A numerical study on the coefficient of restitution of wet collisions

This paper utilizes validated smoothed particle hydrodynamics simulations to investigate the coefficient of restitution in wet collisions within a moderate-to-high Weber number regime, revealing a scaling law dependent on the Stokes number and dimensionless film thickness that exhibits two distinct power-law regimes.

The Gist

Imagine you're playing with a marble. If you drop it on a dry kitchen floor, it bounces back up with almost the same energy it had when it hit the ground. It's a predictable, "dry" bounce.

Now, imagine dropping that same marble onto a thin layer of water on the floor. The bounce is very different. The water acts like a cushion, slowing the marble down, and sometimes the marble gets stuck or bounces much lower. This is a wet collision.

This paper is a deep dive into understanding exactly how and why that wet bounce changes. The researchers used powerful computer simulations (like a super-advanced video game physics engine) to watch marbles hit thin films of liquid, trying to find a simple rule that predicts the bounce height.

Here is the breakdown of their discovery in plain English:

1. The Old Rule vs. The New Reality

For a long time, scientists thought there was one single "magic number" (called the Stokes Number) that could predict how a wet object would bounce. Think of this number as a measure of how "heavy and fast" the object is compared to how "thick and sticky" the liquid is.

• The Old Idea: If you know the Stokes Number, you know the bounce.
• The New Discovery: The researchers found this old rule is incomplete. It's like trying to predict how a car stops by only looking at its speed, but ignoring the thickness of the brakes. They found that the thickness of the liquid layer relative to the size of the object matters just as much.

2. The Two "Bounce Zones"

The most exciting part of the study is that they found two distinct worlds of bouncing, separated by a specific boundary.

• Zone 1: The "Heavy Hitter" Zone (Thick Film / Small Bead)
  Imagine a tiny bead hitting a relatively thick puddle. Here, the liquid acts like a thick, sticky syrup. The bounce depends heavily on how fast the bead is moving and how much it fights against the liquid's resistance. It's a straightforward battle between the bead's momentum and the liquid's stickiness.

  • Analogy: It's like trying to run through waist-deep water. Your speed matters a lot, and the water drags you down predictably.

• Zone 2: The "Chaos" Zone (Thin Film / Large Bead)
  Now, imagine a large marble hitting a very thin film of water. Here, things get weird. The bounce becomes much less dependent on speed and more dependent on the geometry of the setup. Why? Because the liquid starts to swirl, create tiny whirlpools (vortices), and splash in complex ways. The energy isn't just lost to friction; it's lost to creating these chaotic little storms in the water.

  • Analogy: It's like a giant bowling ball rolling over a thin sheet of ice. The ice doesn't just slow it down; it cracks, shatters, and sends shards flying in unpredictable directions. The outcome is chaotic and less about speed, more about the "mess" created.

3. How They Did It (The "Virtual Lab")

Since it's hard to film a marble hitting a liquid film in slow motion without the camera getting in the way, the team built a virtual lab using a technique called Smoothed Particle Hydrodynamics (SPH).

• The Magic Trick: Instead of simulating the whole marble (which is computationally expensive), they only simulated the bottom "shell" of the marble that actually touches the water. It's like only painting the bottom of a shoe to see how it scuffs the floor, rather than painting the whole shoe. This saved them a massive amount of computer power while keeping the physics accurate.
• The Validation: They checked their computer results against real-world experiments done by other scientists, and their numbers matched perfectly.

4. Why Does This Matter?

You might wonder, "Who cares if a marble bounces differently in water?"

Actually, this is huge for industry.

• Pharmaceuticals: Making pills involves mixing wet powders. If the particles bounce too much or too little, the pills won't form correctly.
• Construction: Mudslides and debris flows are essentially giant wet collisions. Understanding how energy is lost helps predict how far a mudslide will travel.
• Manufacturing: Coating objects with paint or ink involves wet impacts.

The Bottom Line

The paper concludes that there is no single "magic number" for wet bounces. Instead, nature has two different rulebooks:

1. One for when the liquid is thick enough to act like a simple drag force.
2. One for when the liquid is thin enough to create chaotic swirls and splashes.

By finding the line that separates these two zones, the researchers have created a new "scaling law"—a mathematical recipe that engineers can use to predict exactly how wet particles will behave, whether they are tiny drug granules or massive mud boulders.

Technical Summary

1. Problem Statement

The coefficient of restitution (COR) is a critical parameter in modeling granular flows, debris flows, and industrial processes like wet granulation and coating. While dry collisions are often modeled with a constant COR or weak velocity dependence, wet collisions involve complex energy dissipation mechanisms due to interstitial liquid films (viscous flow, inertial pressure buildup, and surface deformation).

Current macroscopic models often assume the wet COR depends solely on the Stokes number ($St$), which compares particle inertia to viscous resistance. However, experimental evidence (e.g., Gollwitzer et al.) suggests this single-parameter scaling fails when the ratio of liquid film thickness to particle diameter ($\delta/D$) varies or when inertial effects become significant. The specific gap addressed by this study is the lack of a comprehensive scaling law that accounts for both the Stokes number and the dimensionless film thickness ($\gamma = \delta/D$) across moderate-to-high Weber number regimes where surface tension is negligible.

2. Methodology

The authors employed Incompressible Smoothed Particle Hydrodynamics (ISPH) simulations coupled with rigid-body dynamics to investigate normal impacts of a rigid sphere onto a thin liquid film.

• Numerical Framework:

  • ISPH: Used a projection-based formulation (Cummins and Rudman) with a Cleary–Monaghan viscous model to resolve pressure-driven dissipation and fluid-solid interactions.
  • Rigid Body Modeling: The solid bead was modeled as a free rigid body. To reduce computational cost, only the lower portion of the sphere (a truncated spherical shell of 4 particle widths) interacting with the fluid was discretized using Lagrangian particles. The full sphere's mass, center of gravity (COG), and moment of inertia were used to compute translational and rotational dynamics.
  • Collision Handling: A sub-timestep collision detection algorithm was implemented to precisely capture the impact instant with the solid substrate. A dry coefficient of restitution ($e_r = 0.98$) was applied instantaneously upon substrate contact to isolate liquid-induced dissipation.
  • Adaptive Timestep: The time step was reduced adaptively when the bead approached the wall to ensure numerical stability and resolve rapid momentum changes.

• Simulation Parameters:

  • Regime: Moderate-to-high Weber number ($We \sim 10^1 - 10^2$), allowing surface tension to be neglected.
  • Variables: Impact velocity ($0.1–1.5$ m/s), bead diameter ($2.8–8.0$ mm), and film thickness ($0.40, 0.45$ mm).
  • Fluids: Water and M5 silicone oil.
  • Validation: The model was validated against experimental data from Gollwitzer et al. [9], showing excellent agreement ($R^2 = 0.999$) with a relative $L_2$ error below 1% at sufficient resolution (~178,000 liquid particles).

3. Key Contributions

1. Identification of a Second Governing Parameter: The study demonstrates that the Stokes number alone is insufficient to characterize wet COR. The dimensionless film thickness ($\gamma = \delta/D$) is a critical independent variable.
2. Discovery of Two Distinct Scaling Regimes: The authors identified two distinct regimes (R1 and R2) separated by a boundary in the logarithmic space of $St$ and $\gamma$. Each regime follows a different power-law relationship.
3. Development of a Piecewise Scaling Law: A unified mathematical model was derived that switches between two sets of exponents based on the impact conditions.
4. Physical Mechanism Explanation: The study links the existence of the two regimes to different energy dissipation mechanisms, specifically the presence or absence of secondary flow phenomena (vortices) within the liquid film.

4. Results and Analysis

Scaling Regimes

The wet COR was found to follow a power-law form: $COR \sim St^a \gamma^b$. The analysis revealed two distinct regimes:

• Regime R1 (Low $\gamma$, Large Particles):
  • Characteristics: Strong dependence on both Stokes number and film thickness.
  • Scaling Law: $COR \approx 0.17 \cdot St^{0.17} \cdot \gamma^{-0.16}$
  • Physics: Energy dissipation is dominated by primary viscous flow. The kinetic energy in the fluid decreases monotonically after impact.
• Regime R2 (High $\gamma$, Small Particles):
  • Characteristics: Weak dependence on the Stokes number; dominated by geometric film thickness.
  • Scaling Law: $COR \approx 0.40 \cdot St^{0.01} \cdot \gamma^{-0.22}$
  • Physics: Secondary flow phenomena (e.g., vortices, flow separation) play a significant role. Kinetic energy plots show a secondary peak after substrate contact, indicating complex flow structures that dissipate energy differently than in R1.

Energy Dissipation Analysis

By tracking the total kinetic energy (KE) of the fluid:

• In R1, KE dissipates monotonically after the bead hits the substrate, indicating a primary flow scenario dominated by viscous drag.
• In R2, a distinct second peak in fluid KE is observed after substrate contact. This suggests the generation of secondary flows (vortices) which contribute significantly to energy loss, explaining the weak dependence on $St$ in this regime.

5. Significance and Conclusion

This study provides a robust numerical framework for predicting wet collision outcomes in industrial and natural granular flows where liquid films are present.

• Theoretical Impact: It challenges the prevailing assumption that wet COR is a function of the Stokes number alone, establishing that the ratio of film thickness to particle size is equally critical.
• Practical Application: The derived piecewise scaling laws can be directly implemented in Discrete Element Method (DEM) simulations and other statistical approaches to improve the accuracy of energy dissipation models in wet granular systems.
• Methodological Advance: The use of a truncated spherical shell for rigid-body interaction within an ISPH framework offers a computationally efficient method for simulating fluid-solid interactions without sacrificing physical accuracy.

In summary, the paper successfully bridges the gap between experimental observations and theoretical modeling by identifying a dual-regime behavior in wet collisions, governed by the interplay between inertial forces ($St$) and geometric confinement ($\gamma$).