Unveiling crown-finger instability of a non-spherical drop impacting a liquid surface

This study employs three-dimensional numerical simulations and linear stability analysis to reveal how non-spherical droplet morphology critically influences crown evolution and splash regimes, demonstrating that oblate drops promote finger fragmentation via enhanced rim deceleration while prolate drops favor canopy formation, with the resulting finger count being governed primarily by Rayleigh-Plateau instability and amplified by Rayleigh-Taylor instability.

The Gist

Imagine dropping a single drop of water onto a calm puddle. Usually, we think of that drop as a perfect sphere, like a tiny marble. But in this study, the researchers asked: What happens if the drop isn't a perfect ball? What if it's squashed flat like a pancake, or stretched out like a rugby ball?

Here is a simple breakdown of what they found, using everyday analogies.

The Setup: The Shape-Shifting Drop

The researchers used powerful computer simulations to watch drops hit a pool of water. They didn't just use round drops; they used drops with different shapes:

• Oblate: Flattened like a hamburger patty or a pancake.
• Prolate: Stretched out like a football or a rugby ball.
• Spherical: The standard round ball.

They also changed how hard the drops hit the water (the speed), which they call the "Weber number." Think of this as the difference between gently placing a drop on the water versus throwing it like a dart.

The Four Main Outcomes

Depending on the shape of the drop and how fast it hit, four different things happened:

1. Spreading (The Quiet Splash): The drop hits, flattens out, and spreads smoothly across the water like a drop of ink on paper. No big explosion, just a gentle ripple.
2. Splashing Type-1 (The "Hole" Explosion): The drop hits, creates a crown of water, but then tiny holes appear in the thin sheet of water just below the rim. These holes burst open, shooting out tiny secondary droplets. It's like a bubble popping, but in reverse—the water sheet tears apart.
3. Splashing Type-2 (The "Finger" Explosion): This is the dramatic version. The rim of the water crown slows down so fast that it becomes unstable. It sprouts long, wavy "fingers" of water that eventually break off into many droplets.
4. Canopy Formation (The Umbrella): Instead of spreading out sideways, the water shoots straight up and then curls back over itself, forming a hollow, upside-down bowl or "canopy" that looks like a falling umbrella.

The Big Discovery: Shape Matters

The most important finding is that the shape of the drop dictates the drama:

• The Flattened Drops (Oblate): These are the troublemakers. Because they are wide, they hit the water with a broad surface. This causes the rim of the splash to slow down very quickly. Think of it like a car slamming on its brakes; the sudden stop makes the water unstable. This leads to Splashing Type-2, where lots of fingers and droplets fly everywhere.
• The Stretched Drops (Prolate): These are the smooth operators. Because they are narrow, they hit the water with a smaller area. They don't slow down as abruptly. Instead of spreading out and breaking apart, they shoot straight up and often form that Canopy (the umbrella shape). They are less likely to shatter into a million pieces.

The "Hole" Mystery

The researchers noticed something weird: before the water crown breaks into fingers, tiny holes often appear in the thin film of water just under the rim.

• Analogy: Imagine a thin sheet of plastic wrap being pulled tight. If you poke a hole in it, the tear spreads.
• The Finding: These holes aren't caused by air bubbles getting trapped (a common theory). Instead, they happen because the water sheet gets so thin and unstable that it rips on its own. These holes are the starting gun for the splash.

The Math Behind the Magic

The team also used a mathematical tool called "Linear Stability Analysis" to predict how many fingers would form.

• The Theory: They treated the rim of the splash like a long, wiggly snake. They asked: "How many waves will fit on this snake?"
• The Result: They found that two invisible forces are at play:
  1. Rayleigh-Plateau: This force decides how many fingers will form (the pattern). It's like deciding how many ripples fit in a pond.
  2. Rayleigh-Taylor: This force decides how fast those fingers grow. It's the engine that makes the ripples get bigger and break off.
• The Twist: The math showed that while the "pattern" is set early on, the number of fingers actually decreases over time. Why? Because the rim gets thicker as it gathers more water, causing some fingers to merge back together.

The Bottom Line

This paper tells us that droplet shape is a secret controller of splashes.

• If you want a big, messy explosion with lots of tiny droplets, use a flattened (oblate) drop.
• If you want a tall, clean splash that might form a canopy, use a stretched (prolate) drop.

The researchers created a "map" (a chart) that predicts exactly which of these four outcomes will happen based on the drop's shape and speed. This helps us understand the complex dance of water when it hits water, proving that even a simple splash is full of hidden physics.

Technical Summary

Technical Summary: Unveiling Crown-Finger Instability of a Non-Spherical Drop Impacting a Liquid Surface

Problem Statement
While the impact of spherical droplets on liquid surfaces is well-documented, the splashing dynamics of non-spherical droplets remain largely unexplored. Previous studies have primarily focused on coalescence and air entrapment for non-spherical drops, neglecting the complex interplay of shape anisotropy with splashing regimes, crown evolution, and finger formation. This study addresses the gap in understanding how droplet morphology—specifically the aspect ratio ($Ar$)—influences the transition between spreading, splashing, and canopy formation, as well as the underlying instability mechanisms governing rim breakup.

Methodology
The authors conducted a comprehensive three-dimensional numerical study using the Volume of Fluid (VOF) method within the OpenFOAM framework to simulate the impact of non-spherical droplets on a quiescent liquid film.

• Governing Equations: The incompressible Navier-Stokes and continuity equations were solved, incorporating surface tension via the Continuum Surface Force (CSF) model and gravity.
• Geometry and Parameters: Droplets were modeled as ellipsoids with varying aspect ratios ($Ar = a/b$, where $a$ and $b$ are horizontal and vertical axes). $Ar > 1$ represents oblate drops, $Ar = 1$ spherical, and $Ar < 1$ prolate. The study covered a wide range of Weber numbers ($We$) and fixed the liquid film thickness at $h = 2R_{eq}$.
• Numerical Validation: The solver was validated against experimental data from Dandekar et al. (2025), Wang & Chen (2000), and Cossali et al. (1997) for spherical droplets, confirming accurate capture of crown propagation, secondary droplet ejection, and rim instabilities. Grid independence and adaptive mesh refinement (AMR) tests ensured the resolution of thin lamellae and finger structures.
• Stability Analysis: A linear stability analysis (LSA) was developed to predict the number of fingers. The authors extracted time-dependent physical parameters (rim radius $r_0$, rim acceleration $\dot{V}_0$, and lamella thickness) directly from the 3D simulations to solve a dispersion relation. This approach accounted for both Rayleigh–Plateau (RP) and Rayleigh–Taylor (RT) instabilities.

Key Results

1. Regime Map and Morphologies: A regime map in the $Ar$–$We$ space delineates four distinct impact outcomes:

   • Spreading: Occurs at low $We$ or specific $Ar$ combinations without rim instability.
   • Splashing Type-1: Characterized by hole rupture in the thin lamella beneath the crown rim, leading to finger formation and droplet detachment, but with weak rim deceleration.
   • Splashing Type-2: Occurs at higher $We$ and $Ar$, featuring strong rim deceleration that amplifies RT instabilities, leading to pronounced rim undulations and elongated fingers.
   • Canopy Formation: Predominantly observed in prolate drops ($Ar < 1$) at high $We$, where vertical expansion outpaces radial expansion, creating a hollow, inward-falling structure.

2. Role of Droplet Shape:

   • Oblate Drops ($Ar > 1$): Exhibit broader crown bases and higher rim deceleration. This promotes rapid fluid accumulation at the rim, enhancing RT instability and leading to earlier and more intense finger growth and fragmentation.
   • Prolate Drops ($Ar < 1$): Possess narrower contact areas and slower rim deceleration. This delays instability growth and favors vertical expansion, often resulting in canopy formation rather than traditional splashing.
   • Hole Instability: Holes form in the thinnest region of the lamella just beneath the rim. These ruptures are physical phenomena (confirmed via mesh refinement) that initiate finger splitting and provide momentum for droplet ejection.

3. Instability Mechanisms:

   • The linear stability analysis reveals that Rayleigh–Plateau (RP) instability is the primary determinant of the spatial pattern (wavelength and number) of the initial undulations along the rim.
   • Rayleigh–Taylor (RT) instability does not independently cause node formation but significantly amplifies the growth rate of these perturbations, particularly when rim deceleration is high.
   • The number of fingers is not constant; it decreases over time due to fluid accumulation increasing the rim radius ($r_0$), which lengthens the most unstable wavelength. Nonlinear effects, such as hole-induced merging and splitting, further modulate the final number of secondary droplets.

Significance and Claims
The paper claims to provide the first systematic investigation of splashing dynamics for non-spherical drops, bridging the gap between spherical drop theory and complex morphological impacts. Key contributions include:

• Novel Regime Map: Establishing a comprehensive map of impact outcomes based on aspect ratio and Weber number, highlighting the critical role of shape anisotropy in determining whether a drop spreads, splashes, or forms a canopy.
• Mechanistic Insight: Demonstrating that oblate drops are more prone to splash-induced fragmentation due to enhanced rim deceleration, while prolate drops are prone to canopy formation.
• Theoretical Framework: Presenting a novel theoretical framework that integrates numerical simulation data with linear stability analysis to predict finger numbers. This approach successfully extends existing analytical models (developed for spheres) to non-spherical geometries by accounting for the coupled effects of RP and RT instabilities.
• Hole Dynamics: Identifying and characterizing hole instability as a distinct physical mechanism driving finger splitting and secondary droplet detachment, particularly prominent at higher Weber numbers and in oblate drops.

The study concludes that droplet morphology is a critical parameter in splash dynamics, with significant implications for applications involving droplet impact where shape cannot be assumed spherical.