Oscillatory collective motion in viscoelastic and elastic active fluids and solids under circular confinement

Andreas M. Menzel; Henning Reinken

REVIEW 3 major objections 4 minor 16 references

Oscillatory collective motion in viscoelastic and elastic active fluids and solids under circular confinement

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Two oscillatory modes of collective motion seen in elastic active solids also emerge in viscoelastic active fluids, making the phenomena generic to confined active biomaterials.

desk verdict Plausible numerical extension of Xu et al. modes to viscoelastic fluids, but the genericity claim is overbroad and the dropped advection term in Eq. (3) needs a defense. read the letter →

arxiv 2502.06312 v1 pith:JBQND6YZ submitted 2025-02-10 cond-mat.soft physics.bio-phphysics.flu-dyn

Henning Reinken , Andreas M. Menzel This is my paper · ORCID

classification cond-mat.softphysics.bio-phphysics.flu-dyn

keywords activematterviscoelasticfluidssolidscollectivemotionoscillatorymodesbiofilmscontinuumfieldtheorycircularconfinement

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 sets out to show that two oscillatory modes of collective motion observed in confined active biofilms—a rotation that periodically reverses direction and a uniform translation whose direction rotates—are not a special feature of elastic active solids. Working with a minimal continuum theory that couples polar order, flow, and a displacement field with a finite memory-relaxation time, the authors find both modes in viscoelastic active fluids over a broad parameter range, with elastic solids recovered only in the infinite-memory limit. The practical significance is that these phenomena should be expected in a wide class of biological fluids, soft materials, and biomaterials, and that a biofilm's solid-looking dynamics do not by themselves demonstrate solid elasticity.

What carries the argument

The load-bearing object is the displacement field $\mathbf{u}$ with relaxation dynamics $\partial_t \mathbf{u} = \mathbf{v} - \tau_d^{-1}\mathbf{u}$. The parameter $\tau_d$ is the memory-relaxation time: $\tau_d \to \infty$ recovers a perfectly elastic solid whose history never fades, finite $\tau_d$ describes a viscoelastic fluid that ultimately flows, and $\tau_d \to 0$ is a purely viscous fluid. This single field lets one equation interpolate between solid and fluid behavior: it enters an extended Stokes equation balancing viscous stress, elastic stress from $\mathbf{u}$, substrate friction and elastic restoring forces, and active driving. Together with a confinement force on the normal displacement at the circular boundary, this is what generates the two oscillatory modes and sets their frequency and amplitude; the paper's central move is to show that varying $\tau_d$ at fixed activity leaves the qualitative mode structure intact well into the fluid regime.

What would settle it

A rheological measurement of a living biofilm that shows the oscillatory rotational mode persisting while the independently measured stress-relaxation time is shorter than the model's threshold (about $\tau_d = 0.7$ in the units of Fig. 2) would disprove the claim that this mode is generic to viscoelastic fluids.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the two global oscillatory modes reported for living elastic active solids are generic to active media with memory. The authors solve their coupled fields—polar order $\mathbf{P}$ (the locally averaged migration direction), incompressible velocity $\mathbf{v}$, and displacement $\mathbf{u}$ (the deformation history)—under circular confinement with an elastic restoring force at the boundary, and observe the same rotation-reversing and translation-rotating modes when the displacement relaxes at a finite rate ($\tau_d = 1$ in rescaled units), i.e., for a genuinely viscoelastic fluid. As the activity $\nu_p$ is varied, a transition between the two modes appears for essentially all relaxation times studied, from $\tau_d \to \infty$ (elastic solid) down to strongly fluid-like values; only when $\tau_d$ drops below roughly $0.7$ does the oscillatory rotational mode disappear while the translational mode may still occur. The authors conclude that elastic behavior is only one limiting case of a broader class of oscillatory viscoelastic active matter.

Load-bearing premise

The entire prediction rests on a phenomenological theory in which one relaxation time for the material's memory, together with a linear restoring force at the boundary, captures the essential physics of a real biofilm.

Editorial extensions

If this is right

The observed oscillatory modes in bacterial biofilms do not require solid elasticity; a viscoelastic fluid description already suffices.
The activity-driven transition from rotation-reversing motion to translation-rotating motion is robust across a wide range of memory-relaxation times, so it should be a common feature of confined active materials.
As the material becomes more fluid-like (smaller $\tau_d$), the oscillation frequency and amplitude decrease, and below a threshold ($\tau_d \approx 0.7$ for the parameters shown) the rotational mode disappears while the translational mode can persist.
For this class of collective phenomena, biofilms can be treated as complex viscoelastic media, and the single-relaxation-time model is a minimal description of their large-scale motion.

Reading between the lines

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

The continuous interpolation between fluid and solid suggests the same framework could describe three-dimensional confinement or films with spatially varying rheology, which the paper does not test.
Replacing the single relaxation time with a full relaxation spectrum is a natural next step; the expectation would be that the modes persist as long as the slowest relaxation component is not much shorter than the oscillation period, which could be checked in rheologically characterized biofilms.
The boundary force in the model is linear and grows with distance from the center; numerically probing nonlinear or asymmetric confinement profiles could reveal how sensitively the two modes depend on the confining mechanism.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The paper is a short numerical note showing that the two oscillatory modes Xu et al. observed in active elastic biofilms also appear in a continuum model of viscoelastic fluids with finite relaxation time. That is a genuine, if modest, extension: earlier theoretical treatments used elastic solids, and here the solid limit is shown not to be necessary. The result is probably right in broad strokes: for tau_d = 1 the authors see both modes, and the transition between them persists down to tau_d ~ 0.7, with the rotational mode disappearing below that. The observation that oscillations slow down as tau_d decreases is a nice, testable trend.

The main thing that needs attention is Eq. (3). As written, it is d_t u = v - tau_d^{-1} u, but if u is an Eulerian displacement field, the kinematic relation should involve the material derivative: D_t u = d_t u + (v·grad)u = v - tau_d^{-1} u. The authors drop the v·grad u term without comment. In the oscillatory rotational mode the velocity and displacement are both order one, and with a domain diameter of 6, the omitted term is of the same order as d_t u and the relaxation term, especially near the transition where the frequency is small. So the demonstrated modes, and the tau_d threshold in Fig. 2, could be an artifact of this linearization. I do not think the paper is fatally wrong, but the authors need to either include the advective term or give a convincing argument for why it is negligible in the relevant states.

The genericity claim also outruns the evidence. Only tau_d is varied; all other parameters are held fixed. That shows the modes are not special to the elastic limit, but it does not establish that they appear for a 'broad range' of viscoelastic fluids and solids. The confinement force is introduced ad hoc to mimic Xu et al., with a particular radial profile; that is fine for a first pass, but again it limits the generality.

The model itself is taken from the authors' own companion paper (Ref. [8], also an arXiv preprint), so the framework is not independently established here. The equations are plausible, but that puts more weight on the numeric demonstration. A more practical problem is that there is no code, no data, no error bars, and no convergence test. The SI describes the numerical scheme, but nothing that lets the reader check whether the reported amplitudes and frequencies are resolved. For a paper whose only evidence is simulation, that is a real gap, and one a referee should ask to close.

Overall: the central claim is plausible and the paper is worth engaging with, but it is not, as written, a definitive demonstration of genericity. I would send it out, with instructions to address the kinematic equation, broaden or temper the genericity claim, and provide reproducibility data. These fixes are within reach for a short paper.

Best, [Name]

Referee Report

3 major / 4 minor

Summary. The manuscript studies a previously introduced continuum field theory for active media (Eqs. (1)-(3)) under circular confinement. The authors numerically solve these equations for parameters representing a viscoelastic fluid and observe two oscillatory collective modes that were reported experimentally for active elastic solids by Xu et al.: an oscillatory rotational mode and a translating mode with rotating velocity direction. By varying the displacement relaxation time tau_d, they find these modes for finite tau_d (down to tau_d approximately 0.7) as well as in the elastic limit tau_d -> infinity, and conclude that the phenomena are generic for viscoelastic active fluids and solids.

Significance. If the finding is robust, it meaningfully extends the scope of the experimental observations of Xu et al. from elastic solids to viscoelastic fluids, which is relevant for understanding biofilms and other biological materials. The paper provides a clear numerical demonstration for one representative parameter set (tau_d = 1) and a systematic trend in tau_d, with the qualitative distinction between the two modes well illustrated in Fig. 1. However, the claim of genericity is supported only by a limited parameter sweep, and the kinematics in Eq. (3) are linearized without explicit validation, so the current evidence is suggestive rather than conclusive.

major comments (3)

[Eq. (3) and Fig. 1] The displacement dynamics in Eq. (3) is written as partial_t u = v - tau_d^{-1} u, which omits the advective term v dot grad u that appears in the material derivative D_t u = partial_t u + v dot grad u. In the simulated oscillatory states, the velocity and displacement fields are not infinitesimal: Fig. 1 shows |v| up to about 6 in the same scaled units, and the advective term is estimated to be of the same order as the retained terms near the transition. Since Eq. (3) is the only memory mechanism that produces the oscillations, the demonstrated modes may be an artifact of this linearization rather than a genuine property of the viscoelastic model. The Supplemental Information provides no convergence test or comparison against the full material-derivative form. I request that the authors either justify the linearization by a scale analysis for the simulated amplitudes or repeat the key simulations with the advective term included.
[Fig. 2 and SI Eq. (S1)] The central claim of genericity for 'a broad range of viscoelastic fluids and solids' is based on a one-parameter sweep in tau_d with all other material parameters fixed (gamma_a = 1, eta = 1, mu = 1, nu_v = 1, nu_d = 10). The confinement force in SI Eq. (S1), with strength nu_conf_d0 = nu_d/2 and a linear spatial profile, is introduced ad hoc and is not tested for sensitivity. A generic statement would require at least a demonstration that the two modes persist when other parameters (e.g., gamma_a, nu_d, or the confinement strength) are varied, and that the threshold and frequencies do not depend critically on the chosen confinement profile.
[Supplemental Information (numerical methods)] The numerical results are presented without error bars, grid-convergence tests, or validation of the pseudo-time-stepping scheme. In particular, the claimed threshold tau_d approx 0.7 and the precise frequency curves in Fig. 2 are quantitative results; the authors should report at least a grid-resolution check (e.g., 128 vs 256 grid points) and a check of the convergence of the artificial-compressibility iteration, and ideally provide the code and data for reproducibility.

minor comments (4)

[Eq. (1)] The definition of Omega = [(grad v)^T - (grad v)] is missing the conventional factor 1/2; the dot product in the alignment term v dot [(2 + P dot P)I/3 - PP] is ambiguous and should be written with an explicit tensor contraction.
[Fig. 2] The curves for tau_d = 0.5 and tau_d = 0.7 are hard to distinguish in a grayscale print; consider using markers or different line styles.
[Text following Eq. (1)] The text says the 'first two terms' on the right-hand side of Eq. (1) reflect orientational and translational diffusion, but the term -P is a decay, not a diffusion, term; rephrase for accuracy.
[References] The reference to the companion paper [8] is to an arXiv preprint; if published, the citation should be updated, and the relation of Eqs. (1)-(3) to that paper should be summarized more explicitly for a self-contained reading.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the oscillatory modes are emergent results of direct numerical integration, not quantities fitted to the experiment or defined into the model.

full rationale

The paper's central claim is that the two oscillatory collective modes observed by Xu et al. also emerge in viscoelastic fluids with finite relaxation time tau_d, not only in elastic solids. This is established by numerically integrating Eqs. (1)-(3) for fixed, hand-chosen material parameters (e.g., tau_d = 1, gamma_a = 1, eta = 1, mu = 1, nu_v = 1, nu_d = 10) and reading off the resulting oscillation amplitudes and frequencies from the simulated time series (Fig. 2). No parameter is fitted to the Xu et al. experimental data, and the transition threshold tau_d approximately 0.7 is an emergent, nontrivial feature rather than a built-in input. The theoretical framework is taken from the authors' companion paper (Ref. [8]), so there is a self-citation that is structurally load-bearing: the model equations carry the viscoelastic mechanism. However, the present paper does not claim to prove the model itself, and the numerical results are not equivalent to the model's inputs by construction. There is no uniqueness theorem imported from prior work, no ansatz disguised as external support, and no known result merely renamed. The omission of the v . grad u advection term in Eq. (3), noted by a skeptic, is a modeling or correctness concern about the linearized kinematics, not a circularity: it does not make the output equal to an input. Overall, the derivation chain is self-contained with respect to the claimed demonstration, and the self-citation does not reduce the central result to a tautology.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the authors' own continuum model (Ref. [8]) and a set of hand-picked parameters; the 'generic' statement is supported by varying only tau_d with all other parameters fixed. No parameters are fitted to the Xu et al. data, but the model itself is self-cited and the confinement term is ad hoc.

free parameters (7)

relaxation time tau_d = 0.5, 0.7, 1.0, 2.0, infinity
Scans from viscoelastic fluid to elastic solid; controls elastic memory decay in Eq. (3). Not fitted to data.
active alignment strength gamma_a = 1
Set to 1 in rescaled units; chosen by hand.
viscosity eta = 1
Set to 1 in rescaled units.
shear modulus mu = 1
Set to 1 in rescaled units.
substrate friction coefficient nu_v = 1
Set to 1 in rescaled units.
elastic restoring force coefficient nu_d = 10
Set to 10; gives strong elastic response.
confinement force strength nu_conf_d0 = nu_d/2 = 5
Ad hoc value chosen to mimic the circular confinement; not derived or fitted.

assumptions (4)

domain assumption The active polar field theory in Eqs. (1)-(3) from Ref. [8] captures the dynamics of migrating active units in viscous, viscoelastic, or elastic media.
The entire simulation rests on this model, which is the authors' own companion work; if the model is wrong, the conclusions fail.
domain assumption Low-Reynolds-number Stokes flow and incompressibility hold in the film.
Eq. (2) assumes Stokes flow; appropriate for thin bacterial films.
ad hoc to paper The confinement is represented by a linear elastic restoring force acting only on the normal displacement component, Eq. (S1).
This term is added to mimic the experiment; its form and strength (nu_conf_d0 = nu_d/2) are chosen ad hoc.
domain assumption Neumann boundary conditions for all fields at the circular boundary.
Gradients vanish at the boundary; a modeling choice not derived from microscopics.

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Cite this review

Pith. "Pith review of Oscillatory collective motion in viscoelastic and elastic active fluids and solids under circular confinement." pith.science (2026). https://pith.science/paper/JBQND6YZ

@misc{pith2026250206312,
  author       = {Pith},
  title        = {Pith review of: Oscillatory collective motion in viscoelastic and elastic active fluids and solids under circular confinement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JBQND6YZ}},
  note         = {Machine review of arXiv:2502.06312}
}

read the original abstract

In an inspiring recent study, Xu et al. [Nat. Phys. 19, 46 (2023)] observed for a living active biofilm under circular confinement two emergent dynamic modes of collective motion in the film. One corresponds to global rotational motion of oscillating sense of rotation, the other one to uniformly translating motion of rotating migration direction. The authors reproduced these features in a discretized theoretical model for elastic active solids. We here demonstrate that the discovered fundamental phenomena are generic and emerge abundantly for a broad range of viscoelastic fluids and solids. Elastic solids represent only one limiting case.

Figures

Figures reproduced from arXiv: 2502.06312 by the authors.

**Figure 1.** FIG. 1. Emergent oscillatory modes of active collective motion under circular confinement for a viscoelastic fluid. (a) and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. Transition between oscillatory rotational modes of [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [8]

Reinken and A

H. Reinken and A. M. Menzel, Unified description of viscous, viscoelastic, or elastic thin active films on substrates, arXiv preprint arXiv:2502.04802 (2025)

work page arXiv 2025

[2] [2]

Ferrante, A

E. Ferrante, A. E. Turgut, C. Huepe, A. Stranieri, C. Pinciroli, and M. Dorigo, Self-organized flocking with a mobile robot swarm: A novel motion control method, Adapt. Behav. 20, 460 (2012)

work page 2012

[3] [3]

H. Shen, P. Tan, and L. Xu, Probing the role of mobility in the collective motion of nonequilibrium systems, Phys. Rev. Lett. 116, 048302 (2016)

work page 2016

[4] [4]

Baconnier, D

P. Baconnier, D. Shohat, C. H. L´ opez, C. Coulais, V. D´ emery, G. D¨ uring, and O. Dauchot, Selective and collective actuation in active solids, Nature Phys. 18, 1234 (2022)

work page 2022

[5] [5]

Zheng, C

Y. Zheng, C. Huepe, and Z. Han, Experimental capabilities and limitations of a position-based control algorithm for swarm robotics, Adapt. Behav. 30, 19 (2022)

work page 2022

[6] [6]

Ferrante, A

E. Ferrante, A. E. Turgut, M. Dorigo, and C. Huepe, Elasticity-based mechanism for the collective motion of self-propelled particles with springlike interactions: A model system for natural and artificial swarms, Phys. Rev. Lett. 111, 268302 (2013)

work page 2013

[7] [7]

Scheibner, A

C. Scheibner, A. Souslov, D. Banerjee, P. Sur´ owka, W. T. Irvine, and V. Vitelli, Odd elasticity, Nature Phys. 16, 475 (2020)

work page 2020

[8] [9]

Brotto, J.-B

T. Brotto, J.-B. Caussin, E. Lauga, and D. Bartolo, Hydrodynamics of confined active fluids, Phys. Rev. Lett. 110, 038101 (2013)

work page 2013