Effect of velocity, fluid properties and drop shape on coalescence and neck oscillation

This study employs axisymmetric numerical simulations to investigate how impact velocity, fluid properties, and drop shape influence the coalescence dynamics of a liquid drop in a deep pool, revealing that prolate drops most readily form secondary droplets, that Rayleigh-Plateau instability is insignificant in this context, and that a transition regime with multiple neck oscillations exists between partial and complete coalescence.

The Gist

The Big Picture: The "Droplet Dance"

Imagine you are holding a single drop of water above a bathtub. You let it go. What happens when it hits the water?

Sometimes, it just splashes and disappears instantly. Other times, it does something magical: it hits the water, creates a tiny tower of liquid that shoots up, and then snaps off a brand-new, smaller "baby" droplet that flies into the air. This is called partial coalescence.

This paper is like a high-speed camera study of that exact moment. The researchers wanted to figure out:

1. Why does the drop sometimes make a baby droplet and sometimes just merge completely?
2. What happens if the drop isn't perfectly round (like if it's squashed or stretched)?
3. How do speed, stickiness (viscosity), and gravity change the outcome?

They used powerful computer simulations (like a super-advanced video game physics engine) to watch these drops in slow motion and build a "rulebook" for how they behave.

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The Main Characters (The Forces at Play)

To understand the drop's behavior, think of four invisible characters wrestling for control:

1. Inertia (The Speedster): This is the drop's momentum. If the drop is falling fast, it wants to smash into the pool and keep going down.
2. Surface Tension (The Elastic Band): Water molecules like to hold hands. This force tries to pull the drop into a perfect sphere and snap the "neck" of the liquid column tight.
3. Viscosity (The Honey): This is how "thick" or sticky the liquid is. Honey moves slower and resists changing shape more than water.
4. Gravity (The Heavy Hand): This just pulls everything down.

The researchers used three special "scorecards" (numbers) to measure how strong each character is:

• Weber Number (We): How strong is the speed compared to the surface tension? (High speed = High Weber).
• Ohnesorge Number (Oh): How sticky is the liquid? (High Oh = Very sticky).
• Bond Number (Bo): How much does gravity matter compared to surface tension? (Big drops = High Bond).

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The Three Main Findings

1. The Shape Matters (The "Squash vs. Stretch" Effect)

Most people imagine drops as perfect spheres. But in reality, falling drops can be squashed flat (like a pancake) or stretched long (like a cigar).

• The Pancake (Oblate): If the drop is flat, it hits the water like a wide lid. It drains its water into the pool very quickly. It's like pouring a bucket of water; it merges fast. Result: It usually just disappears (Complete Coalescence).
• The Cigar (Prolate): If the drop is stretched out, it hits the water with a narrow point. It drains slowly. The water column that shoots up is tall and thin. Result: It is much more likely to snap off a baby droplet (Partial Coalescence).

Analogy: Think of a pancake hitting a pool vs. a spear hitting a pool. The pancake spreads out and sinks immediately. The spear punches a hole, creates a tall splash, and might break off a piece of itself.

2. The "Neck Oscillation" (The Bouncing Neck)

When the drop hits, a neck forms. Sometimes, this neck doesn't just snap once. It wiggles!

• The Wiggle: The neck expands, then shrinks, then expands again.
• The Discovery: The researchers found that this "wiggling" is the key to understanding the transition.
  • If the neck wiggles once and snaps: You get a baby droplet (Partial Coalescence).
  • If the neck wiggles twice and then snaps: You might get a baby droplet, or it might just merge.
  • If the neck wiggles twice and doesn't snap: It just merges completely (Complete Coalescence).

Analogy: Imagine a rubber band being stretched. If you pull it too hard (too much speed or gravity), it snaps immediately. If you pull it gently, it might stretch, bounce back, stretch again, and then snap. The number of bounces tells you if a baby droplet will be born.

3. The "Baby Droplet" Factory

The study found that the shape of the drop determines how many babies you get.

• Long drops (Prolate) are the best at making baby droplets. In fact, if the conditions are just right, one long drop can create a chain reaction, breaking into multiple tiny droplets.
• Fast drops actually make fewer babies. If you throw the drop too hard, it smashes into the pool so fast that it merges before it has a chance to form a nice tower.

Analogy: Think of a long, thin rope vs. a short, fat rope. If you drop the long rope, it's easier to snap a piece off the end. If you drop the short rope, it just lands and stays put. Also, if you throw the rope really hard, it just tangles and merges with the ground.

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What They Discovered About "Instability"

For a long time, scientists thought the "baby droplet" was created by a specific instability called the Rayleigh-Plateau instability (basically, a long column of water naturally wanting to break into beads, like water dripping from a tap).

The Twist: This paper says, "Not so fast!"
They found that for the drops they studied, this instability wasn't the main villain. The real reason the drop snaps is a battle between vertical momentum (pulling down) and horizontal momentum (squeezing in). The drop doesn't break because it's unstable; it breaks because the horizontal squeeze wins the race against the vertical pull.

The Final Verdict (The "Rulebook")

The researchers built a 3D map (a phase diagram) that acts like a weather forecast for drops.

• If you know the speed, the stickiness, the gravity, and the shape, you can predict exactly what will happen.
• Low speed + Low stickiness + Round/Long shape = High chance of a baby droplet.
• High speed + High stickiness + Flat shape = The drop just merges and disappears.

Why Does This Matter?

This isn't just about watching water drops. Understanding this helps in:

• Inkjet Printing: Making sure the ink lands exactly where it should without splattering.
• Oil Recovery: Helping oil and water separate better in pipes.
• Rain and Soil: Understanding how rain hits the ground and causes erosion or sprays microplastics into the air.
• Spray Cooling: Making sure engines cool down efficiently.

In short, the authors took a simple phenomenon—a drop hitting water—and used math and computers to reveal a complex, beautiful dance of forces that determines whether a drop survives to have a baby or just merges into the crowd.

Technical Summary

1. Problem Statement

The paper investigates the complex dynamics of a liquid drop impacting a deep liquid pool, specifically focusing on the transition between partial coalescence (where a secondary droplet pinches off) and complete coalescence (where the drop merges entirely). While previous studies have established the roles of viscosity (Ohnesorge number, $Oh$) and gravity (Bond number, $Bo$) in low-inertia regimes, the explicit influence of inertial forces (Weber number, $We$) and drop shape (aspect ratio, $AR$) on these transitions remains under-explored.

Key gaps addressed include:

• The lack of a unified regime map considering the interplay of $We$, $Oh$, and $Bo$.
• The mechanism behind neck oscillations observed during the transition regime.
• The debate regarding the role of Rayleigh-Plateau instability versus capillary wave competition in driving pinch-off.
• The effect of non-spherical drop shapes (oblate, prolate) on coalescence outcomes.

2. Methodology

• Numerical Approach: The authors performed axisymmetric numerical simulations using Gerris, an open-source finite-volume flow solver.
• Governing Equations: The study solves the incompressible Navier-Stokes equations coupled with a Volume-of-Fluid (VoF) method to track the air-liquid interface. Surface tension is modeled using a height-function-based balanced force formulation.
• Parameters:
  • Dimensionless Numbers: Weber ($We = \rho_l V^2 D_{eq}/\sigma$), Ohnesorge ($Oh = \mu_l / \sqrt{\rho_l \sigma D_{eq}}$), and Bond ($Bo = \rho_l g D_{eq}^2 / \sigma$) numbers.
  • Drop Shape: Defined by Aspect Ratio ($AR = b/a$), where $AR < 1$ (oblate), $AR = 1$ (spherical), and $AR > 1$ (prolate). Volume is kept constant across shapes.
  • Domain: A cylindrical domain of $7D_{eq} \times 7D_{eq}$ with a pool depth of $4D_{eq}$.
• Validation: The solver was validated through:
  • Grid convergence studies (mesh size $\Delta x = D_{eq}/293$ selected).
  • Comparison with experimental data from Blanchette & Bigioni (2006) regarding pinch-off timing and morphology.
  • Comparison of entrapped bubble volumes with experimental and numerical data from Tran et al. (2013) and Hendrix et al. (2016).

3. Key Contributions

1. 3D Regime Map: Construction of a comprehensive three-dimensional phase diagram in the $We$–$Oh$–$Bo$ space, delineating boundaries between partial and complete coalescence.
2. Classification of Coalescence Regimes: Identification and systematic classification of four distinct coalescence regimes based on neck oscillation behavior:
   • First-stage partial coalescence: Pinch-off occurs during the first neck oscillation.
   • Second-stage partial coalescence: Pinch-off occurs during the second oscillation.
   • Second-stage complete coalescence: Full merging occurs during the second oscillation.
   • First-stage complete coalescence: Full merging occurs during the first oscillation.
3. Mechanism Elucidation: Clarification that the transition is driven by the competition between vertical collapse rates (driven by gravity and inertia) and horizontal collapse rates (driven by surface tension), rather than solely by Rayleigh-Plateau instability.
4. Shape Dependence: Demonstration that drop shape significantly alters the critical Weber number ($Wec$) required for transition, with prolate drops being most prone to partial coalescence.

4. Key Results

A. Influence of Dimensionless Numbers

• Weber Number ($We$): Increasing $We$ (higher impact velocity) enhances inertial effects, accelerating the drainage of liquid into the pool and increasing the vertical collapse rate. This suppresses the formation of the liquid column, promoting a transition from partial to complete coalescence.
• Ohnesorge Number ($Oh$): Higher viscosity ($Oh$) suppresses capillary waves and perturbations. The study identifies a critical threshold $Oh_c \approx 0.021$ (for air-liquid interfaces), above which complete coalescence occurs regardless of $We$ or $Bo$.
• Bond Number ($Bo$): Increased gravity ($Bo$) accelerates vertical drainage, reducing the time available for horizontal retraction and favoring complete coalescence.
• Interplay: The critical Weber number ($Wec$) required to trigger complete coalescence decreases as $Oh$ and $Bo$ increase.

B. Role of Drop Shape (Aspect Ratio, $AR$)

• Prolate Drops ($AR > 1$): Most prone to partial coalescence. Their narrower initial contact neck restricts drainage, delaying vertical collapse and allowing the liquid column to form and pinch off. They are also most likely to form multiple secondary droplets.
• Oblate Drops ($AR < 1$): Least prone to partial coalescence. Their flatter base creates a wider neck opening, facilitating rapid drainage and vertical collapse, leading to complete coalescence.
• Critical $Wec$ Trend: For a fixed $Oh$, $Wec$ decreases as $AR$ decreases (i.e., prolate drops require lower impact velocities to undergo partial coalescence compared to oblate drops).

C. Neck Oscillations and Transition Mechanism

• The transition from partial to complete coalescence is not abrupt but mediated by neck oscillations.
• Mechanism: The outcome depends on the competition between the horizontal inward momentum (pinching the neck) and the vertical downward momentum (draining the column).
  • If horizontal collapse dominates first $\rightarrow$ Partial coalescence.
  • If vertical collapse dominates first $\rightarrow$ Complete coalescence.
• Oscillation Cycles: The number of oscillations correlates with the regime. Higher $We$, $Oh$, or $Bo$ generally reduces the number of oscillations, pushing the system toward complete coalescence.

D. Rayleigh-Plateau Instability (RPI)

• The study challenges the hypothesis that RPI is the primary driver of partial coalescence.
• Analysis of the height-to-neck ratio ($\psi = h / \pi w_n$) shows that for many cases, $\psi$ does not exceed the critical threshold for RPI instability until after the pinch-off has already been determined by the collapse dynamics.
• Conclusion: RPI is insignificant in driving the initial pinch-off in the explored parameter space. However, it may play a role in the fragmentation of elongated secondary droplets (specifically from prolate drops) after they have formed.

E. Multiple Secondary Droplets

• The formation of multiple secondary droplets is most prominent for prolate drops.
• This occurs when the primary drop forms an extended liquid column that is subsequently broken up by varicose capillary waves (RPI) after the initial pinch-off.
• Increasing $We$, $Oh$, or $Bo$ reduces the likelihood of multiple droplet formation by shortening the liquid column and increasing drainage.

5. Significance

This work provides a fundamental understanding of drop coalescence by integrating inertial effects and drop morphology into the classical framework of viscous and gravitational forces.

• Theoretical Impact: It refines the criteria for partial vs. complete coalescence, moving beyond simple $Oh$ thresholds to a multi-parameter regime map. It clarifies that the competition between collapse rates, rather than RPI, is the dominant mechanism for pinch-off.
• Practical Applications: The findings are relevant for optimizing processes in inkjet printing, spray atomization, microfluidics, and enhanced oil recovery, where controlling droplet merging and secondary droplet formation is critical.
• Methodological Value: The identification of neck oscillation cycles as a robust indicator for coalescence regimes offers a new diagnostic tool for analyzing interfacial dynamics.