the motion of the droplet in a potential well created through spatial contact angle variations. We solve the full Navier-Stokes equations inside the drop domain, and use the Arbitrary Lagrangian-Eulerian method to keep track of the droplet surface; the contact angle is included by using the Frennet-Serret equations. In situation a), we investigate the behavior of the drop velocity as a function of the slip length and compare with experimental results found in the literature. Furthermore, we quantify the energy associated with center of mass translation and internal fluid motion. The model predicts trajectories for tracer particles deposited inside the drop, and satisfactorily describes the sliding motion of steadily accelerating droplets. The model can be used for determining a characteristic slip parameter, associated with slip lengths and drag reduction for hydrophobic surfaces. In situation b), we observe that the droplet oscillations (frequency, amplitude and decay time) in the potential is not linear with respect to the forcing, i.e. the strength of the potential, and contribute this to preferred eigenmodes of the droplet oscillation. The second part of this thesis deals with a droplet bouncing on a vertically vibrated fluid bath of the same liquid, a system which is the first known macroscopic example of pilot-wave dynamics. An introduction to the experimental set-up is given, followed by a description of the mathematical models governing the vertical and horizontal motion of the drop. Two in-depth studies are then presented. In the first, the results of a comprehensive series of experiments are presented. The most detailed characterisation to date of the system’s dependence on fluid properties, droplet size, and vibrational forcing is provided. A number of new bouncing and walking states are reported, including complex periodic and aperiodic motions. Specific attention is given to the first characterisation of the different gaits arising within the walking regime. In addition to complex periodic walkers and limping droplets, we highlight a previously unreported mixed state, in which the droplet switches periodically between two distinct walking modes. The experimental results are compared to previously developed theoretical predictions. In the second study, wo consider the case where the fluid bath is also rotated around its center-line. The drop then experiences an effective Coriolis force, and previous studies have made a comparison between emerging unstable radii in this system and Landau levels for a charged particle in a magnetic field. The system is treated numerically, and the results are compared to experiments. We provide, again, the most detailed regime diagram of the possible orbits depending on the forcing and the rotation rate of the fluid bath. We highlight each class of orbit, and analyze in depth the wobbling state, precessing orbits, wobble-leap dynamics, exotic trajectories and the emergence of statistical behavior when the forcing is near the Faraday threshold.