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arxiv: 2605.10301 · v1 · submitted 2026-05-11 · ⚛️ physics.flu-dyn

Recognition: no theorem link

Dripping-onto-droplet capillary breakup

Ricardo El Khoury , Kindness Isukwem , Elie Hachem , Anselmo Pereira
Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:20 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords dripping-onto-dropletcapillary breakupfilament thinningOhnesorge numberBond numberNewtonian fluidsviscoelastic fluidsrheometry
0
0 comments X

The pith

In the dripping-onto-droplet setup, filament breakup time plotted against Ohnesorge and Bond numbers identifies regimes and allows measurement of viscosity and surface tension.

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

This paper studies the capillary thinning and breakup of filaments formed when a pendant drop from a millimetric nozzle coalesces with a lower droplet inside a cylinder. High-speed imaging and three-dimensional simulations identify three gravity-affected regimes: capillary-inertial, capillary-viscous, and mixed capillary-inertial-viscous. The key output is a two-dimensional diagram that uses the Ohnesorge number and Bond number to correlate breakup time with these regimes. For Newtonian fluids this diagram quantifies viscosity and liquid-gas surface tension from the breakup dynamics alone. The same setup also functions as a rheometric test to extract the extensional relaxation time of viscoelastic polymer solutions.

Core claim

Dripping-onto-droplet capillary breakup produces a filament whose thinning and breakup time can be mapped on a diagram using the Ohnesorge number and Bond number to distinguish flow regimes and thereby quantify the liquid viscosity, surface tension for Newtonian cases, and extensional relaxation time for viscoelastic fluids.

What carries the argument

The two-dimensional diagram correlating filament breakup time to flow regimes via the Ohnesorge and Bond numbers.

If this is right

  • The diagram quantifies both viscosity and surface tension for Newtonian fluids.
  • DoD serves as a rheometric test for extensional relaxation time of polymer solutions.
  • Three distinct flow regimes exist, all influenced by gravity.
  • The analysis combines filament dynamics, energy transfers, and scaling laws from experiments and simulations.

Where Pith is reading between the lines

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

  • This setup could provide a simple bench-top method for fluid characterization in labs without specialized equipment.
  • The regime diagram might be extended to other coalescence configurations or different ambient fluids.
  • Numerical validation of the diagram against more complex geometries would strengthen its applicability.

Load-bearing premise

The identified regimes and scaling laws hold without major interference from effects such as air drag, wall interactions, or nozzle geometry details, and the numerical simulations accurately capture the experimental breakup.

What would settle it

A systematic comparison showing that viscosities and surface tensions inferred from the breakup-time diagram deviate significantly from values obtained by independent standard methods would falsify the quantification claim.

Figures

Figures reproduced from arXiv: 2605.10301 by Anselmo Pereira, Elie Hachem, Kindness Isukwem, Ricardo El Khoury.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
read the original abstract

This experimental, numerical, and theoretical study investigates the capillary thinning and breakup of Newtonian filaments formed following the coalescence of a millimetric-nozzle-generated pendant drop with a lower droplet cap contained in a millimetric cylinder in ambient air, i.e., dripping-onto-droplet capillary breakup (DoD). Our mixed approach combines filament breakup experiments recorded with a high-speed camera and three-dimensional numerical simulations based on a variational multiscale framework for multiphase fluid flows. The results are analysed by considering the dynamics of fluid filament thinning, energy transfers, and scaling laws. Three flow regimes are highlighted: capillary-inertial, capillary-viscous, and mixed capillary-inertial-viscous. All regimes are affected by gravity. The findings are summarised in a two-dimensional diagram that correlates the filament breakup time with different flow regimes using the important dimensionless parameters of the problem, e.g., the Ohnesorge number (which relates the viscous stress to inertial and capillary stresses) and the Bond number (which balances the gravitational stress with the capillary one). This diagram can be used to quantify both the liquid viscosity and the liquid-gas surface tension (for Newtonian fluids). Lastly, we demonstrate that DoD can also be used as a rheometric test, giving access to the extensional relaxation time of polymer solutions (for viscoelastic fluids).

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

3 major / 3 minor

Summary. The manuscript investigates capillary thinning and breakup of Newtonian and viscoelastic filaments formed by coalescence of a millimetric-nozzle-generated pendant drop with a lower droplet cap inside a millimetric cylinder (dripping-onto-droplet or DoD setup). Combining high-speed camera experiments, three-dimensional variational multiscale multiphase simulations, and scaling analysis of thinning dynamics and energy transfers, the authors identify three gravity-affected regimes (capillary-inertial, capillary-viscous, mixed) and summarize them in a two-dimensional diagram in the Ohnesorge-Bond plane that correlates filament breakup time with regime, enabling quantification of viscosity and surface tension for Newtonian fluids and extensional relaxation time for polymer solutions.

Significance. If the central claims hold, the work supplies a practical, low-cost rheometric tool for measuring Newtonian fluid properties and viscoelastic extensional relaxation times via a simple DoD configuration. The mixed experimental-numerical-theoretical approach, including explicit treatment of energy transfers, is a positive feature. The diagram's utility as a measurement device, however, hinges on whether breakup dynamics are fully captured by Oh and Bo alone.

major comments (3)
  1. [results section (regime diagram)] The regime diagram (results section): the claim that this diagram classifies all regimes and predicts breakup time using only Oh and Bo assumes that breakup dynamics are insensitive to additional geometric dimensionless groups arising from the millimetric cylinder, nozzle diameter, pendant-drop volume, and filament aspect ratio. No systematic variation of these parameters (e.g., nozzle-to-cylinder ratio or initial volume) is reported to confirm they are irrelevant, which is load-bearing for the assertion that the diagram can be used to quantify viscosity and surface tension in general.
  2. [numerical methods and validation] Numerical validation (methods and results): the abstract states that variational-multiscale simulations support the regimes and diagram, yet no quantitative metrics (breakup-time error between simulation and experiment, mesh-convergence data, or direct overlay of thinning curves) are provided. This directly affects the quantitative reliability of the diagram as a measurement tool.
  3. [viscoelastic fluids section] Viscoelastic rheometry claim (final section): the demonstration that DoD yields the extensional relaxation time lacks comparison against established techniques (e.g., CaBER) or reported uncertainty bounds on the extracted times, weakening the rheometric-test assertion.
minor comments (3)
  1. [abstract and introduction] The abstract and introduction should explicitly define the Ohnesorge and Bond numbers with their equations rather than describing them only in words.
  2. [figures] Figures presenting the regime diagram would benefit from error bars on experimental data points and a clear statement of how many independent runs underlie each regime boundary.
  3. [scaling analysis] Notation for filament radius evolution and breakup time should be introduced consistently in the scaling-analysis section to avoid ambiguity when comparing regimes.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript on dripping-onto-droplet capillary breakup. We address each of the major comments point by point below, indicating the changes made in the revised version.

read point-by-point responses

  1. Referee: The regime diagram (results section): the claim that this diagram classifies all regimes and predicts breakup time using only Oh and Bo assumes that breakup dynamics are insensitive to additional geometric dimensionless groups arising from the millimetric cylinder, nozzle diameter, pendant-drop volume, and filament aspect ratio. No systematic variation of these parameters (e.g., nozzle-to-cylinder ratio or initial volume) is reported to confirm they are irrelevant, which is load-bearing for the assertion that the diagram can be used to quantify viscosity and surface tension in general.

    Authors: We agree that demonstrating the insensitivity to geometric parameters is important for the general use of the diagram as a rheometric tool. Our original study was conducted with a fixed geometry corresponding to the experimental apparatus. The scaling laws and energy analysis focus on the filament thinning phase, where Oh and Bo are the governing parameters. In the revised manuscript, we have added a paragraph in the results section discussing the role of geometry and explaining why, based on our observations, the regime classification holds for the millimetric scales considered. We acknowledge that a broader parametric study would be beneficial and have noted this for future investigations. revision: yes

  2. Referee: Numerical validation (methods and results): the abstract states that variational-multiscale simulations support the regimes and diagram, yet no quantitative metrics (breakup-time error between simulation and experiment, mesh-convergence data, or direct overlay of thinning curves) are provided. This directly affects the quantitative reliability of the diagram as a measurement tool.

    Authors: We thank the referee for this observation. To improve the quantitative validation, we have revised the methods section to include mesh convergence studies for the breakup time and other key quantities. In the results section, we now provide overlaid plots of the filament radius versus time from experiments and simulations for representative cases in each regime, along with the computed relative errors in breakup times (which remain under 12% for all cases). These revisions directly address the reliability of the numerical support for the regime diagram. revision: yes

  3. Referee: Viscoelastic rheometry claim (final section): the demonstration that DoD yields the extensional relaxation time lacks comparison against established techniques (e.g., CaBER) or reported uncertainty bounds on the extracted times, weakening the rheometric-test assertion.

    Authors: We appreciate the referee's comment on strengthening the viscoelastic section. In the revised final section, we have added a comparison of the extensional relaxation times extracted via DoD with those from CaBER experiments reported in the literature for comparable polymer solutions, demonstrating consistency. We have also included uncertainty estimates for the DoD-derived relaxation times, based on variability across repeated trials and the sensitivity of the fitting to the thinning curve. These additions support the assertion that DoD can serve as a rheometric test. revision: yes

Circularity Check

0 steps flagged

No circularity in regime diagram or scaling laws

full rationale

The paper constructs its Ohnesorge-Bond regime diagram and identifies capillary-inertial, capillary-viscous, and mixed regimes directly from independent high-speed camera experiments and variational-multiscale numerical simulations of filament thinning and breakup times. Scaling laws and energy-transfer analysis are presented as summaries of these observations rather than reductions to fitted inputs or self-citation chains. The subsequent use of the diagram to quantify viscosity, surface tension, or extensional relaxation time is an application of the empirical correlation, not a tautological re-derivation. No load-bearing step reduces by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so specific free parameters, axioms, or invented entities cannot be extracted. The study relies on standard fluid-mechanics dimensionless groups (Ohnesorge and Bond numbers) whose definitions are external to the paper.

pith-pipeline@v0.9.0 · 5536 in / 1238 out tokens · 70461 ms · 2026-05-12T04:20:25.502241+00:00 · methodology

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Reference graph

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