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Drop sloshing on vibrating compliant substrates

This study investigates the vibratory dynamics of water drops on compliant substrates through experiments and modeling, revealing that liquid drops dominate the system's natural frequency and that drop deformation modes depend on contact line behavior while overall sloshiness scales with the Womersley number.

Original authors: MD Emazuddin Alif, Ahmad Naseri Karimvand, Jeffrey Lynn Kauffman, Andrew Keith Dickerson

Published 2026-08-12

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CC BY 4.0

Original authors: MD Emazuddin Alif, Ahmad Naseri Karimvand, Jeffrey Lynn Kauffman, Andrew Keith Dickerson

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where tiny puddles of water don't just sit there; they dance, jump, and sometimes even fly off a surface all on their own. This is the fascinating realm of fluid dynamics, specifically the study of how liquid droplets behave when they are shaken. To understand this, you need to know a few simple things. First, there's the contact line, which is the invisible ring where the water meets the solid surface. Sometimes this ring is stuck (pinned), and sometimes it can slide around (mobile). Second, there's surface tension, the "skin" on the water that tries to keep it in a perfect sphere, fighting against gravity and shaking. Finally, there's compliance, which just means how squishy or flexible a surface is. Think of a stiff table versus a rubber band. When you shake a stiff table, the water just wobbles. But when you shake a rubber band, the surface bends, and the water wobbles back, creating a complex dance between the liquid and the solid. Scientists care about this because if we can master how to make water jump off surfaces without touching them, we could create self-cleaning wings for drones, keep medical equipment sterile, or dry off solar panels in space without using a single drop of soap or a single wipe.

In this study, researchers at the University of Tennessee and the University of Central Florida decided to play with this dance. They took tiny 5-microliter drops of water (about the size of a small raindrop) and placed them on the tips of flexible beams made of two different materials: a Kapton plastic with a contact angle of 60–70 degrees and a PTFE plastic with a contact angle of 100–110 degrees. They attached these beams to a machine that vibrated them up and down, like a tiny, high-speed trampoline. Their goal was to see how the water drops reacted when the surface they were sitting on started to wiggle. They discovered that the water drop and the flexible beam act as a single team. When the beam vibrates, the drop vibrates at the exact same frequency. In fact, the presence of the water drop actually slows down the natural vibration of the beam, making the whole system "heavier" and slower to move than the beam would be on its own.

The researchers introduced a new way to measure how much the water was "sloshing" around, which they whimsically called "sloshiness." Imagine taking a photo of the drop's shape at the very beginning and then taking a photo of it while it's vibrating. If you subtract the first photo from the second, the remaining shape shows you how much the drop has stretched, squished, or changed. They calculated this change from two different angles to get a complete picture. They found that the amount of sloshiness depends on a specific combination of how fast the beam is shaking and how far it moves. They called this combination the Womersley number. The more you shake (higher frequency or bigger movement), the more the drop sloshes, and this relationship holds true for both types of surfaces.

One of the most interesting findings was how the drop moves depending on whether its edges are stuck or free to slide. If the drop's edges are pinned (stuck) to the surface, the drop tends to shoot up and down like a jet, getting taller and thinner. If the edges are free to move, the drop tends to spread out, getting wider and flatter. Despite these different shapes, the overall amount of "sloshiness" was surprisingly similar for both types of drops. The researchers also built a mathematical model to explain this, treating the drop and the beam like two weights connected by springs and shock absorbers. While this model is a simplified version of reality, it helps predict how the drop will behave based on the shaking.

The study suggests that these liquid drops could be incredibly useful as "dampers" to absorb vibrations in machines, especially in situations where traditional metal weights might stop moving. Because the water inside the drop keeps flowing and sloshing even with tiny vibrations, it can absorb energy that a solid weight might miss. Furthermore, understanding these movements helps scientists figure out how to make drops fly off a surface completely, which is the key to self-cleaning technology. The researchers found that by tuning the vibration just right, they could make the drops "jet" off the surface or "spread" and slide off, offering a new way to control liquids without ever touching them. This work doesn't just explain a cool physics trick; it lays the groundwork for designing surfaces that can clean themselves, keeping everything from airplane wings to hospital tools dry and free of contaminants.

Technical Summary: Drop Sloshing on Vibrating Compliant Substrates

Problem Statement

The study addresses the complex physics governing the interaction between sessile liquid drops and vibrating, compliant (elastic) substrates. While the dynamics of drops on rigid surfaces and the vibration of cantilever beams are well-characterized individually, the coupled dynamics of a sloshing drop on a vibrating elastic surface remain largely unexplored. This problem is critical for applications involving surface decontamination and engineered systems where precise drop control is required, particularly for small drops (below the capillary length) that are resistant to removal by gravity or simple wind shear. The research seeks to establish mathematical and analytical approaches for these deformable, two-phase systems, specifically investigating how base vibration induces drop deformation (sloshing) and potential ejection.

Methodology

The researchers employed a combination of high-speed experimental observation and reduced-order mathematical modeling.

Experimental Setup:

Substrates: Elastic cantilever beams were fabricated from hydrophilic Kapton (0.003-inch thick) and hydrophobic polytetrafluoroethylene (PTFE) (0.005-inch thick).
Actuation: Beams were affixed to an electrodynamic shaker, which imposed vertical sinusoidal vibrations at the fixed base.
Specimen: 5-µL water drops were placed at the free tip of the cantilevers.
Measurement: High-speed videography (12,000 fps) captured interactions from two orthogonal views. A sine wave generator controlled the shaker, testing frequencies (80, 90, 100, 110, 115, and 140 Hz) and increasing amplitudes until drop ejection occurred.
Analysis: Custom code tracked post/beam positions, drop height, and spread. Temporal deflection data was analyzed via Fast Fourier Transform (FFT) to determine natural frequencies and damping coefficients.

Modeling Approach:

The system was modeled as a two-degree-of-freedom (2-DOF) mass-spring-damper system on a moving base.
The cantilever beam and the liquid drop were represented as point masses ( $m_1$ and $m_2$ ) with associated stiffness ( $k$ ) and damping ( $c$ ) coefficients.
The model treated the drop centroid displacement as a physical analog for the extent of drop deformation (sloshing).

Key Contributions and Definitions

1. Characterization of Coupled Natural Frequencies:
The study determined that the natural frequency of the drop-cantilever system is significantly lower than that of the individual drop on a rigid surface or the bare cantilever beam. This reduction is attributed to the "added-mass" effect of the liquid load and the participation of the drop's sloshing degree of freedom in the coupled dynamics.

2. Definition of "Sloshiness" ( $S$ ):
The authors introduced a novel metric, Sloshiness ( $S$ ), defined as the normalized change in the projected area of the drop from its initial sessile shape. Calculated from two orthogonal binary image projections, $S$ quantifies the extent of deformation over time.

Jetting Mode: Vertical elongation of the drop dome (higher than initial height).
Spreading Mode: Lateral widening of the drop (wider than initial width).
Metric: The study utilized the Root Mean Square (RMS) of temporal sloshiness ( $S_{RMS}$ ) to represent the overall deformation intensity of a trial.

3. Scaling with the Womersley Number:
The paper proposes that drop sloshing scales with the Womersley number ( $\alpha = L(2\pi f\rho/\mu)^{1/2}$ ), which captures the ratio of oscillatory inertia to viscous effects. This dimensionless group successfully collapsed data across various frequencies and amplitudes for both beam materials.

Key Results

Dynamic Behavior:

Frequency Dominance: In the coupled system, the liquid drop dominates the natural frequency. For freely vibrating systems, the drop deformation and cantilever vibration oscillate at the same frequency.
Damping: The presence of the drop significantly alters the system's damping characteristics. Intermediate viscosity drops were noted in prior literature to provide optimal damping; this study confirms the system behaves as an underdamped harmonic oscillator where the drop's internal flow contributes to energy dissipation.

Deformation Modes and Contact Lines:

Pinned vs. Mobile:
- Pinned Contact Lines: Promote a "jetting" mode, characterized by a taller apex and symmetric vertical distortion (analogous to odd-k zonal modes).
- Mobile Contact Lines: Promote a "spreading" mode, characterized by lateral flattening (analogous to even-k zonal modes).
Overall Sloshiness: Despite the distinct morphological differences between pinned and mobile contact lines, the overall $S_{RMS}$ values were found to be similar for both conditions. The energy loss due to contact line friction in mobile drops appears to balance the increased deformation amplitude seen in pinned drops.

Scaling Laws:

$S_{RMS}$ increases linearly with the Womersley number ( $\alpha$ ).
Resonance (near 85 Hz for Kapton and 90 Hz for PTFE) resulted in higher $S_{RMS}$ values and less data noise compared to off-resonance frequencies.
Material Differences: While PTFE beams have lower bending stiffness (less favorable for sloshing) and hydrophobic surfaces (easier drop deformation), the resulting $S_{RMS}$ values were comparable to Kapton beams when plotted against $\alpha$ . However, PTFE beams exhibited a lower maximum $S_{RMS}$ before ejection, attributed to their lower structural stiffness rather than surface wettability.

Significance and Claims

The paper claims to provide a foundational framework for understanding and predicting drop behavior on compliant, vibrating surfaces. Its primary significance lies in:

New Metric: The introduction of "sloshiness" ( $S$ ) offers a quantitative, spatial measure of drop deformation that complements traditional frequency-based modal analysis.
Predictive Scaling: The demonstration that sloshing scales with the Womersley number allows for the prediction of drop deformation based on base acceleration and fluid properties, independent of specific frequency/amplitude combinations.
Low-Acceleration Damping: The study suggests that liquid drops can serve as effective dampers in low-acceleration environments where traditional Particle Impact Dampers (PIDs) fail (due to the cessation of particle movement). Drops maintain damping capacity even at minimal excitations because internal fluid flow is induced by the smallest disturbances.
Decontamination Applications: The work supports the development of self-decontamination strategies for hard-to-reach surfaces (e.g., in healthcare or food preparation) by harnessing surface properties and periodic excitation to remove resting droplets without chemical or mechanical intervention.

The authors maintain a modest stance, noting that their 2-DOF model is a reduced-order approximation that does not resolve the full spectral eigenvalue problem of sessile drop oscillations. They posit that future work could utilize this framework to measure unknown liquid properties (viscosity, surface tension) for small volumes and further explore the relationship between specific sloshing behaviors and ejection modes.