REVIEW 3 major objections 8 minor 3 references
Spinning Twisted Ribbons: When Two Holes Meet on a Curved Liquid Film
T0 review · 3 major / 8 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read On a curved liquid film, two meeting holes do not collide head-on; their rims cross laterally and the film winds into a spinning twisted ribbon that breaks into droplets.
desk verdict Solid experimental first look at spinning twisted ribbons on curved films; the phenomenon is real, the mechanism is plausible but rests on an untested premise about rim departure. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The carrying object is the parametric ruled-surface model of Eq. (3.2). Each rim is a collection of segments labelled by azimuthal angle $\beta$; after a segment reaches the conjunction plane, it rotates about the axis with angular speed $\omega = u_c\cos\beta/R$ while travelling axially at speed $u_z = u_c\sin\beta$, and the ribbon surface is the ruled surface joining corresponding points on the two rims, with rotation radius $R$, retraction speed $u_c$ and hole-to-plane distance $d$ all measured from experiment, with no fitting parameters. This model produces the conjunction-point positions of Eq. (3.3), the small-mean-curvature estimate of Eq. (3.6), the asymmetric-holes extension of Eq. (3.14), and the outward axial flow speed $u_z/u_c=|z|/(u_c t+d)$ of Eq. (4.1). A complementary two-body central-force model, Eq. (3.15), treats a thin strip of the ribbon as two circular rims pulled together by surface tension $2\gamma$ per unit length, with reduced mass per length $m_\mu=\rho\pi r_1^2 r_2^2/(r_1^2+r_2^2)$, giving open non-circular orbits whose angular and radial periods match measurements. Together, these models carry the argument from the initial lateral crossing to the final droplet ejection.
What would settle it
Use two orthogonally placed high-speed cameras to track individual rim points on a curved film with known thickness and curvature, and check whether the rims depart tangentially from the local surface and then cross laterally; if the measured paths follow the film's curvature and the rims collide head-on, the premise fails. A cleaner variant is to nucleate two holes at controlled staggered times on a curved soap film and test the predicted conjunction-point spacing $\pi R/u_c$ and the axial flow $u_z/u_c=|z|/(u_c t+d)$.
Extended reading notes
Core claim
The central claim is that when two holes meet on a curved liquid film, the film evolves into a spinning twisted ribbon before breaking into droplets, a post-rupture pathway that does not occur on planar films. The rims of the two expanding holes, moving at the Taylor–Culick speed $u_c=\sqrt{2\gamma/(\rho\delta)}$, leave the curved surface because surface tension alone supplies no centripetal force to hold them on it; the rims therefore cross each other laterally rather than colliding head-on. Surface tension of the connecting sheet then makes the crossed rims rotate about the crossing axis, winding the intervening film into a helicoid-like ribbon about one hundred micrometres wide that spins at rates up to roughly 5000 Hz. The paper derives the ribbon's geometric evolution from a ruled-surface kinematic model with no fitting parameters, matches the measured positions of the visible conjunction points over time, and reproduces the open, shrinking rim orbits with a two-body central-force model. It further shows that the spinning motion turns Plateau–Rayleigh or Rayleigh–Taylor corrugations on the rims into ligaments that eject secondary droplets, at lower Weber numbers than the rim-splashing route on flat films.
Load-bearing premise
The load-bearing premise is that a rim of an expanding hole leaves the curved film along a straight tangential path because there is initially no centripetal force to keep it on the curve; the paper infers this departure from the very ribbon it is trying to explain and does not directly measure the rim's trajectory.
Editorial extensions
If this is right
- On curved films, a pair of holes can avoid head-on rim collision altogether; instead the rims cross laterally and the intervening sheet winds into a spinning twisted ribbon, which then fragments into droplets.
- The ribbon surface, the positions of its conjunction points, and the outward axial flow are fixed by the measured Taylor–Culick speed, hole spacing, and rotation radius, so the same equations predict ribbon shape in other curved-film systems.
- Conjunction points are born at regular intervals $\pi R/u_c$ and move toward a limiting speed $u_c$, giving a quantitative clock for how fast the sheet is twisting up.
- The rim orbits are open and non-circular because the central force is distance-independent (a linear potential), placing the spinning ribbon among systems whose orbits differ from both Kepler and harmonic-oscillator orbits.
- Spinning promotes droplet ejection: corrugations with Plateau–Rayleigh or Rayleigh–Taylor wavelength grow into ligaments and pinch off at local Weber numbers around 58, below the threshold above 120 reported for rim splashing on planar films.
Reading between the lines
- Beyond the paper: the same lateral-crossing picture should apply to any curved film with staggered hole nucleation, such as bursting bubbles, liquid shells, or fan-spray sheets, and the scaling $\omega_0 = u_c/R$ gives a direct prediction for the spin rate in those systems.
- Beyond the paper: because simultaneous, mirror-symmetric holes collide head-on instead of forming a ribbon, controlling nucleation timing or local curvature could be a practical switch between collision-dominated and spin-dominated droplet breakup in sprays.
- Beyond the paper: the paper attributes the observed shrinking of the rim orbits to air resistance and changing rim mass; measuring the rim radius and mass loss over time in a single ribbon would test that attribution and could turn the central-force model into a predictive tool for ribbon lifetime.
- Beyond the paper: a direct numerical simulation of two holes on a curved sheet, with the rim treated as a free surface, would test the assumed straight-tangential departure of the rims and reveal the curvature or hole-asymmetry threshold at which ribbons replace head-on collisions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies multiple-hole rupture of the curved crown film in corona splash using high-speed imaging (31,000 fps; 21 drop-impact experiments; 198 holes). The central observation is that when two expanding holes meet on the curved film, their rims cross laterally rather than colliding head-on, and the film between them evolves into a spinning, twisted, helicoid-like ribbon (spin rates up to about 5200 Hz) that subsequently develops corrugations, ligaments, and droplet ejection. The authors attribute the lateral crossing to the absence of an initial centripetal force that would keep the rims on the curved surface (Section 3.1). They then propose three models: a kinematic model (Eq. 3.2) in which the rims expand on two parallel planes separated by 2R and are connected by a ruled surface, reproducing the observed ribbon shape and conjunction-point trajectories (Eq. 3.3, Fig. 4); an asymmetric variant (Eq. 3.14) for unequal holes with a hyperbolic meeting line (Fig. 5); and a two-body central-force model (Eq. 3.15) yielding open, non-circular rim orbits compared with stereoscopic measurements (Figs. 6-7). The corrugations are attributed to Plateau-Rayleigh and/or Rayleigh-Taylor instabilities (Fig. 8), and the outward axial flow predicted by the kinematics (Eq. 4.1) is confirmed by tracking the motion of corrugations (Fig. 9).
Significance. The spinning twisted ribbon is, to my knowledge, a genuinely new rim-interaction morphology; Dombrowski and Fraser (1954) noted a possible 'twist' only in passing, and the present study supplies the first systematic kinematics, instability analysis, and droplet-formation consequences. If the proposed mechanism is confirmed, the result identifies a route to droplet formation distinct from the head-on rim collisions and rim splashing characterised by Neel et al. (2020) and Tang et al. (2024), with plausible implications for sprays, bursting bubbles, and atomization. The paper's strengths should be credited: a solid observational base (198 holes, 21 experiments, stereoscopic imaging for the orbits); checkable claims (supplementary movies, the explicit parametric surface (3.2), an internally verified parallax correction in Appendix B); one genuinely parameter-free prediction, the axial-flow relation (4.1), confirmed in Fig. 9(b); and honest treatment of both the orbit-shrinkage discrepancy (Section 3.4) and the degeneracy between the two instability mechanisms (Section 4.2).
major comments (3)
- [Section 3.1 (with Eq. 3.2 and Section 5)] The causal premise of the paper is asserted rather than tested. Section 3.1 attributes the lateral crossing to rims departing from the curved surface because 'initially, there is no centripetal force to maintain the rims on a curved path', but no measurement of the rim trajectory relative to the original curved surface is reported (the single-hole example in supplementary movie 5 is not quantified), and the kinematic model (3.2) builds the crossing in by prescribing rims that expand on two parallel planes separated by 2R. The conjunction-point agreement in Fig. 4 therefore confirms that a crossing-ribbon kinematics describes the observed geometry, but it does not test why the rims cross: any mechanism producing the same crossing geometry would satisfy Eq. (3.3) equally well. Two consequences follow. First, the claim in Section 5 that the phenomenon 'does not occur in planar films' rests on literature comparison rather than on a control experiment within the same apparatus. Second, the effect size makes the premise quantitatively delicate: with d/R_f < 0.1 (Section 3.1), the out-of-plane sag of the surface between the holes is d^2/(2R_f) of order 50 microns, which is the same order as the measured rotation radii R of 83-111 microns, so the premise is as large as the effect it is invoked to explain. In addition, all experiments are conducted in a corona splash, where the crown is simultaneously moving and stretching, leaving the non-uniform velocity field as an untested alternative cause of lateral crossing. I would ask for (i) direct rim-trajectory measurements relative to the curved surface in the frame of the film (the stereoscopic data used for Fig. 6 may already permit this), or (ii) a flat-film control experiment, or (iii) at minimum an explicit softening of the causal claim to a labelled hypothesis in Sections 3.1 and 5.
- [Section 3.2 (Eq. 3.3, Fig. 4)] The kinematic validation is partly in-sample. The inputs u_c and d are measured from the same video frames that are compared with the model, and R is not measured geometrically but is inferred from the central spin frequency omega_0 = u_c/R, measured from the very ribbon whose shape and conjunction points are then reproduced. The statement 'no fitting parameters are involved' is correct in a narrow sense, but the agreement in Fig. 4 is closer to a self-consistency check of the kinematic description than to an independent test. The non-trivial content of the prediction is the functional form of Eqs. (3.3)-(3.4), including the emergence times t_n = n pi R/u_c and the asymptotic speed u_c; this deserves an out-of-sample check, for example parameters measured at early times applied to later times, or an independent geometric measurement of R (for instance from the ribbon width in the orthogonal stereo view), together with a sensitivity analysis in R and d. The one truly parameter-independent prediction in the paper is the axial-flow relation (4.1), which contains neither R nor an in-sample spin measurement and is confirmed in Fig. 9(b); presenting that relation as the primary validation of the kinematics would be more convincing. The asymmetric model of Section 3.3 is compared with experiment only by visual resemblance (Fig. 5): no quantitative comparison of the hyperbola parameters, of the drift direction, or of the off-midplane structure is provided, and the text itself notes that the drift cannot be measured directly; a quantitative metric would strengthen this section.
- [Section 3.4 (Eq. 3.15, Figs. 6-7)] The corroborative value of the central-force model is limited by its inputs and by the acknowledged discrepancy. The initial conditions (r(0), v_theta(0), v_r(0)) and the reduced linear mass m_mu are measured from the same orbits that the model is asked to reproduce, and the measured orbits shrink with time (Fig. 7b) in a way that the conservative two-body model cannot produce; the text attributes the shrinking to air resistance and rim-mass variation without quantifying either effect. The period comparison in Fig. 7(a) uses three experiments, and the angular and radial periods are computed from a system whose radius drifts by tens of percent over the observation window; without a sensitivity analysis it is unclear how robust the 'reasonably good agreement' is. I would ask the authors to quantify the radial shrinkage (for instance a fitted drift rate), to state how the drift affects T_theta and T_r, to restrict the quantitative comparison to a time window in which the orbit is approximately stationary, or to extend the model with a slow drag or mass-accumulation term. As it stands, the section's main contribution is conceptual, namely the demonstration via Bertrand's theorem that a constant central force yields the open, non-circular orbits observed, and the quantitative period comparison should be softened accordingly.
minor comments (8)
- [Abstract and Section 1] The text twice uses the plural 'this phenomena' where 'this phenomenon' is intended, notably in the sentence 'We compare and contrast this phenomena with its counterpart on planar films'; this should be corrected.
- [Section 3.4] The printed definition of the angular period (T_theta = t_theta 2 pi / Delta theta) appears to be a typesetting error for T_theta = 2 pi Delta t_theta / Delta theta; please correct it for readability.
- [Section 3.2 (Eq. 3.6)] The mean-curvature formula (3.6) appears to contain an exponent error in the printed equation (the placement of alpha^2 and the 3/2 power); please verify the printed expression against the derivation, since the bound H < 1/d in (3.7) is used to support the minimum-surface interpretation.
- [Section 4.1] The claimed facilitation of droplet formation at lower Weber numbers than rim splashing rests on a single local value (We_loc about 58) with an explicit statement that the threshold was not explored; the hedged wording of Section 4.1 is appropriate, and the corresponding sentence in Section 5 should retain the same hedge rather than stating the lower-Weber-number result as a firm finding.
- [Section 3.1] Please clarify the 'ribbons per hole' statistic (0.8 +/- 0.1 over 198 holes): what exactly is counted as a ribbon, and is the quoted uncertainty across experiments or across holes? A brief breakdown of the observed outcomes (ribbon, head-on collision, no interaction) tied to the two stated causes (hole distance, simultaneity) would make the phenomenology of Section 3.1 checkable.
- [Section 2] The film thickness delta is deduced from the measured retraction speed by inverting the Taylor-Culick relation, so any statement that the measured u_c agrees with sqrt(2 gamma/(rho delta)) is partly by construction; this should be acknowledged where u_c is used as an independent input.
- [Section 4.2 (Fig. 8c)] In Fig. 8(c), data for two surface tensions and for spinning and non-spinning conditions are pooled into a single comparison with lambda_PR = 9.0 R_rim; stating the number of segments and the definition of the error bars for each symbol class would help the reader judge the agreement.
- [Section 1 and Section 4.1] The 1954 observation by Dombrowski and Fraser of a twisting ribbon 'at the instant before coalescence of the two rims' is cited as the earliest sighting; since fan-spray sheets are curved, a sentence explicitly reconciling that historical observation with the present claim that the phenomenon is unique to curved films would be useful.
Circularity Check
No significant circularity: the paper's models are parameterized by measured quantities and compared with data and external theory; the Section 3.1 premise is an unverified physical assumption, not a circular reduction.
full rationale
The central derivation chain is not circular. The kinematic model (3.2) assumes the lateral crossing geometry and then predicts the ribbon surface and conjunction-point positions (3.3) using u_c, d, and R measured from the same video frames. This is an in-sample consistency check rather than a fully independent prediction, but it is not a forced reduction: the functional form of (3.3) is nontrivial and is not used to determine R or d, and the measured parameters are stated explicitly with parallax corrections. Similarly, the central-force model (3.15) uses measured initial conditions and liquid properties to predict open, non-circular orbits, and the periods are then compared with separately measured values; the constants are not fitted to the period data. Independent support includes the axial-flow relation (4.1), the Plateau–Rayleigh and Rayleigh–Taylor wavelength comparisons, and the use of Bertrand's theorem from standard external mechanics texts. Self-citations in the paper (e.g., Thoroddsen et al. 2006; Aljedaani et al. 2018) are background references for corona splash and are not load-bearing for the ribbon mechanism. The Section 3.1 claim that rims depart tangentially from the curved film because there is 'no centripetal force' is an unmeasured physical premise, but the kinematic model does not purport to derive that premise from itself; it assumes the crossing and is then tested against observation. That is a scientific limitation that should be evaluated as a correctness or evidence concern, not as circularity by construction. No self-citation uniqueness theorem, ansatz-smuggling, or renaming of a known result is present.
Assumptions & free parameters
free parameters (2)
- Rotation radius R (inferred from central spin frequency omega_0) =
82.8 um (Fig. 2); 111 um (Fig. 5)
- Central-force model initial conditions and linear mass =
e.g., r(0)=841 um, v_theta(0)=4.45 m/s, v_r(0)=0, m_mu=586 ug/m (Fig. 6e)
assumptions (6)
- domain assumption Holes in thin free films expand at constant Taylor-Culick speed u_c = sqrt(2*gamma/(rho*delta)).
- ad hoc to paper Rims depart from the curved surface along straight tangential paths because surface tension acts tangentially and provides no initial centripetal force.
- ad hoc to paper Each rim segment, once at the conjunction plane, rotates at constant radius R with angular speed omega = u_c cos(beta)/R and axial speed u_z = u_c sin(beta).
- ad hoc to paper The ribbon surface is a ruled surface obtained by linearly connecting corresponding points of the two rims.
- domain assumption The central attractive force per length between the rims is 2*gamma (two liquid-air interfaces), independent of separation.
- standard math Classical two-body reduction to relative motion with reduced mass per length m_mu.
Cite this review
Pith. "Pith review of Spinning Twisted Ribbons: When Two Holes Meet on a Curved Liquid Film." pith.science (2026). https://pith.science/paper/3XFXECLH
@misc{pith2026241110562,
author = {Pith},
title = {Pith review of: Spinning Twisted Ribbons: When Two Holes Meet on a Curved Liquid Film},
year = {2026},
howpublished = {\url{https://pith.science/paper/3XFXECLH}},
note = {Machine review of arXiv:2411.10562}
}
read the original abstract
The rupture of a liquid film, where a thin liquid layer between two other fluids breaks and forms holes, commonly occurs in both natural phenomena and industrial applications. The post-rupture dynamics, from initial hole formation to the complete collapse of the film, are crucial because they govern droplet formation, which plays a significant role in many applications such as disease transmission, aerosol formation, spray drying nanodrugs, oil spill remediation, inkjet printing, and spray coating. While single-hole rupture has been extensively studied, the dynamics of multiple-hole ruptures, especially the interactions between neighboring holes, are less well understood. Here, this study reveals that when two holes 'meet' on a curved film, the film evolves into a spinning twisted ribbon before breaking into droplets, distinctly different from what occurs on flat films. We explain the formation and evolution of the spinning twisted ribbon, including its geometry, corrugations, ligaments, and orbits, and compare the experimental observations with models. We compare and contrast this phenomena with its counterpart on planar films. While our experiments are based on the multiple-hole ruptures in corona splash, the underlying principles are likely applicable to other systems. This study sheds light on understanding and controlling droplet formation in multiple-hole rupture, improving public health, climate science, and various industrial applications.
Figures
Figures from the paper (9 more)
Reference graph
Works this paper leans on
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[4]
and decreases towards the two ends. Focusing on the region near the centre plane, we take ω → ω 0 as the angular speed at z = 0 measured before the corrugations are observed. Correspondingly, the measured wavelengths are calculated using one or two segments nearest to the centre. The plot of the measured wavelengths versus the capillary length l c = γ /(ρ...
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[6]
The blue arrows highlight the newly emerged corrugations at later times
of the twisted ribbon, confirming the existence of the outward axial flow. The blue arrows highlight the newly emerged corrugations at later times. ( b ) The speed of the outward axial flow u z measured at different conditions as shown in the legend. See also supplementary movie 7 . Bo = ρ aR 2 rim /γ is the local rim Bond number, in which the rim radius R r...
work page 2018
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[259]
The measured ratio and the calculated ratio are very close, verifying that ( B4 ) is correct. R EFERENCES A GBAGLAH , G.G. 2021 Breakup of thin liquid sheets through hole–hole and hole–rim merging. J. Fluid Mech. 911 , A23. A LJEDAANI , A.B., W ANG , C., J ETLY , A. & T HORODDSEN , S.T. 2018 Experiments on the breakup of drop- impact crowns by Marangoni h...
Reviewed August 12, 2026 · model on record in the stance chip above.
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