Droplets sliding on single and multiple vertical fibers

TL;DR

This work analyzes gravity-driven droplets sliding on vertical fibers and grooved bundles, focusing on how a trailing liquid film modifies speed and remaining volume. It combines a novel real-time measurement setup for $v$ and $\Omega$ with a simple force-balance model incorporating gravity, film capillarity, and dissipation to explain the observed linear $v$–$\Omega$ relation and the role of substrate substructure. The results show that the film thickness $\delta$ (typically $10$–$50\,\mu$m) grows with viscosity and grooves, and that grooves can boost speed by up to about 20% by enhancing coating; a Landau–Levich regime describes single-fiber cases while groove effects deviate from this scaling. The findings have practical implications for water-harvesting devices and suggest the measurement apparatus could serve as a rheometer for coating films and complex liquids.

Abstract

From microfluidics to fog-harvesting applications, tiny droplets are transported along various solid substrates including hairs, threads, grooves, and other light structures. Driven by gravity, a droplet sliding along a vertical fiber is a complex problem since it is losing volume and speed as it goes down. With the help of an original setup, we solve this problem by tracking in real-time droplet characteristics and dynamics. Single fibers as well as multiple fiber systems are studied to consider the presence of grooves. On both fibers and grooved threads, droplet speed and volume are seen to decay rapidly because the liquid entity is leaving a thin film behind. This film exerts a capillary force able to stop the droplet motion before it is completely drained. A model is proposed to capture the droplet dynamics. We evidence also that multiple vertical fibers are enhancing the droplet speed while simultaneously promoting increased liquid loss on grooves.

Figures (6)

• Figure 1: (a) Experimental pictures of droplets of $\rm{3 \mu l}$ sliding down 1, 2 and 3 fibers of $d= \rm{140 \mu m}$ from left to right. A liquid film is seen behind the droplet. Droplet shape characteristics : height $L$ and width $\ell$ are emphasized in red. (b) Sketch of three horizontal cuts of the system : dry fibers in front of the droplet (orange), the droplet cross section (light blue) and the liquid film of thickness $\delta$ after the passage of the droplet (dark blue).
• Figure 2: (a) Sketch of the experimental setup where a camera records the droplet placed on a moving thread. Back illumination allows contrasted images. (b) Typical measurements of a droplet sliding on a single vertical fiber ($d=140 \, {\rm \mu m}$) : both volume $\Omega$ and speed $v$ are decreasing rapidly over time
• Figure 3: (a) Height of the droplet $L$ as a function of the width $\ell$ of the droplet. All data points are colored as a function of viscosity $\eta$ and fiber number $n$. See the color legend given as an inset. (b) Lengths are rescaled by the effective diameter $d_e$. The region colored in red corresponds to the $\Omega_c < 1$ criterion such that a droplet there is no longer in a barrel configuration.
• Figure 4: (a) Volume loss rate $\dot \Omega$ as a function of droplet speed $v$. Lines are linear fits used to extract the slopes for determining $\bar{\delta}$ from Eq.(\ref{['eq_volume']}). (b) Average Liquid film thickness $\bar{\delta}$ as a function of $n$. Lines are fitted on the data to emphasize $\eta$ and $n$ dependencies, but should be considered as guides for the eye. (c) Average film thickness normalized by $\eta^{2/3}$ emphasizing that the effect of viscosity is captured by the Landau-Levich model only for $n=1$.
• Figure 5: (a) Droplet speed as a function of $\Omega$ emphasizing the gravity driven mechanism. Each color is associated to viscosity $\eta$ and fiber number $n$. Error bars are given. Data are fitted by a linear model of Eq.(\ref{['eq_speed']}). (b) Rescaled speed $v$ and volume $\Omega$ according to Eq.(\ref{['eq_speed']}). (c) From the slopes extracted in the top graph, the inverse dissipation factor $1/\xi$ is estimated. It is plotted as a function of $n$ with the same color code. One observes that the presence of grooves favors the droplet motion, i.e. $1/\xi$ increases with $n$.