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The silhouette of a vibrated sessile drop encodes its internal viscous dissipation

This paper demonstrates that the volumetric viscous dissipation of a vibrating sessile drop can be accurately estimated solely from its external silhouette deformation, enabling the prediction of internal energy transfer and dissipation across a wide range of viscosities without requiring optical access to the drop's interior.

Original authors: Ahmad Naseri Karimvand, Daren Watson, Andrew Dickerson

Published 2026-09-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ahmad Naseri Karimvand, Daren Watson, Andrew 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

When a drop of liquid sits on a surface and that surface begins to vibrate, the drop does not simply bounce up and down like a solid object. Instead, it sloshes, deforms, and churns internally. This internal churning is where the energy of the vibration goes. As the liquid moves against itself, friction between the moving layers converts the mechanical energy of the shake into heat, a process known as viscous dissipation. Understanding exactly how much energy is lost to this internal friction is crucial for many practical tasks, from designing better mixing systems for biological reagents to predicting how a drop will behave when it hits a surface. However, measuring this energy loss has traditionally been a difficult problem. To see the friction, scientists usually need to see the speed of the liquid at every point inside the drop. This requires expensive, high-tech equipment to track tiny particles floating in the fluid, and even then, the curved surface of the drop often distorts the view, making the measurements unreliable or impossible for opaque liquids.

A team of researchers at the University of Tennessee and Florida Polytechnic University has found a way to bypass this difficulty entirely. They discovered that the changing shape of the drop's outline, visible from the outside, contains a complete record of the energy being lost inside. By filming a vibrating drop from two angles and analyzing how its silhouette changes over time, they can calculate the internal energy dissipation without ever needing to look inside the liquid or measure its internal flow. The researchers focused on drops made of water mixed with glycerol to create fluids with different thicknesses, or viscosities. They placed these drops on a vibrating platform and used high-speed cameras to record the motion at 12,000 frames per second. From these videos, they extracted a simple number that quantifies how much the drop's shape deviates from its resting state, a measure they called "sloshiness."

The core of their discovery is a direct link between the rate at which this sloshiness changes and the rate at which energy is lost to heat. They found that the faster the drop's outline changes shape, the more energy is being dissipated, and this relationship holds true regardless of the fluid's thickness. To prove this, they compared their image-based calculations against independent measurements of how quickly the drop's motion died out after the vibration stopped. The results matched closely, confirming that the visible deformation is a reliable proxy for the invisible internal friction. This approach works because the way the surface of the drop moves is tightly coupled to the flow of the liquid beneath it. If the surface is churning, the inside is churning too, and the energy lost to friction is encoded in that motion.

Beyond just measuring loss, the team developed a model to predict how much energy is transferred from the vibrating surface into the drop while the vibration is still happening. They treated the drop like a mass on a spring that is being shaken, but they added a correction to account for the fact that the liquid near the bottom moves with the surface while the liquid at the top might not move as much. This difference in movement creates a shear layer, a zone of sliding fluid where most of the friction occurs. The researchers identified a specific ratio—the thickness of this friction zone compared to the height of the drop—as the key factor determining how efficiently energy is transferred. They found that this efficiency is not a simple straight line; it is non-monotonic. If the friction zone is too thin, it only affects a tiny layer of the drop, leaving most of the liquid untouched and wasting the opportunity to dissipate energy. If the friction zone is too thick, the entire drop moves as a single solid block, and without relative motion between layers, there is no friction to speak of. Maximum energy transfer happens at an intermediate point where the friction zone is thick enough to engage a large portion of the drop but thin enough to maintain the necessary sliding motion.

The researchers tested these predictions using a separate experiment involving a vibrating metal strip, or cantilever, with drops placed on its tip. By measuring how quickly the strip stopped vibrating when different drops were added, they confirmed that smaller drops dissipate more energy per unit of volume than larger ones, and that there is an optimal viscosity for maximum energy loss. Too little viscosity, and the drop flows too easily; too much, and it becomes too stiff to slosh. The model successfully predicted the energy transfer for a fluid it had never seen before, provided the vibration frequency was below the drop's natural resonant frequency. Above that frequency, the drop could not keep up with the rapid shaking, and the model's predictions broke down. This limitation highlights that the method works best when the drop has enough time to respond to the vibration.

The significance of this work lies in its simplicity and accessibility. By relying only on the visible outline of the drop, the method removes the need for complex internal measurements that are often impossible to perform. This opens the door to studying energy dissipation in liquids that are opaque, such as crude oil or liquid metals, or in extreme environments where adding tracking particles is not feasible. It also avoids the need to know the surface tension of the liquid beforehand, a property that can be difficult to measure and changes over time. The researchers showed that the dynamic silhouette of a sessile drop is not just a visual curiosity but a rich source of data, encoding the complex physics of internal flow and energy loss in a form that can be captured with a standard high-speed camera. This approach transforms a difficult measurement problem into a straightforward visual one, allowing scientists to understand the hidden energetics of a vibrating drop just by watching its shape change.

Technical Summary: The Silhouette of a Vibrated Sessile Drop Encodes Its Internal Viscous Dissipation

Problem Statement
Quantifying viscous dissipation in deforming liquid drops is critical for understanding fundamental drop dynamics, predicting impact outcomes, and optimizing engineering applications such as mixing and decontamination. However, direct measurement of internal viscous dissipation is challenging because it depends on velocity gradients within the liquid bulk. Conventional techniques like Particle Image Velocimetry (PIV) require high spatial resolution, specialized instrumentation, and optical access to the drop interior. These methods face significant hurdles with oscillating drops due to optical refraction at curved interfaces, the trade-off between interrogation window size and particle count, and the impracticality of ensemble averaging for transient events. Furthermore, many practical applications involve opaque fluids (e.g., crude oils, liquid metals) or extreme conditions where internal flow measurement is impossible. Existing analytical models often assume inviscid bulk fluids or rely on simplified potential flow approximations that fail to capture true rotational, no-slip flow dynamics.

Methodology
The authors propose a method to infer volumetric viscous dissipation and energy transmission rates solely from the visible 2D silhouette of a sessile drop undergoing vertical oscillation. The approach involves three main components:

Image-Based Scalar ("Sloshiness"): The study introduces a dimensionless scalar, termed "sloshiness" ( $\mathcal{S}$ ), defined as the exclusive difference ratio between the instantaneous drop silhouette and its equilibrium reference shape. This is calculated by binarizing images from two orthogonal high-speed cameras and computing the pixel-wise difference relative to the equilibrium area.
Data-Driven Modal Decomposition: Using Singular Value Decomposition (SVD), the authors decompose the deformation field into spatial and temporal modes. They demonstrate that for the tested conditions, a single dominant mode captures 75–80% of the deformation energy. This allows the reconstruction of a reduced-order interior velocity field and strain-rate tensor based on the time derivative of the dominant temporal coefficient.
Theoretical Modeling:
- Dissipation Scaling: By substituting the reconstructed velocity field into the viscous dissipation equation, the authors derive that the volumetric dissipation rate ( $\Phi$ ) is proportional to the dynamic viscosity ( $\mu$ ) and the square of the sloshiness rate ( $\dot{\mathcal{S}}^2$ ), expressed as $\Phi \propto \mu \dot{\mathcal{S}}^2$ .
- Energy Transmission Model: For steady-state oscillations, the authors develop a semi-empirical model equating the time-averaged power transmitted from the substrate to the drop with the viscous dissipation rate. This model combines a base-excited spring-mass-damper analogy (using experimentally measured natural frequency and damping ratio) with a Stokes boundary-layer correction. The correction accounts for the non-uniformity of the internal velocity field, governed by the ratio of the Stokes penetration depth ( $\delta_v$ ) to the drop height ( $H_0$ ).

Experimental Validation
Experiments were conducted using water-glycerol mixtures with viscosities spanning a 33-fold range (0.88 to 393.5 mPa·s) and drop volumes of 3, 5, and 7 $\mu$ L.

Ring-Down Tests: To validate the dissipation scaling, the vibrating substrate was impulsively stopped, and the decay of the drop's center-of-mass oscillation was tracked. The independently measured dissipation rate (derived from kinetic energy decay) was plotted against the squared sloshiness rate.
Steady-State Tests: Drops were subjected to continuous vertical oscillation. The model was calibrated using data from two fluid viscosities and tested against a withheld fluid ( $\mu = 2.3$ mPa·s).
Cantilever Experiments: A plucked cantilever with drops deposited on its tip was used to independently verify predictions regarding the non-monotonic dependence of energy transfer on drop size and viscosity.

Key Results

Dissipation Scaling: A strong linear correlation ( $R^2 = 0.89$ ) was found between the normalized viscous dissipation rate and the normalized squared sloshiness rate across the tested viscosity and volume ranges. This confirms that the silhouette deformation rate encodes the internal viscous dissipation.
Transmission Model Performance: The semi-empirical model accurately predicts transmitted power in the pre-resonance regime ( $f \leq f_n$ ), achieving an $R^2$ of 0.77 for the withheld test fluid. However, the model fails to predict the reduction in energy transmission observed in the post-resonance regime, where the drop cannot fully respond to the high-frequency forcing.
Non-Monotonic Energy Transfer: Both the theoretical model and cantilever experiments confirm that volumetric energy transfer is non-monotonic with respect to viscosity and drop size.
- Viscosity: Energy transfer increases with viscosity up to a threshold where the fluid becomes too viscous to deform significantly, after which transfer decreases.
- Drop Size: Smaller drops exhibit higher dissipation per unit volume than larger drops of the same fluid, provided the Stokes penetration depth is not negligible compared to the drop height.
Governing Parameter: The ratio of Stokes penetration depth to drop height ( $\delta_v / H_0$ ) is identified as the governing parameter for energy transfer efficiency. Maximum efficiency occurs when the shear layer is thick enough to engage a significant portion of the drop volume but thin enough to maintain velocity gradients.

Significance and Claims
The paper claims to establish a robust physical link between the visible deformation of an oscillating sessile drop and its internal viscous dissipation. The primary significance lies in the ability to estimate bulk dissipation and energy transmission using only high-speed video of the drop silhouette, eliminating the need for internal flow field measurements or optical access to the drop interior. This approach is particularly valuable for opaque fluids, extreme environments, or systems where adding tracers is prohibited.

The authors modestly note that the method relies on the dominance of a single deformation mode and assumes axisymmetry. The model's validity is restricted to the pre-resonance regime, and the current single-mode approximation leaves approximately 11% of the variance in dissipation unexplained, likely due to contact-line effects, higher-mode mixing, and image-processing noise. Nevertheless, the framework provides a practical, image-based tool for characterizing the energetics of sessile drops in systems where traditional rheological or velocimetric methods are inaccessible.

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