End-pinching and inertial-capillary reopening in viscoplastic ligaments at low Ohnesorge number

This study demonstrates that for Herschel-Bulkley viscoplastic ligaments at low Ohnesorge numbers, shear-rate-dependent rheology enables two distinct mechanisms—vorticity detachment in shear-thickening fluids and curvature-induced pressure gradients in shear-thinning fluids—to prevent end-pinching and facilitate inertial-capillary reopening, thereby challenging the Newtonian assumption that break-up is the inevitable outcome in the inviscid limit.

The Gist

Imagine you have a long, thick string of honey or toothpaste hanging in the air. Gravity isn't pulling it down; instead, surface tension (the "skin" of the liquid) wants to pull it back into a perfect, round ball because that's the shape that uses the least amount of energy.

In the world of simple water (Newtonian fluids), we know exactly what happens to this string. As it shrinks, the ends get bulbous and the middle gets thin, like a figure-eight. Eventually, the thin neck snaps, and a droplet pops off. This is called "end-pinching." It's like a magician snapping a rubber band; the tension builds until snap!

But what happens if the liquid isn't just water? What if it's something weird, like ketchup, paint, or toothpaste? These are viscoplastic fluids. They act like solids until you push them hard enough to make them flow, and once they flow, they might get thicker (shear-thickening) or thinner (shear-thinning) depending on how fast you move them.

This paper is a deep dive into what happens to these "weird" liquid strings when they try to shrink. The researchers used powerful computer simulations to watch these strings in slow motion and discovered some surprising tricks they play to avoid snapping.

Here are the four main stories the paper tells, explained with everyday analogies:

1. The "Solid Block" (The Motionless Regime)

Imagine your liquid string is made of cold, stiff clay. If the surface tension tries to pull it, but the clay is too stiff to move, nothing happens. The string just sits there, frozen.

• The Science: If the "yield stress" (the force needed to make it flow) is too high compared to the surface tension, the liquid refuses to budge. It's like trying to push a boulder with your finger; the boulder wins.

2. The "Smooth Squeeze" (The No-Neck Regime)

Now, imagine the clay is a bit softer, like warm play-doh. When you squeeze it, it doesn't get thin in the middle and snap. Instead, the whole thing just shrinks uniformly, like a deflating balloon, until it becomes a single, round blob.

• The Science: The liquid is stiff enough that it doesn't allow a "neck" (a thin spot) to form. The whole string retracts smoothly into one droplet without ever breaking.

3. The "Traffic Jam" (Shear-Thickening Escape)

This is where it gets really cool. Imagine the liquid is like a crowd of people running through a hallway.

• The Scenario: As the string tries to pinch off, the neck gets very thin, and the liquid has to rush through that tiny gap very fast.
• The Twist: For some fluids (shear-thickening), moving fast makes them act like solid concrete. As the neck tries to thin, the liquid suddenly gets super thick and sticky right at the bottleneck.
• The Result: It's like a traffic jam forming in the neck. The liquid gets so thick that it creates a swirling vortex (a whirlpool) that pushes the fluid back out. The neck stops thinning, swells up, and the string survives! It's like the liquid says, "Nope, I'm too thick to break," and reopens.

4. The "Pressure Wave" (Shear-Thinning & The Inviscid Surprise)

This is the paper's biggest discovery. Imagine a fluid that gets thinner and more slippery when you squeeze it fast (like ketchup).

• The Expectation: You'd think, "If it gets thinner, it should snap even faster!"
• The Reality: The researchers found that if the liquid is extremely slippery (almost like water with zero viscosity), something magical happens. As the neck gets tiny, the shape of the curve creates a weird pressure wave.
• The Analogy: Think of a trampoline. If you push down in the middle, the edges pop up. In this liquid, the pressure at the neck changes so drastically that it pushes the liquid away from the center instead of pulling it in.
• The Result: The neck stops shrinking, reverses direction, and the string "reopens" into a smooth shape again.
• Why it matters: Scientists used to think that if a liquid was slippery enough, it would always snap. This paper proves that's wrong. Even with zero viscosity, the string can save itself using pure physics (inertia and surface tension) without needing any "thickness" to help.

The Big Picture

The researchers mapped out a "menu" of outcomes. Depending on how stiff the liquid is and how its thickness changes when it moves, the liquid string will either:

1. Snap (End-pinching).
2. Refuse to move (Motionless).
3. Shrink smoothly (No-neck).
4. Save itself (Reopening).

Why should you care?
This isn't just about physics homework. This helps engineers design better:

• Inkjet printers: Making sure the ink drops cleanly without leaving messy tails.
• Spray painting: Ensuring the paint covers evenly.
• 3D printing: Making sure the "goo" holds its shape or flows correctly.
• Agriculture: Understanding how pesticide sprays break up into droplets to hit leaves effectively.

In short, this paper shows that liquid strings are smarter than we thought. They have secret ways to dodge breaking, and by understanding these tricks, we can control how liquids behave in our technology.

Technical Summary

1. Problem Statement

The paper investigates the capillary retraction dynamics of liquid ligaments, a fundamental process in free-surface flows relevant to inkjet printing, spray coating, and droplet formation. While the retraction of Newtonian ligaments is well-understood (dominated by the "end-pinching" mechanism where a droplet detaches from the tip), the behavior of viscoplastic fluids remains poorly explored.

Specifically, the authors address a gap in understanding how Herschel-Bulkley fluids (which possess both a yield stress and shear-rate-dependent viscosity) behave in the low-viscosity (low Ohnesorge number, $Oh_K$) regime. Previous studies on Newtonian fluids suggested that as viscosity approaches zero ($Oh_K \to 0$), end-pinching and breakup are the inevitable asymptotic outcomes. The authors challenge this assumption by exploring whether viscoplasticity and shear-thinning/thickening rheology can alter this fate, potentially leading to reopening (escape from breakup) even in the inviscid limit.

2. Methodology

The study employs Direct Numerical Simulations (DNS) using the open-source finite-volume solver Basilisk.

• Governing Equations: The incompressible two-phase Navier-Stokes equations are solved using a one-fluid formulation with a Volume-of-Fluid (VOF) method to track the interface.
• Constitutive Model: The ligament is modeled as a Herschel-Bulkley fluid, defined by:
  $$\tilde{\mu}_{eff} = \frac{J}{2\|\tilde{D}\| + \tilde{\epsilon}} + Oh_K (2\|\tilde{D}\| + \tilde{\epsilon})^{n-1}$$
  Where $J$ is the plastocapillary number (yield stress vs. surface tension), $Oh_K$ is the Ohnesorge number (viscous vs. inertial-capillary), and $n$ is the flow behavior index ($n>1$ for shear-thickening, $n<1$ for shear-thinning).
  • Key Distinction: Unlike Bingham models (constant post-yield viscosity), this model introduces spatially varying effective viscosity, creating new stress gradients.
• Numerical Setup:
  • Axisymmetric simulations of a ligament with aspect ratio $L=15$ retracting in air.
  • Adaptive Mesh Refinement (AMR) is used, reaching grid levels up to $l=13$ (approx. 328 cells across the radius) to resolve interfacial singularities.
  • An $\epsilon$-regularization technique is used to handle the yield stress singularity, with sensitivity analysis confirming results are independent of $\epsilon$ for $\epsilon \le 10^{-4}$.
• Validation: The solver is validated against analytical solutions for Poiseuille flow and previous Newtonian results (Notz and Basaran), showing excellent agreement.

3. Key Contributions

1. Regime Map Construction: The authors delineate a comprehensive regime map in the $(n, J)$ parameter space, identifying four distinct dynamical outcomes: Pinch-off, End-pinching Escape, No-neck formation, and Motionless arrest.
2. Discovery of Shear-Thinning Reopening: They identify a novel mechanism where strongly shear-thinning fluids ($n \lesssim 0.6$) escape end-pinching via curvature-induced pressure gradients, even when yield stress is negligible.
3. Inviscid Limit Reopening: The most significant theoretical contribution is the demonstration that purely inertial-capillary reopening occurs as $Oh_K \to 0$ in Newtonian fluids. This contradicts previous literature suggesting breakup is the unique inviscid limit, showing instead that end-pinching is not the asymptotic outcome for sufficiently low $Oh_K$.
4. Differentiation from Bingham Models: The study proves that shear-rate-dependent viscosity (Herschel-Bulkley) generates flow regimes and escape mechanisms that cannot be captured by simpler constant-viscosity yield-stress models (Bingham).

4. Key Results

A. Phenomenological Regime Map

At $Oh_K = 10^{-3}$, the dynamics are governed by the interplay of $J$ (yield stress) and $n$ (shear dependence):

• Pinch-off (End-pinching): Occurs for low $J$ and moderate $n$. The ligament thins and breaks, following the classical inertial-capillary scaling $r_{min} \sim (t_b - t)^{2/3}$.
• No-Neck Regime ($0.06 \lesssim J \lesssim 0.45$): Yield stress prevents the formation of a localized neck. The ligament retracts smoothly into a single droplet because the unyielded core shields the center from capillary localization.
• Motionless Regime ($J \gtrsim 0.45$): Yield stress dominates capillary forces entirely, arresting the ligament immediately.
• Escape (Reopening): Occurs via two distinct mechanisms depending on $n$.

B. Mechanisms of Escape from End-Pinching

1. Shear-Thickening Regime ($n > 1$):
   • As the neck thins, the local strain rate increases, causing a sharp rise in effective viscosity ($\mu_{eff} \sim \dot{\gamma}^{n-1}$).
   • This high viscosity generates strong vorticity and a backflow (vortex ring) that opposes the capillary singularity, forcing the neck to reopen.
2. Shear-Thinning Regime ($n < 1$) & The Newtonian Limit:
   • In strongly shear-thinning fluids, the effective viscosity drops drastically in the neck, weakening viscous resistance.
   • Surprisingly, this leads to reopening driven by curvature-induced pressure gradients.
   • Mechanism: In the low $Oh_K$ limit, viscous diffusion is too weak to smooth out axial variations in interface curvature. This creates a non-monotonic pressure gradient ($-\partial_z p \sim -\sigma \partial_z \kappa$) with opposing axial accelerations. This local force imbalance reverses the neck dynamics, driving fluid away from the minimum and causing the neck to expand rather than collapse.
   • Inviscid Limit ($Oh_K \to 0$): The authors show that this reopening mechanism persists in the Newtonian limit ($J=0, n=1$) as $Oh_K$ drops below $10^{-5}$. Previous studies only observed breakup down to $Oh_K \approx 10^{-4}$; this work reveals that true inviscid dynamics favor reopening.

C. Scaling Laws

• No-Neck Threshold: Derived analytically as $J_{end} \approx 0.055$, based on the balance where the yielded envelope fails to span the axial extent of the neck.
• Inertial-Capillary Scaling: Confirmed $r_{min} \sim (t_b - t)^{2/3}$ for breakup cases, but observed a plateau in $r_{min}$ (reopening) for very low $Oh_K$.

5. Significance and Implications

• Theoretical Advancement: The paper fundamentally corrects the understanding of the inviscid limit in free-surface flows. It establishes that breakup is not the unique asymptotic outcome as viscosity vanishes; instead, a purely inertial-capillary pathway for reopening exists.
• Material Design: For industrial applications (e.g., inkjet printing of complex fluids), the results suggest that tuning the shear-thinning index ($n$) or yield stress ($J$) can prevent unwanted droplet breakup (satellite drops) or control ligament retraction without relying solely on viscosity.
• Modeling Limitations: The study highlights the inadequacy of Bingham models for predicting the dynamics of complex viscoplastic fluids in low-viscosity regimes, as they miss the critical shear-rate-dependent viscosity effects that drive the reopening mechanisms.
• Experimental Relevance: While reaching $Oh_K < 10^{-5}$ is experimentally challenging, the findings provide a theoretical baseline for understanding why certain low-viscosity viscoplastic fluids (like certain gels or suspensions) might exhibit unexpected stability or retraction behaviors not predicted by standard models.

In conclusion, the authors demonstrate that the fate of a retracting viscoplastic ligament is a complex interplay of capillarity, inertia, yield stress, and shear-dependent rheology, revealing a rich landscape of dynamical behaviors that extend far beyond the classical end-pinching paradigm.