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REVIEW 3 major objections 3 minor 66 references

Drawing of Weakly Viscoelastic Fluid Tubes

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper establishes that weak polymer elasticity in tube drawing mostly hastens hole closure and, for negligible inertia, always destabilizes the process.

desk verdict Careful asymptotic derivation of viscoelastic tube drawing with a credible hole-closure mechanism; the 'always destabilizing' stability claim is not checkable from the manuscript without the omitted eigenvalue details or code. read the letter →

arxiv 2411.18117 v1 pith:ZXI7LTDD submitted 2024-11-27 physics.flu-dyn

Diandian Gu , Jonathan J. Wylie , Dongdong He , Yvonne M. Stokes This is my paper · ORCID
classification physics.flu-dyn MSC 76A1076D4576E15
keywords viscoelastictubedrawingGiesekusmodelweaklyasymptoticssecondnormalstressdifferenceholeclosuredrawresonancemicrostructuredopticalfibreslinearstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether adding polymer elasticity to the fluid used to draw microstructured ('holey') optical fibres helps or hurts the two manufacturing goals: keeping the internal hole open and avoiding oscillatory draw resonance. Working in the weakly viscoelastic limit of the Giesekus constitutive model, it derives one-dimensional long-wave equations as small corrections to the well-studied Newtonian tube-drawing flow. The main result is a reversal of intuition: elastic stresses generally make the outlet hole smaller than the Newtonian prediction, and they are always destabilizing when inertia is negligible, unlike viscoelastic solid threads. A small regime—very large inlet holes or draw ratios close to one—does benefit from elasticity, preserving a larger hole. If the paper is right, polymer selection and draw-protocol design for holey fibres must be based on this sign change, not on the familiar pinching-suppression behaviour of elastic jets.

What carries the argument

The load-bearing machinery is a double asymptotic expansion: the usual slender-tube (long-wave) expansion in the aspect ratio $\epsilon$, and then a low-Deborah-number expansion $\psi = \psi^{(0)} + De\,\psi^{(1)}$ around the Newtonian state. The leading-order system reduces to the Newtonian tube equations, and the first-order system determines the corrections $u^{(1)}$, $h^{(1)}$, and $H^{(1)}$. The central diagnostic identity is $h^{(1)}(z=1) = \frac{1}{h^{(0)}(z=1) D} \int_0^1 C^{(1)} dz$, which shows that the outlet hole correction is the integrated elastic correction to the radial-flow strength $C$. The paper rewrites $C^{(1)}$ in Eq. (4.7) as four terms: axial advection of the leading-order second normal stress difference, radial advection, a Giesekus mobility term, and a geometric response term; term 2 dominates for moderate holes, while term 1 dominates near the inlet for very large holes. The stability part uses the adjoint (Fredholm solvability) condition on the linearized perturbation equations to compute the elastic growth-rate correction $\omega_1$.

What would settle it

Run a full two-dimensional numerical simulation of the Giesekus equations for $De \approx 0.1$, $D \approx 1.5$, $Ca \approx 1.8$, $\varphi \approx 0.5$, $Re = 0$, $\alpha = 0$, $\beta = 0.1$: if the outlet hole radius is larger than the Newtonian value, the predicted sign $h^{(1)}(z=1) < 0$ is wrong. Separately, at the critical draw ratio for $Re=0$, compute the real part of the elastic growth-rate correction $\omega_1$: a negative value would disprove the claim that elasticity is always destabilizing when inertia is negligible.

Watch

Extended reading notes

Core claim

To first order in the Deborah number $De$ (the ratio of polymer relaxation time to device transit time), the elastic correction to the outlet hole radius, $h^{(1)}(z=1)$, has the opposite sign from what elasticity-driven pinching suppression would suggest. The sign is set by the elastic correction $C^{(1)}$ to the strength of the surface-tension-driven radial flow, and ultimately by the second normal stress difference induced by the Giesekus stresses. For a tube whose leading-order hole stays open, $C^{(1)}$ is typically negative, so the hole closes faster than in the Newtonian case; the opposite occurs only for large inlet hole ratio $\varphi$ or draw ratio $D$ close to 1, where axial advection of stress near the inlet produces a strong outward flow. In the linear stability problem, the first-order elastic correction $\omega_1$ to the growth rate is positive at the Newtonian critical draw ratio when $Re=0$, so elasticity always destabilizes; for non-zero inertia, $\omega_1$ can be negative or positive depending on $Ca$, $\varphi$, $\alpha$, and $Re$.

Load-bearing premise

The central results are first-order corrections in the Deborah number (relaxation time divided by transit time through the device), so the conclusions hold only when that number is small enough that the neglected quadratic corrections cannot reverse the sign of $h^{(1)}$ or $\omega_1$, and only for inlet hole ratios large enough that the leading-order hole does not close (the paper sets aside $\varphi < 0.5$ and strong surface tension).

Editorial extensions

If this is right

  • For moderate inlet hole ratios, a weakly viscoelastic polymer melt will produce a smaller outlet hole than a Newtonian melt under the same draw conditions, so manufacturing compensation must account for elasticity.
  • For tubes with very large inlet holes and draw ratios close to one, elasticity can enlarge the outlet hole, offering a possible benefit for extrusion of large-air-hole microstructured fibres.
  • At negligible inertia, viscoelasticity lowers the stability margin: the critical draw ratio for draw resonance is reached earlier than the Newtonian prediction.
  • With inertia, the effect is parameter-dependent: elasticity stabilizes for sufficiently large capillary number or sufficiently small inlet holes, and destabilizes otherwise.
  • As the solvent fraction $\beta$ approaches one, all elastic corrections vanish and the Newtonian tube-drawing behaviour is recovered, so the results interpolate cleanly between polymer and Newtonian processing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the paper's mechanism implies that rheological characterisation for holey-fibre polymers should prioritise the second normal stress difference, since hole evolution is independent of the first normal stress difference at this order.
  • Editorial inference: the sign reversal between solid threads and tubes suggests a crossover as hole size shrinks; testing very small $\varphi$ (where the leading-order hole remains open) could locate where elastic destabilisation turns into stabilisation.
  • Editorial inference: direct numerical simulation of the full Giesekus equations for $De$ around 0.05 to 0.1 would test whether the long-wave first-order corrections remain accurate away from the asymptotic limit.
  • Editorial inference: for manufacturing, the paper implies a regime map in $(\varphi, D, Re)$ where elasticity is beneficial; one could optimise toward large-hole, low-draw conditions while using inertia to restore stability.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper derives long-wave asymptotic equations for the drawing of an axisymmetric Giesekus tube at small Deborah number, starting from the full slender geometry and treating De as a perturbation about the Newtonian tube-drawing state of Fitt et al. (2002). The O(De) correction fields are computed from the leading-order solution, and the authors show that the outlet hole correction h(1)(z=1) is typically negative (enhanced hole closure) except for large inlet hole ratios phi or draw ratios close to unity, which they explain through the second normal stress difference and its axial/radial advection. The paper then presents a linear stability analysis of the same base state, reporting that for negligible inertia (Re=0) the elastic correction to the growth rate omega1 has positive real part at the Newtonian critical draw ratio, while for non-zero inertia it can be stabilizing or destabilizing depending on parameters.

Significance. If the results are correct, this is a valuable contribution to the fluid mechanics of microstructured fibre drawing: it gives the first systematic weakly-viscoelastic tube-drawing model, identifies the second normal stress difference as the controlling mechanism, and produces a falsifiable prediction about the sign of the hole-size correction. The steady-state part is transparent and self-contained, and the leading-order reduction correctly matches the known Newtonian equations; the paper also makes a clear contrast with solid-thread viscoelastic drawing, which is mechanistically interesting. The main limitation is that the central stability claim is not independently checkable from the manuscript because the first-order eigenvalue system and the solvability condition for omega1 are omitted, with no code, tables, or explicit adjoint supplied. The steady-state derivation and the physical explanation in Section 4 are strong and, in my assessment, the more reliable part of the paper.

major comments (3)
  1. [§5 (linear stability analysis)] The central claim that "elastic effects are always destabilizing for negligible inertia" (abstract and Section 6) rests entirely on the sign of Re(omega1) at the Newtonian critical draw ratio. However, Section 5 states that "the detailed expressions of these equations are omitted here for the sake of brevity" and that the eigenvalue problem can be derived using Mathematica. The paper supplies neither the first-order eigenvalue equations, the adjoint problem, the Fredholm/solvability algebra, nor any numerical data (tables, convergence studies, code) that would allow the reader to verify the sign of Re(omega1). Because the first-order correction involves a non-self-adjoint eigenvalue problem and integration by parts with boundary terms, a sign error in any boundary contribution would reverse the conclusion. This is a load-bearing verification gap: the abstract's contrast with solid-thread drawing cannot be supported by the manuscript as it stands. I request that the omitted expressions be written out, or the relevant code/data be made available, at least for the representative parameter sets plotted in Figures 12-15.
  2. [§4.1, Figures 3-6] The paper's abstract and conclusions state that elastic effects "enhance hole closure for most parameter values," but the analysis explicitly excludes the hole-closure regime phi < 0.5 and small capillary numbers (text below Eq. (4.3) and Section 4.1). Within the stated domain the claim is supported by the plotted sweeps, but the phrase "most parameter values" is broader than the computed and physically regular regime. The authors should either soften the universality language or, better, state precisely the parameter region over which h(1)(z=1)<0 has been verified, including the De-sufficiency condition needed to prevent O(De^2) terms from changing the sign.
  3. [§5, Eqs. (5.1)-(5.6)] The expansion omega = omega0 + De*omega1 assumes that the eigenvalue perturbation is analytic in De and that the leading-order operator has a simple eigenvalue at the critical point. This is plausible for the plotted cases but not demonstrated; if the eigenvalue is defective or if the critical draw ratio is not isolated, the first-order correction to the growth rate would not be given by the stated solvability condition. The manuscript should at least state the non-defectiveness assumption or verify it numerically for the reported eigenvalues.
minor comments (3)
  1. [§2.1, around Eq. (2.14)] The scaling for Ca in Eq. (2.25) uses the total viscosity eta0 and the small parameter epsilon; this is fine, but the reader should be reminded that Ca depends on epsilon, so the statement that Ca is O(1) in the asymptotic limit is an ordering assumption; a one-line clarification would help.
  2. [§4.2, Eq. (4.7)] The labels 1-4 in Eq. (4.7) are very helpful, but the text describes terms 1 and 2 as arising from axial and radial advection without giving the corresponding signs explicitly for term 2. Since the sign of the integral of C(1) is the crux of the hole-size mechanism, a short sign table or a direct inequality would make Section 4.2 easier to follow.
  3. [Throughout] There are several typographical artifacts, e.g. the LaTeX commands "/greaterorsimilar" in Sections 4.1.1 and 5 should be rendered as symbols, and the reference "Geyling, F. and GM, H. (1980)" appears to have an incomplete author name. A final proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the asymptotic derivation is self-contained; the stability claim rests on omitted algebra, a checkability gap, not a circular step.

full rationale

The derivation chain is self-contained. Section 3 substitutes the two-term expansion "ψ = ψ(0) + Deψ(1) + O(De^2)" into the dimensionless Giesekus equations and boundary conditions, solves the O(De^0) system (3.9)-(3.13), and then inserts those leading-order solutions as known coefficients into the O(De) equations (3.15)-(3.17) and (3.22)-(3.28). The steady-state quantities h(1)(z=1) are obtained by numerically integrating this linear system, and Eq. (4.5) follows from integrating the kinematic condition rather than from assuming the desired output. No parameter is fitted to any target result, and no elastic prediction is used as an input. Section 5 is less transparent: the first-order eigenvalue equations are omitted, with the paper stating "The detailed expressions of these equations are omitted here for the sake of brevity," and omega_1 is asserted to be obtainable via a Mathematica derivation and the Fredholm alternative; this is a verification and reproducibility gap, not a circular step. The self-citations (Stokes et al. 2014, 2019; Wylie et al. 2023) are background Newtonian tube-modelling references and do not supply the viscoelastic conclusion. The scope caveat excluding hole closure for small phi is a stated limitation, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation is self-contained apart from standard constitutive and asymptotic assumptions. No free parameters are fitted to data; the dimensionless groups (Re, Ca, De, beta, alpha, phi, D) are inputs describing the physical setup. No new entities are postulated; the second normal stress difference is a standard rheological quantity.

assumptions (5)
  • domain assumption Slender geometry: epsilon = Hin sqrt(1-phi^2)/L << 1, with Re, Ca, alpha, beta treated as O(1).
    Introduced in Section 2.1 scaling; the entire long-wave reduction relies on this separation of scales.
  • domain assumption Weakly viscoelastic limit: Deborah number De << 1, with a regular perturbation expansion psi = psi(0) + De psi(1) + O(De^2).
    Section 3; every O(De) result depends on this expansion being uniformly valid.
  • domain assumption Giesekus constitutive model (2.4) with mobility factor alpha < 0.5, reducing to Oldroyd-B or UCM in special limits.
    The stress closure is assumed; the physical mechanism is interpreted through second normal stress differences within this model.
  • ad hoc to paper The leading-order Newtonian hole does not close: phi >= 0.5 and capillary number above the hole-closure threshold.
    Explicitly stated in Section 4.1; the perturbation equations are only solved where h(0) remains positive.
  • standard math The linear-stability operator at the critical draw ratio has a simple eigenvalue, so the Fredholm alternative determines omega1.
    Used implicitly in Section 5; no degeneracy check is reported.

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Cite this review

Pith. "Pith review of Drawing of Weakly Viscoelastic Fluid Tubes." pith.science (2026). https://pith.science/paper/ZXI7LTDD

@misc{pith2026241118117,
  author       = {Pith},
  title        = {Pith review of: Drawing of Weakly Viscoelastic Fluid Tubes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZXI7LTDD}},
  note         = {Machine review of arXiv:2411.18117}
}
read the original abstract

We explore the drawing of an axisymmetric viscoelastic tube subject to inertial and surface tension effects. We adopt the Giesekus constitutive model and derive asymptotic long-wave equations for weakly viscoelastic effects. Intuitively, one might imagine that the elastic stresses should act to prevent hole closure during the drawing process. Surprisingly, our results show that the hole closure at the outlet is enhanced by elastic effects for most parameter values. However, the opposite is true if the tube has a very large hole size at the inlet of the device or if the axial stretching is very weak. We explain the physical mechanism underlying this phenomenon by examining how the second normal stress difference induced by elastic effects modifies the hole evolution process. We also determine how viscoelasticity affects the stability of the drawing process and show that elastic effects are always destabilizing for negligible inertia. This is in direct contrast to the case of a thread without a hole for which elastic effects are always stabilizing. On the other hand, our results show that if the inertia is non-zero, elastic effects can be either stabilizing or destabilizing depending on the parameters.

Figures

Figures reproduced from arXiv: 2411.18117 by the authors.

Figure 1
Figure 1. Schematic of the drawing process for a viscoelasti [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Left column: (a) the velocity u (0) 0 , (c) the hole radius h (0), and (e) the outer radius H(0) for a Newtonian fluid versus the axial distance z. Right column: the corrections (b) u (1) 0 , (d) h (1), and (f) H(1) to the Newtonian flow due to elastic effects. The parameters are D = 1.5, Ca = 1.8, Re = 0, φ = 0.5, α = 0, and β = 0.1. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. The effect of elasticity on the hole size at the outle [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: The effect of elasticity on the hole size at the outle [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: The effect of elasticity on the hole size at the outle [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: The effect of elasticity on the hole size at the outle [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: The effect of elasticity on the hole size at the outle [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: The effect of elasticity on the hole size at the outle [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: The effect of elasticity on the hole size at the outle [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: (a) The quantity C (1), (b) the terms ○1 − ○4 in (4.7) , and (c) h (1) versus z for D = 1.5, Ca = 1.8, φ = 0.5, Re = 0, α = 0, β = 0.1. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: (a) The quantity C (1), (b) the terms ○1 − ○4 , and (c) h (1) versus z for D = 1.5, Ca = 1.8, φ = 0.95, Re = 0, α = 0, β = 0.1. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]
Figure 12
Figure 12. Figure 12: (a) Real(ω0) versus draw ratio D for various values of the Reynolds number Re. (b) The elastic contribution to the growth rate Real(ω1) versus D for various Re. Other parameters are Ca = 2, φ = 0.5, α = 0, β = 0. For each curve, there is a value of D marked by a black…
Figure 13
Figure 13. Figure 13: Real(ω0) versus draw ratio D for various values of the capillary number Ca for (a) Re = 0 and (c) Re = 0.01, respectively. The elastic contribution to growth rate Real(ω1) versus D for various Ca for (b) Re = 0 and (d) Re = 0.01, respectively. Other parameters are φ =…
Figure 14
Figure 14. Figure 14: Real(ω0) versus draw ratio D for various values of φ for (a) Re = 0 and (c) Re = 0.1, respectively. The elastic contribution to growth rate Real(ω1) versus D for various φ with (b) Re = 0 and (d) Re = 0.1, respectively. Other parameters are Ca = 2, α = 0, β = 0. For e…
Figure 15
Figure 15. Figure 15: Real(ω1) versus draw ratio D for various values of the mobility factor α for (a) Re = 0 and (b) Re = 0.1. Other parameters are φ = 0.5, Ca = 2, β = 0. Besides, with these parameters, Real(ω0) are plotted against D in [PITH_FULL_IMAGE:figures/full_fig_p035_15.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.