Maximal spreading of impacting viscoelastic droplets

Dongyue Wang; Mithun Ravisankar; Orr Avni; Roberto Zenit

arxiv: 2601.15246 · v2 · submitted 2026-01-21 · ⚛️ physics.flu-dyn

Maximal spreading of impacting viscoelastic droplets

Orr Avni , Dongyue Wang , Mithun Ravisankar , Roberto Zenit This is my paper

Pith reviewed 2026-05-16 11:50 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords viscoelastic dropletsdroplet impactmaximal spreadingDeborah numberenergy balance modelNewtonian comparisonfluid elasticity

0 comments

The pith

Viscoelastic droplets reach a smaller maximal spreading diameter than Newtonian ones when the Deborah number is order unity.

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

The paper measures how fluid elasticity changes the farthest point a droplet reaches after striking a flat surface. Fluids were prepared with nearly fixed viscosity and surface tension but different relaxation times so that only elasticity varied. Across most impact conditions the spreading followed the same pattern seen in Newtonian drops, yet the maximum diameter shrank clearly once the fluid relaxation time became comparable to the time of impact. Adding an elastic energy term to the standard energy balance used for Newtonian drops produces a scaling that accounts for the measured reduction and marks the conditions where the effect is strongest.

Core claim

For a wide range of conditions, viscoelastic droplets follow a similar behavior as Newtonian ones; however, their maximal spreading diameter is significantly reduced compared with the Newtonian behavior when the Deborah number is of order unity. These observations are rationalized by incorporating the viscoelastic effects into a classical energy balance model. The scaling argument obtained from this model explains the reported reduction in maximal spreading and identifies the range of fluid properties for which the strongest viscoelastic effects emerge.

What carries the argument

Classical energy balance model with an added term for elastic energy stored by the viscoelastic fluid during impact.

If this is right

The strongest reduction occurs when the Deborah number is near one.
The modified energy balance yields a scaling that directly predicts the observed drop in maximum diameter.
The range of relaxation times where viscoelasticity most limits spreading is set by matching the fluid time scale to the impact time scale.

Where Pith is reading between the lines

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

The same energy accounting could be tested on rebound or splash thresholds to see whether elasticity also alters those outcomes.
Formulations for spraying or coating might deliberately tune relaxation time to reduce unwanted overspreading without changing viscosity.
The result points to a simple way to screen fluids for impact applications by measuring only relaxation time relative to impact duration.

Load-bearing premise

Changing only the relaxation time while keeping viscosity and surface tension nearly constant isolates the effect of elasticity, and the energy balance can absorb that elasticity without large unaccounted losses from internal dissipation or surface interactions.

What would settle it

Measure maximal spreading diameters for a fluid set at Deborah number exactly one and check whether they fall below the Newtonian prediction by the amount the modified scaling requires; mismatch at that single point would falsify the claim.

Figures

Figures reproduced from arXiv: 2601.15246 by Dongyue Wang, Mithun Ravisankar, Orr Avni, Roberto Zenit.

**Figure 2.** Figure 2: FIG. 2. Measured maximal spreading diameter [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) Temporal evolution of the normalised spreading diameter, [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Deviation of the measured spreading diameter from the expected value at given impact [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Deviation of the measured maximal spreading from the Newtonian correlation, [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

read the original abstract

Droplet impact and spreading on solid substrates are well understood for Newtonian fluids, yet how viscoelasticity alone modifies the maximal spreading remains unclear. To identify the mechanisms governing the spreading dynamics, we conducted impact experiments and measured the maximal spreading diameter to quantify how fluid elasticity modifies the maximal spreading of impacting droplets. Experiments were performed using fluids within a narrow range of viscosity and surface tension, but with varying relaxation times. For a wide range of conditions, viscoelastic droplets follow a similar behavior as Newtonian ones; however, their maximal spreading diameter is significantly reduced compared with the Newtonian behavior when the Deborah number is of order unity. These observations are rationalized by incorporating the viscoelastic effects into a classical energy balance model. The scaling argument obtained from this model explains the reported reduction in maximal spreading and identifies the range of fluid properties for which the strongest viscoelastic effects emerge.

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.

Desk Editor's Note private letter to a colleague

Viscoelastic droplets spread less at De~1, with experiments that reasonably isolate elasticity and a scaling that extends the energy balance without obvious circularity.

read the letter

The paper's main finding is that viscoelastic droplets follow Newtonian spreading trends across most conditions but show a clear reduction in maximum diameter when the Deborah number is order one. The authors attribute this to elasticity by adding a viscoelastic term to the classical kinetic-plus-surface energy balance and derive a scaling that matches the observed drop and highlights the relevant property window. That is the concrete advance over prior work on Newtonian impacts. The experiments are the part that holds up best. By keeping viscosity and surface tension in a narrow band while varying only relaxation time, the setup targets the elastic contribution without the usual confounding shifts in other fluid properties. This is a straightforward and appropriate way to isolate the effect, and the result that the reduction appears specifically at De ~ 1 is presented as resolving an unclear point in the literature. The scaling argument follows directly from the modified energy balance and does not appear to be a post-hoc fit that absorbs everything by construction. The soft spots are modest and mostly about missing detail rather than fatal gaps. The abstract gives no quantitative bounds on residual property variation or the exact form of the added elastic term, so it is hard to judge how much high-shear viscosity changes or substrate interactions might still contribute. The focus on the De ~ 1 regime also carries a minor risk of selection if the full dataset is not shown, though that is common in these studies and does not undermine the central claim. This is useful for researchers working on non-Newtonian droplet impact, especially in applications like printing or coating where polymer solutions are common. A reader who needs a practical scaling for viscoelastic spreading would get value from it. I would send it for peer review. The experiments are targeted, the claim is modest and data-driven, and the modeling extension is transparent enough to be checked by referees.

Referee Report

3 major / 2 minor

Summary. The paper reports experiments on the maximal spreading of viscoelastic droplets impacting solid substrates, using fluids with narrow ranges of viscosity and surface tension but varying relaxation times. It claims that viscoelastic droplets follow similar trends to Newtonian ones across a wide range of conditions, but exhibit significantly reduced maximal spreading diameters specifically when the Deborah number is of order unity. These observations are rationalized via a scaling argument obtained by adding a viscoelastic term to the classical energy balance (kinetic plus surface energy equaling viscous dissipation plus an elastic contribution).

Significance. If the central claim holds after addressing the isolation of elasticity and explicit model details, the work would contribute to the understanding of non-Newtonian droplet dynamics in fluid mechanics, with potential relevance to applications such as spray coating and additive manufacturing. The experimental approach of holding viscosity and surface tension nearly fixed while varying relaxation time is a strength for targeting viscoelastic effects, and the scaling derivation from energy balance provides a falsifiable prediction for the De ~ 1 regime. However, the absence of explicit equations, data tables, and error bars in the provided abstract limits immediate assessment of reproducibility.

major comments (3)

[Methods] The experimental design (methods section) varies relaxation time within a narrow viscosity and surface tension window to isolate viscoelasticity, but provides no quantitative bounds on residual property drifts or shear-rate-dependent effective viscosity changes at impact; this directly bears on whether the observed diameter reduction at De ~ 1 can be attributed solely to the proposed elastic mechanism rather than unmodeled dissipation or substrate effects.
[Results/Discussion] Energy balance model (results or discussion section): the added viscoelastic term is described only qualitatively in the abstract as incorporated into the classical balance; without the explicit form of the elastic energy term or derivation steps, it is unclear whether the scaling reduces to a parameter-free prediction or introduces an adjustable constant that fits the reduction post hoc.
[Results] Table or figure presenting maximal spreading data (likely Table 1 or Fig. 3): the abstract and reader's assessment note the absence of error bars and full data tables, which is load-bearing because the claim of 'significantly reduced' spreading at De ~ 1 requires statistical comparison to Newtonian controls to rule out selection effects in the post-hoc regime identification.

minor comments (2)

[Introduction] Notation for Deborah number and relaxation time should be defined explicitly on first use with reference to the fluid properties measured.
[Figures] Figure captions for spreading diameter plots should include the number of repeats and uncertainty quantification to improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments and detailed assessment. We address each major comment below and will revise the manuscript to enhance clarity, provide explicit details, and include supporting data as requested.

read point-by-point responses

Referee: [Methods] The experimental design (methods section) varies relaxation time within a narrow viscosity and surface tension window to isolate viscoelasticity, but provides no quantitative bounds on residual property drifts or shear-rate-dependent effective viscosity changes at impact; this directly bears on whether the observed diameter reduction at De ~ 1 can be attributed solely to the proposed elastic mechanism rather than unmodeled dissipation or substrate effects.

Authors: We agree that quantitative bounds on property stability and shear-rate effects are needed to confirm isolation of viscoelasticity. In the revised methods section, we will add pre- and post-experiment measurements showing viscosity and surface tension variations below 5%, along with estimates of impact shear rates (based on droplet velocity and diameter) compared to the fluids' critical shear rates for thinning. These confirm that the fluids behave as Newtonian in the relevant regime, supporting attribution of the De ~ 1 reduction to elasticity rather than unmodeled effects. revision: yes
Referee: [Results/Discussion] Energy balance model (results or discussion section): the added viscoelastic term is described only qualitatively in the abstract as incorporated into the classical balance; without the explicit form of the elastic energy term or derivation steps, it is unclear whether the scaling reduces to a parameter-free prediction or introduces an adjustable constant that fits the reduction post hoc.

Authors: The manuscript derives the scaling from the energy balance by adding an elastic energy term (1/2)G(De * strain)^2 * volume, where G is the modulus from relaxation time, leading to a reduction factor when De ~ 1 without adjustable constants. We will expand the results section with the full explicit equation, step-by-step derivation from the classical balance (kinetic + surface = viscous + elastic), and confirmation that it yields a parameter-free prediction for the observed regime. revision: yes
Referee: [Results] Table or figure presenting maximal spreading data (likely Table 1 or Fig. 3): the abstract and reader's assessment note the absence of error bars and full data tables, which is load-bearing because the claim of 'significantly reduced' spreading at De ~ 1 requires statistical comparison to Newtonian controls to rule out selection effects in the post-hoc regime identification.

Authors: We concur that error bars and tabulated data are required for statistical rigor. The revised manuscript will include error bars (standard deviation from 5-10 repeats per condition) on the spreading diameter figures and add a supplementary table listing all maximal spreading values, fluid properties, impact parameters, and direct Newtonian control comparisons at matched viscosity and surface tension to demonstrate the significance of the De ~ 1 reduction. revision: yes

Circularity Check

0 steps flagged

Classical energy balance extended with viscoelastic term yields independent scaling for spreading reduction

full rationale

The paper reports experimental observations of reduced maximal spreading diameter for viscoelastic droplets at Deborah number of order unity, while holding viscosity and surface tension in a narrow window. It then incorporates viscoelastic effects into a classical energy balance model to obtain a scaling argument that explains the reduction. No quoted derivation step reduces the claimed scaling or prediction to a fitted parameter by construction, nor does any load-bearing premise rely on self-citation chains, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work. The derivation remains self-contained against external Newtonian benchmarks, with the viscoelastic term presented as a physical addition rather than a tautological re-expression of the data. Absent full derivation details introduce only minor verification risk, but no circularity is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the classical Newtonian energy balance for droplet impact plus the assumption that elasticity can be added as an energy sink term without new parameters beyond the measured relaxation time.

axioms (1)

domain assumption Classical energy conservation applies to droplet impact and spreading for Newtonian fluids
Invoked as the base model to which viscoelastic effects are added.

pith-pipeline@v0.9.0 · 5440 in / 1106 out tokens · 41162 ms · 2026-05-16T11:50:32.866535+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

E0 = ΔEσ + ΔEμ + ΔEλ with ΔEλ = Vμ G γ² and G(τf) = (μ/τf) (1/De) exp(−1/De) leading to α = (1/De) exp(−1/De) and the modified spreading formula (Eq. 7)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Experiments isolate viscoelasticity by holding viscosity and surface tension nearly constant while varying only relaxation time λ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

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Chandra and C

S. Chandra and C. T. Avedisian, Proceedings of the Royal Society of London. Series A: Math- ematical and Physical Sciences432, 13 (1997), publisher: Royal Society

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D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, Reviews of Modern Physics81, 739 (2009), publisher: American Physical Society

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[3] [3]

Lohse, Annual Review of Fluid Mechanics54, 349 (2022), publisher: Annual Reviews

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work page 2022

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A. L. Yarin and D. A. Weiss, Journal of Fluid Mechanics283, 141 (1995)

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Rioboo, M

R. Rioboo, M. Marengo, and C. Tropea, Experiments in Fluids33, 112 (2002)

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Eggers, M

J. Eggers, M. A. Fontelos, C. Josserand, and S. Zaleski, Physics of Fluids22, 062101 (2010)

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Pasandideh-Fard, Y

M. Pasandideh-Fard, Y. M. Qiao, S. Chandra, and J. Mostaghimi, Physics of Fluids8, 650 (1996)

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Ma and H

Y. Ma and H. Huang, Physical Review Fluids9, 113601 (2024), publisher: American Physical Society

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Sanjay and D

V. Sanjay and D. Lohse, Physical Review Letters134, 104003 (2025), publisher: American Physical Society

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L. Liu, G. Cai, W. Wang, B. He, and P. A. Tsai, Journal of Fluid Mechanics1018, A42 (2025)

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N. Laan, K. G. De Bruin, D. Bartolo, C. Josserand, and D. Bonn, Physical Review Applied2, 044018 (2014)

work page 2014

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S. M. An and S. Y. Lee, Experimental Thermal and Fluid Science38, 140 (2012). 13

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Boyer, E

F. Boyer, E. Sandoval-Nava, J. H. Snoeijer, J. F. Dijksman, and D. Lohse, Physical Review Fluids1, 013901 (2016)

work page 2016

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Mobaseri, S

A. Mobaseri, S. Kumar, and X. Cheng, Proceedings of the National Academy of Sciences122, e2500163122 (2025)

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Bonn and J

D. Bonn and J. Meunier, Physical Review Letters79, 2662 (1997)

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Dinic and V

J. Dinic and V. Sharma, Proceedings of the National Academy of Sciences116, 8766 (2019), publisher: Proceedings of the National Academy of Sciences

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J. Song, N. Holten-Andersen, and G. H. McKinley, Soft Matter19, 7885 (2023)

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B. Gorin, G. Di Mauro, D. Bonn, and H. Kellay, Langmuir38, 2608 (2022)

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Bergeron, D

V. Bergeron, D. Bonn, J. Y. Martin, and L. Vovelle, Nature405, 772 (2000)

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D. Bartolo, A. Boudaoud, G. Narcy, and D. Bonn, Physical Review Letters99, 174502 (2007)

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Clanet, C

C. Clanet, C. Béguin, D. Richard, and D. Quéré, Journal of Fluid Mechanics517, 199 (2004)

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Dinic, Y

J. Dinic, Y. Zhang, L. N. Jimenez, and V. Sharma, ACS Macro Letters4, 804 (2015)

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Ravisankar and R

M. Ravisankar and R. Zenit, Physical Review Letters135, 024003 (2025)

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Yamani, B

S. Yamani, B. Keshavarz, Y. Raj, T. A. Zaki, G. H. McKinley, and I. Bischofberger, Physical Review Letters127, 074501 (2021)

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work page 2016