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Linear Motility Maps in Nonlinear Viscous Fluids

T0 review · 1 major / 0 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Linear motility maps extend to power-law fluids but can be violated in Carreau-Yasuda fluids to allow net motion from reciprocal motions.

desk verdict Linear motility maps hold for power-law fluids and can break in Carreau-Yasuda via a lumped inchworm model, but the model assumptions need checking. read the letter →

arxiv 2606.00063 v1 pith:C7NQDP4I submitted 2026-05-19 cs.RO math-phmath.MPphysics.flu-dyn

Yishun Zhou , Shai Revzen This is my paper · ORCID
classification cs.ROmath-phmath.MPphysics.flu-dyn
keywords motilitymapspower-lawfluidsCarreau-YasudascalloptheoremlowReynoldsnumberlocomotioninchwormmodelnonlinearviscositygeometricmechanics
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 establishes that the linear motility map relating shape-change rates to body-frame velocity continues to hold for any power-law viscosity fluid. This means reciprocal body deformations produce no net displacement in such fluids, extending the scallop theorem to many biological fluids at intermediate shear rates. In contrast, Carreau-Yasuda fluids allow the linearity to break, so an inchworm model with two unequal masses and unequal drag coefficients can achieve net locomotion even when its motions are reciprocal. The direction of that net motion can reverse depending on the speed of the shape changes.

What carries the argument

The linear motility map that relates shape-change rates to body-frame velocity, extended to power-law fluids and shown to be breakable in Carreau-Yasuda fluids.

What would settle it

Measure whether a physical two-mass reciprocal actuator with unequal drags in a Carreau-Yasuda fluid produces net displacement whose sign reverses when actuation speed is changed.

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Extended reading notes

Core claim

We show that linear-in-velocity motility maps extend to any power law viscosity (a.k.a. Ostwald--de Waele fluid), and therefore to many biological fluids in intermediate shear ranges. We also show that the linear-in-velocity property can be violated in Carreau-Yasuda fluids to produce net motion using an inchworm model consisting of two unequal masses with unequal drag coefficients performing reciprocal motions. Interestingly, the direction of motion can be switched by changing speeds.

Load-bearing premise

The inchworm model of two unequal masses with unequal drag coefficients performing reciprocal motions accurately captures the dynamics that allow violation of linearity in Carreau-Yasuda fluids.

Editorial extensions

If this is right

  • The linear motility map can be used to analyze and design locomotion in power-law fluids.
  • Net locomotion becomes possible in Carreau-Yasuda fluids despite reciprocal motions.
  • Direction of net motion can be reversed by changing the speed of the reciprocal motions.
  • Nonlinear drag relationships can be exploited to generate net locomotion that appears to violate the scallop theorem.

Reading between the lines

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

  • Design of microrobots could exploit speed-dependent direction reversal in biological fluids without needing non-reciprocal actuators.
  • The same linearity-breaking mechanism may appear in other fluids whose viscosity depends on shear rate in a non-power-law way.
  • Geometric-mechanics tools remain usable for power-law cases but require nonlinear extensions when fluid response deviates from Ostwald-de Waele form.
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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

1 major / 0 minor

Summary. The paper claims that linear motility maps relating shape-change rates to body-frame velocity in low-Re fluids extend to power-law (Ostwald-de Waele) viscosity fluids, preserving the applicability of geometric mechanics and the scallop theorem. It further claims that this linearity is violated in Carreau-Yasuda fluids, enabling net propulsion from reciprocal motions in a two-mass inchworm model with unequal constant drag coefficients; the direction of net motion can be reversed by changing the speed of the reciprocal cycle.

Significance. If the power-law extension holds, the result supplies a parameter-free generalization that directly supports analysis and design of locomotion in many biological fluids. The Carreau-Yasuda violation claim, if substantiated beyond the lumped model, would demonstrate a concrete mechanism by which fluid nonlinearity can be exploited to break the scallop theorem, with potential implications for micro-robotics.

major comments (1)
  1. [Abstract, final paragraph] Abstract, final paragraph (inchworm model): the claim that the linear-in-velocity property is violated in Carreau-Yasuda fluids rests on a two-mass model with independent, constant, unequal drag coefficients executing reciprocal motions. In the actual Carreau-Yasuda relation the local viscosity is a continuous function of the instantaneous shear-rate magnitude produced by the velocities of both bodies, and the flow fields remain hydrodynamically coupled at low Re; the lumped model therefore does not demonstrably reproduce the regime in which the motility map ceases to be linear.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this key limitation in our presentation of the Carreau-Yasuda results. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract, final paragraph] Abstract, final paragraph (inchworm model): the claim that the linear-in-velocity property is violated in Carreau-Yasuda fluids rests on a two-mass model with independent, constant, unequal drag coefficients executing reciprocal motions. In the actual Carreau-Yasuda relation the local viscosity is a continuous function of the instantaneous shear-rate magnitude produced by the velocities of both bodies, and the flow fields remain hydrodynamically coupled at low Re; the lumped model therefore does not demonstrably reproduce the regime in which the motility map ceases to be linear.

    Authors: We agree that the lumped two-mass model with fixed, unequal drag coefficients is only an illustrative toy model and does not capture the continuous shear-rate dependence of viscosity or the hydrodynamic coupling present in the full Carreau-Yasuda constitutive relation. Consequently the model does not rigorously demonstrate that the motility map itself becomes nonlinear under Carreau-Yasuda rheology. We will revise the abstract to state explicitly that the inchworm example is a simplified illustration of how nonlinear drag can permit net locomotion from reciprocal shape changes, and we will add a paragraph in the discussion section noting that confirmation in a spatially resolved, hydrodynamically coupled simulation remains future work. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; claims rest on explicit model construction and standard low-Re extensions

full rationale

The paper's core results extend the known linear motility map to power-law fluids via direct substitution into the Stokes equations under the stated constitutive relation, and demonstrate a possible violation in Carreau-Yasuda fluids by constructing an explicit two-body reciprocal-motion example. Neither step reduces to a fitted parameter renamed as prediction, a self-definitional loop, nor a load-bearing self-citation. The inchworm model is presented as a lumped illustration whose assumptions are stated outright; any question of its hydrodynamic fidelity is a modeling-validity concern, not a circularity in the derivation chain itself. The work is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Abstract-only; ledger inferred from stated claims. Relies on low-Re regime assumptions and specific rheological models without independent verification shown.

assumptions (2)
  • domain assumption Low Reynolds number regime governs the fluid dynamics
    Stated in opening sentence of abstract as the regime where motility maps apply.
  • domain assumption Power-law (Ostwald-de Waele) viscosity preserves linearity of motility map
    Central claim of the paper; no derivation visible in abstract.

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Cite this review

Pith. "Pith review of Linear Motility Maps in Nonlinear Viscous Fluids." pith.science (2026). https://pith.science/paper/C7NQDP4I

@misc{pith2026260600063,
  author       = {Pith},
  title        = {Pith review of: Linear Motility Maps in Nonlinear Viscous Fluids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C7NQDP4I}},
  note         = {Machine review of arXiv:2606.00063}
}
read the original abstract

Systems moving in low Reynolds number fluid regimes are known to be governed by a ``motility map'' which linearly relates their shape change rates to they body frame velocity moving through the fluid. A consequence of this is ``Purcell's Scallop Theorem'' -- a locomotion system that undergoes shape changes that follow the same path forward and backward in time (reciprocal body deformations) cannot achieve net displacement, regardless of pacing of those changes.We show that linear-in-velocity motility maps extend to any power law viscosity (a.k.a. Ostwald--de Waele fluid), and therefore to many biological fluids in intermediate shear ranges. We also show that the linear-in-velocity property can be violated in Carreau-Yasuda fluids to produce net motion using an ``inchworm'' model consisting of two unequal masses with unequal drag coefficients performing reciprocal motions. Interestingly, the direction of motion can be switched by changing speeds. Our results show that the linear motility map of geometric mechaincs can be used to analyze and design locomotion in power-law fluids, and that some nonlinear drag relationships such as Carreau-Yasuda can be exploited to generate net locomotion in seeming violation of the ``scallop theorem''.

Figures

Figures reproduced from arXiv: 2606.00063 by the authors.

Figure 2
Figure 2. Inchworm shape change profile over time. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 1
Figure 1. Inchworm model schematic. Two masses m1 and m2 are positioned at x1 and x2 respectively. An actuation force f(t) acts on the two masses in op￾posite directions. The drag forces act in the opposite direction of the velocity of each mass with drag co￾efficients µ1 and µ2. The function η represents the friction model we choose. As our minimalistic model, we consider the one￾dimensional dynamics of two masses with dispa… view at source ↗
Figure 3
Figure 3. Inchworm dynamics with linear viscous friction at different Reynolds numbers with different masses, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Inchworm dynamics with power law 1/2. We simulated multiple Reynolds numbers and masses, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Inchworm dynamics with power law 2. We simulated multiple Reynolds numbers and masses, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Inchworm dynamics with Carreau-Yasuda friction with the parameters for synovial fluid [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Inchworm COM net displacement over a cycle versus mass with different drag laws. (a) We plotted results for power law models with different exponents β (solid lines with square markers) and a linear fit to the results for small masses (dashed lines). The slope at small…

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