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arxiv: 2511.18388 · v3 · submitted 2025-11-23 · ⚛️ physics.flu-dyn

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

Shu Yang , Fahim Tanfeez Mahmood , C. Ricardo Constante-Amores This is my paper

Pith reviewed 2026-05-17 06:17 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords viscoplastic ligamentsend-pinchinginertial-capillary reopeningHerschel-Bulkley fluidlow Ohnesorge numbercapillary retractiondroplet detachment
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The pith

Viscoplastic ligaments can reopen through inertial-capillary mechanisms as viscosity approaches zero, avoiding end-pinching.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies retraction of ligaments made from Herschel-Bulkley fluids that combine yield stress with shear-rate-dependent viscosity. In the low-viscosity range where Newtonian liquids break up by end-pinching, these materials follow different paths. One path appears in the shear-thickening regime where rising local viscosity detaches vorticity from the neck. The other, in strong shear-thinning, relies on curvature-driven pressure gradients that push the neck open again. This second route survives into the Newtonian limit of vanishing viscosity.

Core claim

Axisymmetric simulations of retracting Herschel-Bulkley ligaments show that a purely inertial-capillary reopening mechanism appears as the Ohnesorge number Oh_K tends to zero. This indicates that end-pinching is not the asymptotic outcome in the inviscid limit, in contrast to earlier Newtonian results that tracked end-pinching down to Oh_K approximately 10 to the minus four.

What carries the argument

Curvature-induced pressure gradients that drive inertial-capillary reopening in the strongly shear-thinning regime and persist into the vanishing-viscosity limit.

If this is right

  • End-pinching need not occur in the inviscid limit for viscoplastic ligaments.
  • Break-up outcomes depend on the details of shear-thickening or shear-thinning behavior.
  • Yield stress combined with rate-dependent viscosity can stabilize ligaments against capillary break-up.
  • Previous low-viscosity conclusions based on Newtonian fluids require adjustment for viscoplastic cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the reopening mechanism holds in three dimensions it would alter predictions for droplet sizes in sprays of yield-stress fluids.
  • The same pressure-gradient reopening could appear in other shear-thinning materials not modeled by Herschel-Bulkley rheology.
  • Targeted experiments at extremely low viscosity could isolate whether inertial-capillary effects truly dominate over any residual surface instabilities.

Load-bearing premise

The axisymmetric simulations remain accurate and representative of the physical reopening process even as the Ohnesorge number approaches zero without three-dimensional effects or unmodeled surface phenomena taking over.

What would settle it

A simulation or experiment at Ohnesorge numbers below 10 to the minus five that directly checks whether the ligament neck reopens or continues to thin and pinch off.

Figures

Figures reproduced from arXiv: 2511.18388 by C. Ricardo Constante-Amores, Fahim Tanfeez Mahmood, Shu Yang.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) Schematic of the initial configuration of the system (top panel) and the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Regime map in terms of the plastocapillary number [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: End-pinching regime. (a) Temporal sequence of the pressure field [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Shear-thickening reopening regime. (a) Snapshots of pressure [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Shear-thinning reopening regime. (a) Snapshots of pressure [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: No neck formation regime. Snapshots of pressure [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Motionless regime. Snapshots of pressure [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Epsilon independence check at [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: (a) [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

Capillary retraction of liquid ligaments is well understood for Newtonian fluids, whereas viscoplastic effects remain comparatively unexplored. Here, we consider Herschel-Bulkley fluids, which incorporate both yield stress and shear-rate-dependent viscosity, thereby introducing a spatially varying effective viscosity that is absent in simpler yield-stress models (e.g., Bingham models). We focus on the low-viscosity regime, where droplet detachment in Newtonian fluids is controlled by the end-pinching mechanism. Using fully resolved axisymmetric simulations, we show that viscoplasticity and shear-rate-dependent rheology reorganize the routes by which a retracting ligament may pinch off, escape break-up or stay motionless due to large yield stress. We identify two distinct routes by which a retracting Herschel-Bulkley ligament can escape end-pinching. In the shear-thickening regime, increased local viscosity during neck thinning leads to larger vorticity detachment from the curved neck, which opposes the capillary singularity. In the strongly shear-thinning regime, reopening is governed by curvature-induced pressure gradients. We show that this latter mechanism persists in the Newtonian limit of vanishing viscosity, yielding a purely inertial-capillary pathway for reopening. While previous Newtonian studies report end-pinching down to Ohnesorge number $Oh_K \approx 10^{-4}$, suggesting break-up as the asymptotic low-viscosity outcome (Anthony et al. 2019), our results demonstrate that a purely inertial-capillary reopening mechanism can arise as $Oh_K \to 0$, indicating that end-pinching is not the route in the inviscid limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines capillary retraction of Herschel-Bulkley viscoplastic ligaments at low Ohnesorge numbers using fully resolved axisymmetric simulations. It identifies two distinct mechanisms allowing escape from end-pinching: in the shear-thickening regime, increased local viscosity during neck thinning produces larger vorticity detachment opposing the capillary singularity; in the strongly shear-thinning regime, reopening is driven by curvature-induced pressure gradients. The central result is that this inertial-capillary reopening persists into the Newtonian limit as Oh_K → 0, implying that end-pinching is not the asymptotic route for inviscid Newtonian ligaments, in contrast to prior reports of end-pinching persisting down to Oh_K ≈ 10^{-4}.

Significance. If the central claim holds, the work is significant for fluid dynamics of non-Newtonian ligaments, showing how yield stress and power-law rheology reorganize breakup pathways and suggesting a purely inertial-capillary reopening mechanism in the inviscid limit. Credit is due for the use of fully resolved axisymmetric simulations across a range of yield stresses and power-law indices, which enables direct numerical exploration without fitted parameters or self-referential quantities. The findings could influence models of atomization, inkjet processes, and other capillary-driven flows involving complex fluids.

major comments (2)
  1. [Abstract and Newtonian-limit discussion] The claim that a purely inertial-capillary reopening mechanism arises as Oh_K → 0 and that end-pinching is not the route in the inviscid limit (abstract and discussion of Newtonian limit) rests on axisymmetric simulations. At low Ohnesorge numbers, retracting ligaments are susceptible to three-dimensional instabilities (azimuthal Rayleigh-Plateau or inertial modes) that axisymmetric formulations suppress by construction. The manuscript does not examine whether these modes could grow on timescales comparable to the reported reopening and enforce pinch-off, which would undermine the extrapolation from shear-thinning viscoplastic cases to physical Newtonian behavior.
  2. [Numerical methods and results on shear-thinning regime] The handling of the yield surface and its interaction with the neck region during reopening is central to distinguishing the shear-thinning reopening route from end-pinching. Without explicit grid-convergence tests or validation against known Newtonian benchmarks at the lowest Oh_K values simulated, it is difficult to confirm that the observed pressure-gradient-driven reopening is numerically robust rather than an artifact of regularization or mesh resolution near the free surface.
minor comments (2)
  1. [Abstract] The abstract states that viscoplasticity 'reorganize[s] the routes by which a retracting ligament may pinch off, escape break-up or stay motionless due to large yield stress,' but does not quantify the yield-stress thresholds separating these outcomes; a brief parameter map or table would improve clarity.
  2. [Introduction] Notation for the Ohnesorge number (Oh_K) and its relation to the capillary time scale should be defined explicitly on first use in the main text to aid readers unfamiliar with the specific nondimensionalization.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding the limitations of axisymmetric simulations and the need for additional numerical validation are important. We respond to each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract and Newtonian-limit discussion] The claim that a purely inertial-capillary reopening mechanism arises as Oh_K → 0 and that end-pinching is not the route in the inviscid limit (abstract and discussion of Newtonian limit) rests on axisymmetric simulations. At low Ohnesorge numbers, retracting ligaments are susceptible to three-dimensional instabilities (azimuthal Rayleigh-Plateau or inertial modes) that axisymmetric formulations suppress by construction. The manuscript does not examine whether these modes could grow on timescales comparable to the reported reopening and enforce pinch-off, which would undermine the extrapolation from shear-thinning viscoplastic cases to physical Newtonian behavior.

    Authors: We acknowledge that axisymmetric simulations suppress azimuthal instabilities by construction, and this represents a genuine limitation when extrapolating to the Newtonian inviscid limit. Prior Newtonian studies reporting persistent end-pinching (Anthony et al. 2019) likewise employed axisymmetric formulations down to comparable Oh_K values. In our viscoplastic shear-thinning cases, the reopening is driven by curvature-induced pressure gradients that operate within the axisymmetric framework. We will revise the discussion of the Newtonian limit to explicitly note this dimensionality limitation, discuss the possible growth of 3D modes on relevant timescales, and cite relevant literature on three-dimensional capillary instabilities. We will also qualify the central claim to emphasize that the inertial-capillary reopening is demonstrated in axisymmetric geometry and may be subject to 3D effects in physical realizations. This will be presented as an important caveat rather than a fully resolved issue. revision: partial

  2. Referee: [Numerical methods and results on shear-thinning regime] The handling of the yield surface and its interaction with the neck region during reopening is central to distinguishing the shear-thinning reopening route from end-pinching. Without explicit grid-convergence tests or validation against known Newtonian benchmarks at the lowest Oh_K values simulated, it is difficult to confirm that the observed pressure-gradient-driven reopening is numerically robust rather than an artifact of regularization or mesh resolution near the free surface.

    Authors: We agree that explicit demonstration of numerical robustness is necessary, particularly near the free surface and yield surface in the shear-thinning regime. Although the manuscript employs standard regularization and adaptive mesh refinement focused on the neck region, we did not include dedicated convergence studies or direct Newtonian benchmark comparisons at the lowest Oh_K in the current version. We will add these in the revised manuscript: grid-convergence tests showing invariance of neck evolution and pressure fields under successive refinement, plus validation against established Newtonian retraction benchmarks from the literature. These additions will be incorporated into the methods and results sections with supporting figures. revision: yes

standing simulated objections not resolved
  • Conducting fully three-dimensional simulations to assess growth rates of azimuthal instabilities relative to the reported reopening timescales at the lowest Ohnesorge numbers.

Circularity Check

0 steps flagged

No circularity: claims rest on direct numerical simulations without reduction to inputs

full rationale

The paper's central results on inertial-capillary reopening as Oh_K approaches zero are obtained from fully resolved axisymmetric simulations of the Herschel-Bulkley equations for retracting ligaments. No load-bearing step reduces a prediction to a fitted parameter, self-defined quantity, or self-citation chain; the observed curvature-induced pressure gradients and vorticity effects emerge from the numerical solution of the governing equations rather than being presupposed. External citation to Anthony et al. (2019) provides contrast with prior Newtonian findings and does not justify the new viscoplastic routes. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The study rests on the standard incompressible Navier-Stokes equations together with the Herschel-Bulkley constitutive model; no new physical entities are introduced and the rheological parameters are varied rather than fitted to produce the reported mechanism.

free parameters (2)
  • Power-law index n
    Varied across shear-thinning and shear-thickening regimes to identify distinct reopening routes.
  • Yield stress tau_0
    Central parameter controlling whether the fluid remains unyielded or flows in different regions.
axioms (2)
  • domain assumption Herschel-Bulkley constitutive relation accurately describes the viscoplastic fluid
    Invoked to introduce both yield stress and shear-rate-dependent viscosity absent in simpler Bingham models.
  • domain assumption Flow remains axisymmetric throughout retraction
    Used to reduce the simulation to two dimensions.

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