[Paper Review] The role of deformations in the bouncing droplet dynamics

Denis Terwagne, F. Ludewig+2 more|arXiv (Cornell University)|Jan 30, 2013

Fluid Dynamics and Heat Transfer3 citations

TL;DR

This paper proposes a bouncing mass-spring-damper model to simulate deformable droplet dynamics on a vibrated liquid bath, capturing complex periodic and resonant bouncing behaviors observed experimentally. The model successfully reproduces bifurcation diagrams of bouncing trajectories, demonstrating its ability to predict multi-periodic droplet dynamics through elastic energy storage and dissipation mechanisms.

ABSTRACT

Droplets bouncing on a vibrated liquid bath open ways to methods of manipulating droplets, creating double emulsion and performing pilot wave model experiments. Many periodic trajectories and resonant modes are observed when tuning the forcing parameters. This complex dynamics emphasizes the interplay between elastic energy storage and energy dissipation at each bounce. We propose to model droplets using a bouncing mass-spring-damper system that mimics a deformable droplet bouncing on a non-deformable liquid bath. From the experimental measurements, we constructed bifurcation diagrams of the bouncing trajectories and challenged our bouncing spring model. The agreement between experiment and the spring model reveals that this model can be used to rationalize and predict a variety of bouncing droplets behaviors involving multi-periodicities.

Motivation & Objective

• To understand the complex periodic and resonant bouncing behaviors of droplets on a vibrated liquid bath.
• To identify the role of elastic energy storage and energy dissipation in droplet dynamics.
• To develop a minimal mechanical model that captures key features of droplet bouncing without full hydrodynamic simulation.
• To validate the model against experimental bifurcation diagrams of bouncing trajectories.

Proposed method

• Model droplets as a mass-spring-damper system to mimic deformable droplet behavior during bouncing.
• Use experimentally measured forcing parameters to calibrate the spring stiffness and damping coefficients.
• Simulate bouncing trajectories under varying forcing amplitudes and frequencies.
• Construct bifurcation diagrams from simulation results to compare with experimental data.
• Match model dynamics to observed multi-periodicities and resonant modes in droplet motion.
• Use energy-based analysis to link spring deformation to elastic energy storage and damping to energy dissipation.

Experimental results

Research questions

• RQ1 How do elastic energy storage and energy dissipation govern the emergence of periodic bouncing trajectories in droplets?
• RQ2 To what extent can a mass-spring-damper model reproduce experimentally observed bifurcations in droplet bouncing?
• RQ3 What role do deformation dynamics play in enabling multi-periodic and resonant bouncing modes?
• RQ4 Can the spring model predict transitions between different dynamical regimes observed in experiments?

Key findings

• The bouncing spring model accurately reproduces experimentally observed bifurcation diagrams of droplet trajectories.
• The model captures multi-periodic bouncing behaviors arising from the interplay between elastic energy storage and viscous dissipation.
• Resonant modes in droplet dynamics are rationalized by the spring model’s response to periodic forcing.
• Agreement between simulation and experiment confirms that deformation dynamics are central to complex droplet behavior.
• The model provides a predictive framework for droplet manipulation in pilot-wave experiments and emulsion formation.
• Energy-based modeling via spring deformation explains the emergence of stable periodic orbits in bouncing droplets.