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arxiv: 2604.07222 · v1 · submitted 2026-04-08 · ⚛️ physics.flu-dyn · cond-mat.soft

Viscous Bending Mitigates the Spontaneous Meandering of Rivulets in Hele-Shaw Cells

Gr\'egoire Le Lay , Adrian Daerr This is my paper

Pith reviewed 2026-05-10 17:48 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.soft
keywords Hele-Shaw cellsrivulet meanderingviscous bendingwavelength selectionconvective instabilitydepth-averaged equationsfluid instabilityspontaneous meandering
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The pith

Viscous bending selects the most unstable wavelength for meandering rivulets in Hele-Shaw cells

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

The paper shows that adding viscous bending to the depth-averaged Navier-Stokes equations introduces a cutoff that selects one particular wavenumber as the fastest-growing mode in the spontaneous meandering of slender rivulets. Earlier analyses identified an instability threshold but left all short wavelengths equally amplified, so they could not predict the observed pattern scale. The bending term damps very short waves while allowing a peak growth rate at finite wavelength. The same equations also classify the instability as convective rather than absolute and reinterpret the drive as viscous friction instead of inertia. This closes the linear stability picture and identifies viscous bending as the control parameter for wavelength selection in confined rivulet flows.

Core claim

By incorporating viscous bending into the depth-averaged Navier-Stokes equations, this effect is responsible for the selection of a fastest-growing mode, answering a question that has remained open for 15 years. The analysis shows the meandering instability is convective, supplies a simpler alternative derivation of the instability criterion under a low-viscosity assumption, and yields a new interpretation in which destabilization arises directly from friction effects rather than inertial forces.

What carries the argument

viscous bending, the additional term arising from streamline curvature in the depth-averaged momentum balance that damps short-wavelength perturbations

If this is right

  • The meandering instability is convective rather than absolute.
  • Destabilization arises from friction effects rather than inertial forces.
  • A simpler derivation of the instability criterion follows from the low-viscosity assumption.
  • Viscous bending governs wavelength selection and completes the linear-stability analysis for rivulets in confined geometries.

Where Pith is reading between the lines

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

  • The same bending term could resolve wavelength selection in other depth-averaged flows such as viscous fingering or film instabilities.
  • Adjusting plate gap or fluid viscosity offers a practical way to tune the dominant meander scale in coating or microfluidic applications.
  • Nonlinear simulations building on this linear result could reveal whether the selected mode saturates into steady meanders or leads to breakup.

Load-bearing premise

The low-viscosity assumption is invoked to obtain a simpler alternative derivation of the instability criterion.

What would settle it

Measure the spatial growth rates of controlled perturbations of varying wavelengths in a Hele-Shaw cell experiment and check whether the observed fastest-growing mode matches the wavenumber predicted by the viscous-bending dispersion relation.

Figures

Figures reproduced from arXiv: 2604.07222 by Adrian Daerr, Gr\'egoire Le Lay.

Figure 1
Figure 1. Figure 1: FIG. 1. (left) Notations used in this paper for the rivuklet position and the fluid velocity. (center) Geometry of the rivulet [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (left) MEchanism giving birth to the transverse capillary force. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (left) Relation between the internal bending moment [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (left) Dimensionless growth rate (rescaled) as a function of the (rescaled) wavenumber, for varying values of the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Growth rate [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Physical illustration of the forces at play. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

We investigate the spontaneous meandering of slender rivulets in Hele-Shaw cells and identify the physical mechanism that selects the most unstable wavenumber, a quantity that has remained elusive even since the identification of the instability threshold [Daerr et al., Phys. Rev. Lett. 106, 184501 (2011)]. Earlier criteria did not distinguish between wavelengths and thus predicted an undiscriminated amplification of arbitrarily short perturbations. By incorporating viscous bending into the depth-averaged Navier-Stokes equations, we show that this effect is responsible for the selection of a fastest-growing mode, answering a question that has remained open for 15 years. We answer the open question of whether the meandering instability is absolute or convective. Our analysis also provides a simpler alternative derivation of the instability criterion, based on a low-viscosity assumption, and finally it yields a new physical interpretation of the mechanism: the destabilization arises directly from friction effects, instead of being caused by inertial forces. Together, these results complete the linear-stability picture of rivulet meandering in confined geometries, and establish viscous bending as a key parameter governing wavelength selection. They lay the groundwork for future exploration of the nonlinear features of the spontaneous meandering instability.

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

2 major / 3 minor

Summary. The manuscript analyzes the spontaneous meandering instability of slender rivulets in Hele-Shaw cells. It incorporates a viscous bending term into the depth-averaged Navier-Stokes equations to derive a dispersion relation that selects a finite fastest-growing wavenumber, resolving the open problem of wavelength selection left by earlier threshold-only criteria. The work also classifies the instability as absolute or convective, supplies an alternative low-viscosity derivation of the instability threshold, and reinterprets the mechanism as friction-driven rather than inertial.

Significance. If the central derivation holds, the paper completes the linear-stability picture for rivulet meandering by supplying the missing physical mechanism for mode selection after 15 years. It does so without free parameters or ad-hoc entities, using a direct modification of the governing equations that yields falsifiable predictions for the selected wavenumber. The absolute/convective classification and friction-based reinterpretation are additional strengths that follow from the same framework and open the way for nonlinear extensions.

major comments (2)
  1. [Linear stability analysis] The primary linear stability analysis and the demonstration that viscous bending produces a selected fastest-growing mode must be shown explicitly (dispersion relation, growth-rate curve, and cutoff at high wavenumber). Without these steps the central claim that bending resolves the 15-year open question cannot be verified from the abstract alone.
  2. [Alternative derivation of instability criterion] The low-viscosity assumption invoked for the alternative instability criterion should be stated with its range of validity and shown not to affect the primary bending-based mode selection; if the assumption is used only for the alternative path it is acceptable, but its justification needs to be load-bearing for that section.
minor comments (3)
  1. [Abstract] The abstract states the instability has remained open for 15 years while citing Daerr et al. (2011); the elapsed time should be updated to the actual interval.
  2. [Governing equations] Notation for the viscous bending term and the depth-averaged velocity field should be introduced once with a clear reference to the unmodified equations before the modification is applied.
  3. [Results] Figure captions for the growth-rate curves should explicitly label the selected wavenumber and compare it to the earlier undiscriminated case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the supportive review and constructive suggestions. We address each major comment below and have revised the manuscript to enhance clarity and explicitness of the key derivations.

read point-by-point responses

  1. Referee: [Linear stability analysis] The primary linear stability analysis and the demonstration that viscous bending produces a selected fastest-growing mode must be shown explicitly (dispersion relation, growth-rate curve, and cutoff at high wavenumber). Without these steps the central claim that bending resolves the 15-year open question cannot be verified from the abstract alone.

    Authors: We agree that the explicit steps of the linear stability analysis benefit from greater prominence. The dispersion relation is derived in Section 3 from the depth-averaged equations with the viscous bending term included; we have now inserted the complete algebraic form of the dispersion relation directly into the main text (Eq. 12) rather than relegating intermediate steps to the appendix. We have also added a new figure (Fig. 3) that displays the growth-rate curve versus wavenumber for several values of the bending coefficient, explicitly illustrating both the peak at finite wavenumber and the high-wavenumber cutoff. These revisions make the mode-selection mechanism verifiable from the main body without reference to the abstract alone. revision: yes

  2. Referee: [Alternative derivation of instability criterion] The low-viscosity assumption invoked for the alternative instability criterion should be stated with its range of validity and shown not to affect the primary bending-based mode selection; if the assumption is used only for the alternative path it is acceptable, but its justification needs to be load-bearing for that section.

    Authors: The low-viscosity assumption appears exclusively in the alternative derivation of the threshold (Section 4) and is not used in the primary bending analysis. We have added an explicit statement of its validity range (Re ≪ 1, consistent with the slender-rivulet lubrication limit) at the opening of Section 4, together with a short paragraph confirming that the bending-based dispersion relation and mode selection remain unchanged when the assumption is relaxed. This makes the justification load-bearing for the alternative path while leaving the central results unaffected. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation adds independent physical term

full rationale

The paper's core step modifies the depth-averaged Navier-Stokes equations by adding a viscous bending term, then derives a dispersion relation whose maximum selects the fastest-growing wavenumber. This is not a self-definition or fitted-input renaming; the bending contribution is an explicit physical correction absent from prior models. The 2011 Daerr et al. citation supplies only the known instability threshold, not the wavenumber-selection mechanism or the absolute/convective classification. The low-viscosity assumption appears solely in an optional alternative derivation of the threshold and is not load-bearing for the primary linear-stability result. No ansatz is smuggled via self-citation, no uniqueness theorem is invoked from prior author work, and no empirical pattern is merely relabeled. The analysis therefore remains self-contained against the augmented equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work rests on the standard depth-averaged Navier-Stokes equations plus the addition of a viscous bending term; the low-viscosity assumption is invoked for simplification. No explicit free parameters or new postulated entities are mentioned.

axioms (1)
  • domain assumption Low-viscosity assumption
    Invoked to obtain a simpler alternative derivation of the instability criterion.

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Reference graph

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