Self-similar breakup of a liquid ligament with a solid particle

This study demonstrates through numerical simulations and analytical modeling that a solid particle induces universal, self-similar pinch-off dynamics in stretching liquid ligaments, where the subsequent breakup becomes independent of particle size and is governed by the interplay between ligament stretching and Rayleigh-Plateau instability.

The Gist

Imagine you have a long, thin strand of honey or thick syrup hanging in the air. If you pull the ends apart, the strand gets thinner and thinner until, eventually, it snaps and breaks into separate drops. This is a common sight in nature and technology, from rain falling off a leaf to inkjet printers shooting out tiny dots.

Usually, this breaking happens because the liquid is naturally unstable; it wants to turn into spheres (drops) to save energy. But what happens if there is a tiny, solid speck—like a grain of sand or a dust particle—trapped inside that sticky strand?

This paper investigates exactly that scenario. The researchers used computer simulations and math to see how a single solid particle changes the way a stretching liquid strand breaks.

Here is the story of their findings, broken down into simple concepts:

The Setup: A Stretching String with a Knot

Think of the liquid strand as a long, stretchy rope made of honey. The researchers pulled the ends of this rope apart at a steady speed. Inside the rope, they placed a single, solid ball (the particle).

At first, the rope is thick, and the ball is small compared to the rope's width. It's like having a marble inside a thick garden hose. The marble doesn't really do anything; the rope just gets thinner and thinner as it stretches, following a predictable pattern.

The Turning Point: When the "Hose" Shrinks to the Size of the "Marble"

As the rope continues to stretch, it gets narrower. Eventually, the rope becomes so thin that it is almost touching the surface of the marble inside.

This is the critical moment. The paper calls this when the ratio of the particle size to the rope size gets close to 1. Suddenly, the marble acts like a "knot" or a "bump" in the rope. Because the rope is so thin, this bump creates a localized disturbance.

The Surprise: The "Universal" Snap

Here is the most interesting part of the discovery. The researchers tested this with different-sized marbles (some small, some large).

• Before the snap: The larger marbles made the rope break sooner than the smaller ones. This makes sense; a bigger obstacle causes trouble earlier.
• During the snap: Once the rope got thin enough to touch the marble, something magical happened. The speed at which the final break occurred became exactly the same, regardless of whether the marble was small or large.

The researchers call this "self-similar" behavior. It's as if, once the rope gets thin enough to touch the obstacle, the specific size of the obstacle stops mattering. The liquid "forgets" how big the particle was and follows a universal, predictable path to breaking.

The Analogy: The Traffic Jam

Imagine a highway (the liquid strand) where cars are driving away from each other, making the traffic spread out (stretching).

• Early stage: If there is a small pothole (small particle) or a large rock (large particle) in the middle of the road, it doesn't matter much yet because the road is wide.
• Late stage: As the road narrows down to a single lane, both the pothole and the rock become massive obstacles.
• The Break: The moment the traffic gets so squeezed that it hits the obstacle, the way the traffic gridlocks and stops (the "breakup") happens in the exact same way for both the pothole and the rock. The size of the obstacle no longer changes the timing of the final gridlock; only the fact that something is there matters.

The Math and Physics

The researchers didn't just watch this happen; they wrote a mathematical formula to predict exactly when the break would occur.

• They found that the breaking time depends on a battle between two forces: the stretching (pulling the rope apart) and the viscosity (how "thick" or sticky the liquid is).
• In thick, sticky liquids (like the honey in our analogy), the "stickiness" dominates.
• Their formula successfully predicted the break time, matching their computer simulations perfectly.

The Bottom Line

The paper concludes that while a particle changes when the breakup starts (by making the rope thin faster near the particle), once the rope gets thin enough to touch the particle, the final act of breaking follows a universal rule.

In this specific "sticky" regime, the liquid strand behaves like a machine that has a set program for snapping. Once the particle gets close enough to trigger the program, the size of the particle becomes irrelevant, and the strand snaps in a predictable, self-similar way every time.

Technical Summary

Technical Summary: Self-similar breakup of a liquid ligament with a solid particle

Problem Statement
The breakup of thinning liquid ligaments is a fundamental process in applications ranging from spray formation and inkjet printing to microfluidic droplet generation. In the absence of external stretching, this breakup is governed by the Rayleigh–Plateau instability, where surface perturbations grow due to capillary forces. The dynamics are typically characterized by the Ohnesorge number ($Oh$), which represents the ratio of viscous forces to inertial and surface tension forces. While the behavior of clean ligaments is well-understood, many practical systems involve suspensions containing solid particles or impurities. Previous studies have examined particle-laden jets with many particles or dense suspensions, demonstrating altered breakup patterns. However, the fundamental mechanism by which a single solid particle acts as a localized perturbation to modify the breakup dynamics and pinch-off time of a stretching ligament remains poorly understood. Specifically, it is unclear how the interplay between ligament stretching, viscosity, and a localized particle-induced perturbation dictates the final breakup regime.

Methodology
The authors employ a combination of numerical simulations and analytical modeling to investigate this problem.

• Numerical Setup: The study utilizes the Lattice Boltzmann Method (LBM) based on a color-gradient approach for two-phase immiscible fluids. The system models a cylindrical liquid ligament of initial radius $R_i$ and length $L$ (aspect ratio $L/R_i = 28.8$) surrounded by air. A solid spherical particle of radius $r_p$ is placed at the center of the ligament with a no-slip boundary condition.
• Stretching Mechanism: Axial stretching is imposed by applying linear velocity profiles at the ligament ends ($u_x = \frac{2}{L}(x - L/2)u_{max}$), causing mass to exit the domain and the ligament radius to decrease over time. The stretching rate is characterized by a dimensionless outflow rate $\tilde{\dot{\epsilon}}$.
• Parameters: The simulations focus on the viscous regime where $Oh \sim 1.0$. The particle size is varied relative to the initial ligament radius ($r_p/R_i$), and the outflow rate is adjusted to control the stretching time scale relative to the capillary-viscous time scale ($\tau_v = \mu R_i / \sigma$).
• Analytical Approach: The authors derive an analytical expression for the pinch-off time by modeling the ligament radius as the sum of an unperturbed mean radius (governed by mass conservation) and a localized perturbation amplitude induced by the particle. They assume the perturbation grows according to Rayleigh–Plateau instability dynamics, dominated by viscous effects in the $Oh \sim 1$ regime.

Key Results

1. Two-Stage Dynamics: The breakup process exhibits two distinct stages. Initially, when the ligament radius $R(t)$ is significantly larger than the particle radius ($r_p/R(t) < 1$), the ligament thins monotonically according to mass conservation, and the particle's influence is negligible.
2. Onset of Self-Similarity: As stretching proceeds and the ligament surface approaches the particle ($\beta(t) = r_p/R(t) \sim 1$), a localized perturbation is triggered. Once this proximity is reached, the subsequent growth of the perturbation and the final pinch-off dynamics become self-similar and independent of the particle size.
3. Universal Pinch-off Time: While the total time to breakup ($t_p$) depends on the particle size (larger particles trigger earlier breakup), the self-similar pinch-off time ($t_{ps} = t_p - t_{exp}$), defined as the duration from the onset of perturbation growth to breakup, collapses to a single value for different particle sizes at a given stretching rate.
4. Analytical Validation: The authors derive an analytical expression for the pinch-off time ($t_{p,a}$) based on the competition between the stretching rate and the viscous capillary instability growth rate ($\lambda = \sigma / \mu R(t)$). This expression, $t_{p,a} = \frac{1}{\lambda_0 e^{\dot{\epsilon} t_{p,a}} + \dot{\epsilon} \ln(R_i/a_0)}$, shows excellent quantitative agreement with the numerical simulation data across various particle sizes and outflow rates.

Significance and Claims
The paper claims to reveal a universal mechanism by which localized perturbations control the breakup of ligaments containing solid particles in the viscous regime. The primary contribution is the demonstration that once the ligament surface approaches a solid particle, the specific size of the particle becomes irrelevant to the final breakup dynamics; the system enters a universal self-similar regime.

The authors emphasize that this universality arises because the viscous time scale ($t_\mu = \mu R / \sigma$) allows particle-induced perturbations to persist and grow, unlike in the inertial regime ($Oh \ll 1$) where breakup occurs too rapidly for such perturbations to significantly alter the outcome. The proposed analytical framework successfully captures this universal scaling regime for $Oh \sim 1$, providing a predictive tool for the pinch-off time that depends on the stretching rate and fluid properties but is independent of the specific particle size once the perturbation is triggered. The study is limited to symmetric, stationary particle configurations, serving as a minimal setup to isolate these effects, with more complex, advected particle interactions left for future work.