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This paper establishes that elastic deformation below the yield stress re-enables the Rayleigh-Plateau instability in yield-stress filaments, leading to beads-on-a-string breakup.

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2026-08-03 14:13 UTC pith:KU2DILP4

load-bearing objection Sub-yield elasticity re-enabling Rayleigh-Plateau in yield-stress filaments is a solid, internally consistent result; the bead-anatomy classification is honest but non-asymptotic and should be read as suggestive, not definitive. the 2 major comments →

arxiv 2512.21059 v2 pith:KU2DILP4 submitted 2025-12-24 physics.flu-dyn

Rayleigh-Plateau instability of an elasto-viscoplastic filament

James D. Shemilt , Neil J. Balmforth This is my paper
classification physics.flu-dyn MSC 76A0576A1076E17 PACS 47.20.Dr47.50.-d
keywords Rayleigh-Plateau instabilityyield stresselasto-viscoplastic fluidslender-thread modelbeads-on-a-stringWeissenberg numberpinch-offplastocapillarity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper shows that a finite yield stress does not, by itself, protect a fluid filament from the Rayleigh-Plateau instability that breaks liquid threads into drops. If the material can deform elastically below its yield stress, the uniform thread becomes linearly unstable once the Weissenberg number W exceeds a critical value, and this threshold is independent of the yield stress. When elastic stresses at the thinnest sections then exceed the yield stress, the fluid flows plastically there and the thread thins toward pinch-off, producing beads connected by strings. The bead shapes fall into three categories depending on the yield stress: fully yielded, partially yielded, or elastic with plastic collars. The result matters for processing and spraying yield-stress fluids, where breakup is often assumed to be suppressed by the yield stress.

Core claim

The paper's central claim is that the classical Rayleigh-Plateau instability, which a pure viscoplastic description completely suppresses for a uniform filament, reappears when sub-yield elastic deformation is included in the constitutive model. For an elasto-viscoplastic thread, the unyielded filament behaves as a soft elastic solid, and small varicose perturbations grow once W > 6/(1-k^2), with W the Weissenberg number and k the disturbance wavenumber. The yield stress J plays no role in this linear stability threshold because infinitesimal perturbations cannot breach the yield condition. Nonlinear growth concentrates stress at the thinnest sections; when those stresses exceed sqrt(3)J, th

What carries the argument

The central object is a slender-thread model for the thread radius h(z,t), axial velocity w, and the normal polymer stresses tau_rr and tau_zz, derived from a constitutive law that switches between elastic and viscoplastic behaviour according to a yield condition. In the unyielded state the stress difference obeys tau_rr - tau_zz = W^{-1}(h^2 - h^{-4}), an elastic stretch relation, and the final steady profile is set by F{h} = min(sqrt(3)J, W^{-1}(h^2 - h^{-4})), where F contains the full curvature. This relation is what turns the elastic Rayleigh-Plateau instability into a plastic pinch-off mechanism, and it underlies the three bead-anatomy regimes.

Load-bearing premise

The bead shapes at late times are computed with a slender-thread model even though the beads are not slender; the paper itself says any non-slender feature is not reliable and the elasto-plastic structures are 'only suggestive'.

What would settle it

A fully axisymmetric 3D simulation (or experiment on a yield-stress fluid with measurable elasticity) of a thread with W above the critical value and moderate J should show whether varicose perturbations grow and whether the final bead has the predicted case A/B/C anatomy. If no instability appears despite W > 6/(1-k^2), or if the bead shapes differ qualitatively from the one-dimensional model's predictions, the central pathway would be contradicted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A uniform viscoplastic filament is linearly stable to infinitesimal varicose perturbations whenever the yield stress is finite and elastic deformation is ignored; including elasticity makes the same filament linearly unstable for W > 6/(1-k^2).
  • The linear stability threshold depends only on elasticity (via W), not on the yield stress, so even a very stiff yield-stress thread can become unstable if it is elastic enough.
  • Once yielding begins, it starts at the thinnest sections and spreads, with the radius there thinning exponentially as exp(-t/(3W)), so breakup occurs, though only after exponentially long times.
  • The final beads-on-a-string structure is not transient: beads evolve toward steady shapes whose anatomy is set by the yield stress, ranging from fully plastic to elastic-cored.

Where Pith is reading between the lines

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

  • A direct consequence the authors leave implicit: in printing or spraying yield-stress fluids, processing at high Weissenberg numbers may generate satellite drops even when the fluid's yield stress is high enough to suppress breakup under slow flow; the present results give a stability threshold to predict the onset.
  • The exponential string thinning implies that true pinch-off takes exponentially long times; the authors note that adding power-law plastic viscosity or finite polymer extensibility yields finite-time pinch-off. One could test whether those effects also change the bead anatomy categories.
  • The distinction between cases A, B, and C is made within a one-dimensional model that the paper itself flags as unreliable for order-unity aspect ratios; full axisymmetric simulations would test whether real beads show the predicted elastic cores and plastic collars.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper derives a one-dimensional slender-thread model for an elasto-viscoplastic (Saramito) fluid and uses it to study the capillary instability of a uniform cylindrical filament. The central result is that a finite yield stress, which suppresses the Rayleigh–Plateau instability in a purely viscoplastic thread, no longer does so when sub-yield elasticity is present: linear stability analysis gives instability for W > Wc = 6/(1-k^2), independent of J. Nonlinear solutions show that yielding first occurs at the thinnest sections, after which the string thins exponentially as exp(-t/(3W)) and the filament approaches a beads-on-a-string structure. The authors classify late-time bead morphologies (fully plastic, partly plastic, elastic-core), compute a regime diagram in (J,W), and show that low solvent viscosity can trigger yielding through elastic oscillations.

Significance. If correct, the paper's central result resolves a nontrivial mechanistic question: yielding alone stabilizes a capillary thread, but the elastic branch of the Saramito model restores the instability and provides a pathway to pinch-off. The paper has clear strengths: the dispersion relation (3.2) is derived rather than fitted; it reduces to the known soft-solid threshold; the yield-onset criterion (6.3) is a falsifiable prediction that is checked against nonlinear numerics; and Appendix A convincingly shows that the added stress diffusion is not affecting the reported dynamics. The main caveat, acknowledged by the authors, is that the late-time bead shapes are computed with a non-asymptotic slender-thread closure, so the detailed anatomy is only suggestive. The central instability pathway is not threatened by this caveat.

major comments (2)
  1. [§5, Eq. (5.6)] The claimed local solution for the fully plastic case does not appear to satisfy the profile equation. With A = √3J and Δ = Z−z, substituting h ∼ sqrt(−4A Δ log Δ) into (2.19) gives F → 0 as Δ → 0, not F = A as required by (5.4) in case A. The two terms in (2.19) individually tend to zero in this limit, so the constant-A balance is not established. This raises a question about the free-boundary problem F = √3J with h → 0 at finite Z: either (5.6) is incorrect or the problem is not well posed as stated. Since the comparisons in Fig. 3(a) and the diagnostics in Fig. 4 use solutions of (5.4), the asymptotic support for the fully plastic bead profile needs to be corrected or explicitly delimited.
  2. [§5 and Figs. 3–5] The bead-anatomy categorization (cases A, B, C) and the quantitative profiles are computed with the full-curvature closure (2.19), which is explicitly non-asymptotic at order-unity aspect ratios. The manuscript itself states that any non-slender feature is not reliable and that the elasto-plastic structures are only suggestive. Because this section is presented as a main result, I would like to see either (i) a clear downgrading of the anatomy claims to an exploratory/conjectural status, or (ii) at least one 3D axisymmetric validation per bead category. The central instability pathway and the yield-onset criterion are not affected by this concern.
minor comments (4)
  1. [§3 and §8] There are several typos: 'outined' in §3, 'Raleigh-Plateau' in §3, 'whch' in §8, 'en routeto' in §7, and 'sufficently' in §7. A careful proofread is needed.
  2. [§5, Eq. (5.4)] The use of Min[...] with the signed stress difference τrr−τzz is not immediately obvious, especially because the negative branch is relevant in the thinning string. Please explain why only the positive branch is used for the bead profiles and how the sign convention is chosen.
  3. [Fig. 4] The definition of bead half-length as the point where h(z,t_end)=2 min(h)=0.04 depends on the termination criterion; this should be stated explicitly in the caption so the aspect-ratio data are reproducible.
  4. [§6, Eq. (6.3)] The yield-onset criterion is evaluated on elastic steady states, while in the nonlinear evolution yielding occurs during the transient. The agreement with the numerical boundary is good, but a short explanation of why the steady-state criterion is valid for the dynamical boundary would be useful.

Circularity Check

0 steps flagged

No significant circularity: the elastic Rayleigh-Plateau threshold and pinch-off pathway are derived from the model equations, with only minor background self-citations and an explicitly non-asymptotic bead-shape caveat.

full rationale

The central derivation chain is self-contained. Section 2 reduces Saramito's constitutive law (2.6) to the slender-thread model (2.17)-(2.18) by long-wave asymptotics; Section 3 linearizes those equations to obtain the dispersion relation (3.2) and instability threshold W > 6/(1-k^2) (3.3) without fitting any parameter. The yield-onset divider (6.3) is computed analytically from elastic steady states of the same model and then compared with independent numerical solutions of the initial-value problem, so it is a consistency check rather than a fitted input. The comparison with Mora et al. [17] for soft solids is an external benchmark. The only explicit limitation is that the late-time bead anatomy is computed with a slender-thread closure that retains full curvature as 'a non-asymptotic device' and is 'only suggestive'; this undermines confidence in secondary bead-shape details but does not reduce the central instability-yield-pinchoff claim to its inputs. The self-citations (e.g., [11], [13], [14]) supply background on yield-stress suppression of similar instabilities and are not load-bearing for the new threshold or dynamics. No circular step is exhibited.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

No physical data are fitted. The central claim rests on a chain of modeling choices: Saramito's EVP law, slender-thread asymptotics, a non-asymptotic full-curvature closure, and unbounded Oldroyd-B polymeric stress. The numerical stress-diffusion coefficient and the initial amplitude are additional hand-set quantities, one verified as negligible and the other not explored. All are stated openly, but the bead-anatomy claims inherit the non-slender limitation.

free parameters (2)
  • stress-diffusion coefficient D = 10^-4
    Added to the stress equations (A.1)-(A.2) to enable stiff numerical integration. Appendix A shows D=1e-4 changes h_max by 3e-5 and h_min by 5e-6 relative to D=0, so it is a numerical regularization rather than a physically fitted parameter.
  • initial perturbation amplitude = 10^-3
    Seeds the linear instability in (2.21). The paper does not survey amplitude dependence, so the nonlinear bead anatomy could, in principle, depend on this chosen value.
axioms (5)
  • domain assumption Saramito constitutive law (2.6)-(2.7): below yield the material is a Kelvin-Voigt elastic solid; above yield it is elasto-viscoplastic with constant plastic viscosity.
    The central mechanism requires sub-yield elastic deformation. If a real fluid instead creeps or has finite polymer extensibility, the predicted instability pathway and pinch-off behaviour change. Section 2.1.
  • standard math Slender-thread long-wave asymptotics: axial flow is plug-like, radial stress components are uniform and equal (tau_rr = tau_thetatheta), and shear stresses are O(epsilon) smaller.
    This reduction produces the 1D model (2.17) used throughout. It is a standard asymptotic premise but is not valid for non-slender bead shapes. Section 2.2.
  • ad hoc to paper The full curvature expression (2.19) is retained as a non-asymptotic closure.
    The authors state that retaining the full curvature is a 'non-asymptotic device' needed for a well-behaved model and to capture spherical beads; it is not justified by the slender asymptotics. Sections 2.3 and 8.
  • domain assumption Infinitesimal perturbations of the uniform unyielded thread stay below the yield threshold, so Y=0 in the linear stability analysis.
    This makes the linear growth independent of J for any finite yield stress. It is valid for the chosen base state of zero deviatoric stress, but it excludes scenarios starting from a pre-yielded or pre-stressed base state. Section 3.
  • domain assumption The Oldroyd-B-type polymer stress can grow without bound, so strings thin exponentially as exp(-t/(3W)) rather than pinching in finite time.
    The paper itself notes that alternative constitutive laws with finite polymer extensibility lead to finite-time pinch-off (Section 8). Thus the 'pathway to pinch-off' is an asymptotic, model-dependent statement.

pith-pipeline@v1.3.0-alltime-deepseek · 15104 in / 16023 out tokens · 163520 ms · 2026-08-03T14:13:18.622657+00:00 · methodology

0 comments
read the original abstract

A slender-thread model is used to explore the Rayleigh-Plateau instability of a filament of elasto-viscoplastic fluid. Without elasticity, a finite yield stress suppresses any linear instability for a filament of constant radius. Including sub-yield elastic deformation permits an elastic Rayleigh-Plateau instability above a critical Deborah number. If stresses over the thinner sections of the thread breach the yield threshold, viscoplastic deformations then drive the filament towards pinch-off. The thread consequently evolves to a beads-on-a-string structure. The elasto-plastic anatomy of the beads is explored and categorized.

Figures

Figures reproduced from arXiv: 2512.21059 by James D. Shemilt, Neil J. Balmforth.

Figure 1
Figure 1. Figure 1: Sketch of the geometry of a thin thread, showing the main model [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sample solution for {W, L, J, 𝛽, R } = {10, 20, 0.2, 1, 0.1}. (a) Snapshots of the thread at the times indicated. The elastic regions are shaded grey when never previously yielded, and blue otherwise. (b) Radius plotted as a density on a space-time diagram, with superposed contours (black) showing the yield surfaces. The white lines indicate the paths taken by a selection of fluid elements; the red lines i… view at source ↗
Figure 3
Figure 3. Figure 3: Numerical solutions showing examples of (a) case A ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Final maximum radius (red) and aspect ratio (blue) of the beads [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Final (a) minimum and (b,c) maximum radii from computations with a range of values of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Maximum radius, and (b) maximum stress difference, from [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Numerical solutions for smaller solvent viscosity, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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