Morphological transitions of sliding drops: Dynamics and bifurcations

TL;DR

This paper investigates the morphological transitions and bifurcations of sliding droplets on inclined surfaces using a thin-film model, revealing stability, oscillatory behaviors, and scaling laws through analysis and simulations.

Contribution

It introduces a comprehensive bifurcation analysis of 3D sliding drops, including stability, time-periodic behaviors, and scaling laws, using continuation techniques and numerical simulations.

Findings

Identification of existence ranges and instabilities of stationary drops

Discovery of time-periodic breakup-coalescence cycles and hysteresis

Observation of scaling laws governing bifurcation features across droplet sizes

Abstract

We study fully three-dimensional droplets that slide down an incline by employing a thin-film equation that accounts for capillarity, wettability, and a lateral driving force in small-gradient (or long-wave) approximation. In particular, we focus on qualitative changes in the morphology and behavior of stationary sliding drops. We employ the inclination angle of the substrate as control parameter and use continuation techniques to analyze for several fixed droplet sizes the bifurcation diagram of stationary droplets, their linear stability, and relevant eigenmodes. The obtained predictions on existence ranges and instabilities are tested via direct numerical simulations that are also used to investigate a branch of time-periodic behavior (corresponding to repeated breakup-coalescence cycles, where the breakup is also denoted as pearling) which emerges at a global instability, the related hysteresis in behavior, and a period-doubling cascade. The nontrivial oscillatory behavior close to a Hopf bifurcation of drops with a finite-length tail is also studied. Finally, it is shown that the main features of the bifurcation diagram follow scaling laws over several decades of the droplet size.