🔬 applied physics

Elasto-hydrodynamics of droplet-pool-interactions

This study investigates the impact of droplets on liquid pools of elastic fluids, revealing that fluid elasticity—distinct from viscosity or surface tension—drives unique morphological regimes, stores significant kinetic energy during cavity expansion, and governs the exponential decay of the resulting Worthington jet through competitive elastic and capillary stresses.

Original authors: Md Sultan, Purbarun Dhar

Published 2026-08-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Md Sultan, Purbarun Dhar

Original paper licensed under CC BY 4.0 (http://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 a single drop of water hitting a puddle is like a tiny, high-speed trampoline act. When a normal drop of water smashes into a pool, it creates a splash, digs a hole, and then the hole snaps back together, shooting a thin, fast column of water straight up into the air. This is a classic dance of physics involving inertia (the drop's momentum), capillarity (the surface tension that tries to keep the water together like a rubber sheet), and viscosity (how thick or sticky the fluid is). Scientists have studied this "Worthington jet" for over a century because it happens everywhere, from rain hitting a pond to inkjet printers spraying ink.

But what happens if the water isn't just water? What if it's a "Boger fluid"—a liquid that looks and flows like water but is secretly filled with long, stretchy polymer chains, like microscopic rubber bands? In this paper, the authors explore a hidden corner of fluid dynamics: elasto-hydrodynamics. This is the study of how fluids that can store energy by stretching (elasticity) behave when they crash into each other. While we know how sticky fluids (like honey) behave, we didn't really know how these "stretchy" fluids would react when a drop hits a pool. The big question is: does the stretchiness change the splash, the hole, and the upward jet, or is it just a minor detail?

The Stretchy Drop Crash: A New Kind of Splash

In this study, researchers Md Sultan and Purbarun Dhar from the Indian Institute of Technology Kharagpur decided to play a game of "drop and pool" with a twist. They used special fluids made by mixing water with tiny amounts of long-chain polymers (specifically Polyethylene Oxide, or PEO). These fluids are "elastic," meaning when you pull them, they want to snap back, storing energy like a stretched rubber band. They tested what happens when these stretchy drops hit stretchy pools, comparing them to normal water drops hitting normal water pools.

The Big Surprise: It's Not About Stickiness
First, the team had to figure out if the weird things they saw were caused by the fluid being "thicker" (viscous) or "stretchier" (elastic). They ran a clever test: they compared a stretchy fluid to a thick, sticky fluid (water mixed with glycerol) that had the exact same thickness.

The Result: The thick, sticky fluid acted just like normal water, making a round, hemispherical hole. But the stretchy fluid? It made a completely different shape: a flat-bottomed, trapezoidal hole.
The Takeaway: This proves that the weird shapes aren't because the fluid is thick; they are purely because the fluid is elastic. The "rubber bands" inside the fluid are doing the heavy lifting.

The Energy Thief
When a drop hits a pool, it brings a certain amount of kinetic energy (movement energy). In normal water, almost all of that energy goes into digging the hole and shooting the jet up. But in the stretchy fluids, the authors found that the polymer chains act like energy thieves.

As the hole expands, the polymer chains get stretched out.
The researchers calculated that about 30% to 40% of the drop's initial crash energy gets stolen and stored as "elastic energy" inside those stretched chains.
Because so much energy is locked away in the stretching polymers, there is less energy left to dig the hole. This means the holes in elastic fluids are smaller and don't grow as fast as in normal water.

The Jet That Refuses to Break
The most dramatic change happens with the "Worthington jet"—the column of water that shoots up after the hole collapses.

In Normal Water: The jet shoots up, gets thin, and then snaps apart into tiny droplets due to surface tension (like a garden hose breaking into a spray).
In Elastic Fluids: The jet still shoots up, but the stretched polymer chains fight back against the surface tension. Instead of snapping, the jet often stays connected to a satellite droplet by a long, thin, unbreakable thread.
The "Beads-on-a-String" Effect: In some cases, the thread doesn't just stay thin; it forms a string of beads (like a necklace) before finally breaking. This happens because the elastic stress balances the surface tension, delaying the snap.
The Rule of Thumb: The authors found that if the fluid is stretchy enough, the jet might never break at all, even at high speeds. It just keeps stretching, held together by the elastic "rubber bands."

The Math of the Snap
The team didn't just watch; they built a mathematical model to predict exactly how these jets thin out.

For normal water, the jet gets thinner following a specific power law (a standard curve).
For the elastic fluids, they discovered that once the jet enters a certain "elasto-capillary" regime, the radius of the jet shrinks exponentially.
Their theory suggests the jet radius decays as exp(-t / 2λ), where t is time and λ is the relaxation time (how long it takes the polymer to relax). This is a different rate than what is seen in other types of stretching experiments, suggesting the unique way the jet forms from a collapsing cavity changes the physics.

The Computer Simulation
To see what was happening inside the invisible fluid, the authors ran computer simulations. They mapped out the "stress" (the tension) inside the jet.

They found that the polymer chains in the very center of the jet are stretched the most, creating a core of high elastic stress.
This central tension is what holds the jet together, preventing it from snapping like a normal water thread.
The simulations also showed that in elastic fluids, the air rushing into the cavity moves much slower than in water because the elastic fluid resists the expansion, effectively "braking" the crash.

The Regime Map
Finally, the authors created a "map" of what happens based on two main numbers: the Weber number (how hard the drop hits) and the Deborah number (how stretchy the fluid is).

Low Stretchiness/High Speed: You get a normal splash with droplets breaking off.
Medium Stretchiness: You get a "satellite droplet + filament" regime, where a droplet hangs on by a thin thread (sometimes with beads).
High Stretchiness: You get a "no pinch-off" regime, where the jet stretches out forever without breaking, defying the usual rules of splashing.

Why This Matters
This research opens up a new chapter in understanding how fluids behave. It shows that elasticity is a powerful force that can completely rewrite the rules of a splash. Whether you are designing better inkjet printers, coating surfaces with paint, or even understanding how biological fluids (like mucus) behave, knowing that a fluid can "store" energy by stretching and use it to stop a jet from breaking is a game-changer. The authors have shown that in the world of elastic fluids, the splash isn't just a splash; it's a complex dance of stored energy, stretching chains, and delayed snaps.

Technical Summary: Elasto-hydrodynamics of Droplet-Pool-Interactions

Problem Statement

While the hydrodynamics of a droplet impacting a liquid pool are well-characterized for Newtonian fluids—resulting in cavity formation, crown expansion, capillary waves, and Worthington jets—the corresponding behavior for elastic (or Boger) fluids remains largely unexplored. Existing literature focuses heavily on Newtonian systems or viscoelastic filament thinning in controlled geometries (e.g., capillary breakup rheometers). There is a significant gap in understanding the unsteady, spatially evolving dynamics of cavity formation, collapse, and subsequent jet ejection when both the impacting droplet and the target pool are elastic polymeric fluids. Specifically, the interplay between inertial, capillary, and elastic forces in governing the spatio-temporal evolution of these events, and the distinct morphological regimes that arise, requires systematic investigation.

Methodology

The study employs a triad of experimental, theoretical, and computational approaches:

Experimental Setup:
- Fluids: Aqueous solutions of high molecular weight polyethylene oxide (PEO, ~5–6 million Da) at concentrations of 0, 250, 500, 1000, and 2000 ppm were used to create elastic (Boger) fluids. Control fluids included pure water and water-glycerol mixtures (matched viscosity to elastic fluids).
- Procedure: Droplets were generated quasi-steadily and allowed to fall freely onto a deep pool. Impact velocities ranged from 1.5 to 4.3 m/s.
- Diagnostics: High-speed imaging (4000 fps) captured the temporal evolution of cavity expansion, collapse, crown formation, and jet ejection.
- Dimensionless Parameters: The dynamics were analyzed across a range of Weber ($We$) and Deborah ($De$) numbers, with the Bond number ($Bo$) fixed at 0.27 to ensure spherical pre-impact droplets. The Ohnesorge number ($Oh$) was kept low (<0.0046), rendering viscous effects negligible.
Theoretical Framework:
- Cavity Evolution: An energy conservation model was derived within a potential-flow framework. The initial kinetic energy of the droplet was balanced against gravitational potential energy, surface energy, kinetic energy of the pool, and a term for stored elastic energy.
- Jet Thinning: A theoretical model for the Worthington jet radius evolution was developed using the FENE-P (Finitely Extensible Nonlinear Elastic - Peterlin) constitutive model. This model accounts for the finite extensibility of polymer chains and the balance between elastic and capillary stresses in the elasto-capillary regime.
Numerical Simulations:
- Solver: Simulations were performed using COMSOL Multiphysics.
- Governing Equations: The Cahn-Hilliard phase-field model was coupled with the FENE-P constitutive equation to track the interface and model elastic stresses.
- Validation: Simulations were validated against experimental data for both Newtonian and elastic cases, focusing on cavity morphology, jet height, and velocity fields.

Key Contributions and Results

1. Dominance of Elasticity over Viscosity

Comparative experiments between elastic fluids (2000 ppm PEO) and Newtonian fluids with matched viscosity (water-glycerol) demonstrated that the observed morphological changes are driven purely by fluid elasticity, not viscosity.

Cavity Morphology: While Newtonian fluids produce hemispherical cavities, elastic fluids exhibit a distinct trapezoidal cavity shape during the retraction phase. This is attributed to non-uniform polymer stretching and the resulting elastic stress distribution, which delays capillary wave propagation and flattens the cavity floor.
Viscosity Independence: The study confirms that for the tested range, viscosity plays a negligible role in the breakup process, while elastic stress distribution dictates the cavity shape and jet dynamics.

2. Energy Conversion and Storage

The energy conservation model quantified the conversion of kinetic energy into elastic energy.

Elastic Storage: The study indicates that ~35% of the droplet's initial kinetic impact energy may be stored as elastic energy via the stretching of polymer chains during cavity expansion in specific high-concentration cases. Specifically, for the 500D-500P case, the reduction in the energy partitioning factor ( $\beta$ ) from 0.80 (Newtonian) to 0.45 (elastic) corresponds to approximately 35% of the initial kinetic energy being stored as elastic energy.
Impact on Dynamics: This energy storage reduces the energy available for cavity expansion, leading to smaller maximum cavity sizes and slower growth rates in elastic fluids compared to Newtonian counterparts. The model successfully predicted the temporal evolution of the mean cavity radius by adjusting the energy partitioning factor ( $\beta$ ).

3. Regime Maps and Morphological Transitions

The study established four distinct regimes in the $[De, We]$ parameter space:

Pinch-off: Complete detachment of secondary droplets (observed primarily in Newtonian systems).
Satellite Droplet + Filament: Formation of a satellite droplet connected by a filament. This includes:
- Beads-on-a-String (BOAS): Observed when Rayleigh instability growth rates are comparable to or slower than the inverse relaxation time, allowing asymmetry to develop before polymer stretching dominates.
- Thin/Thick Filaments: Observed when elastic stresses balance capillary pressure, preventing complete breakup.
No Jet: Suppression of jet formation at low $We$ or high $De$ due to delayed capillary wave convergence.
No Pinch-off: Formation of a stable Worthington jet that does not break up, occurring at high $De$ where elastic stresses continuously counteract capillary necking.

4. Elasto-Capillary Jet Thinning

Theoretical and experimental analysis of the Worthington jet thinning revealed:

Exponential Decay: In the elasto-capillary regime, the jet radius decays exponentially with time, following the scaling law $r(t) \sim \exp(-t/2\lambda)$ .
Scaling Difference: This differs from the classical $r(t) \sim \exp(-t/3\lambda)$ decay law observed in standard viscoelastic filament thinning, attributed to the specific boundary conditions and axial tensile forces in the Worthington jet.
Stability: Higher elasticity suppresses the Rayleigh-Plateau instability, leading to thicker, more stable filaments and delaying or preventing pinch-off.

5. Numerical Insights into Stress and Velocity Fields

Simulations provided visualization of internal flow fields not accessible experimentally:

Stress Distribution: Elastic stress ( $\tau_{zz}$ ) is maximum at the jet's axis of symmetry and decreases toward the interface, corresponding to the higher axial velocity and greater polymer stretching in the core.
Velocity Suppression: In elastic fluids, the cavity expansion velocity is approximately half that of pure water due to the opposing elastic stresses. Consequently, air entrainment is significantly reduced, and no vortex is observed inside the cavity during retraction (unlike in Newtonian cases).

Significance and Claims

The authors claim that this work significantly advances the understanding of the uncharted paradigm of inertia-capillarity-elasticity interplay in droplet-pool interactions.

Fundamental Physics: The study isolates elastic effects from viscous effects, demonstrating that fluid elasticity alone can fundamentally alter cavity morphology (trapezoidal vs. hemispherical) and suppress jet breakup.
Quantitative Insight: By deriving a theoretical model for cavity radius evolution and jet thinning, the paper provides a method to estimate the extensional relaxation time of elastic fluids directly from droplet-pool interaction studies.
Regime Definition: The identification of four distinct morphological regimes offers a predictive framework for droplet impact outcomes in elastic fluids, which is crucial for applications involving non-Newtonian fluids such as paints, inks, and biological fluids.

The paper concludes that the conversion of kinetic energy into elastic energy is the primary mechanism governing the suppression of crown breakup, reduction of cavity size, and the stabilization of Worthington jets in elastic fluids.