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arxiv: 2604.11678 · v1 · submitted 2026-04-13 · ❄️ cond-mat.soft

Kinematic and rheological equivalence of steady shearing and planar extensional flows

Nicholas King , Gareth H. McKinley This is my paper

Pith reviewed 2026-05-10 15:36 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords steady shearing flowplanar extensional flowrheological equivalenceeffective extension rateviscoelastic fluidsconstitutive modelspolymer solutionskinematic mapping
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The pith

Steady shearing flows can reconstruct the planar extensional viscosity of complex fluids by removing the rotational component of shear.

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

The paper establishes that steady shearing and planar extensional flows share a kinematic equivalence allowing an effective extension rate to be defined for shear. This rate strips away the rotational part of the deformation so that shear viscosity and normal stress data alone can reconstruct the steady planar extensional viscosity. The mapping is shown to hold for both phenomenological and frame-invariant microscopic constitutive models. Experiments on a viscoelastic polymer solution confirm that the reconstructed extensional viscosity matches direct measurements.

Core claim

By leveraging the kinematic equivalence between steady shearing and planar extension, an effective extension rate is derived that removes the rotational component of the shearing flow. This equivalence allows the steady planar extensional viscosity to be reconstructed using only material functions measured in steady shearing flow. The result holds for both phenomenological and microscopically motivated frame-invariant constitutive models and is confirmed experimentally with a viscoelastic polymer solution.

What carries the argument

The effective extension rate derived from the kinematic equivalence between steady shearing and planar extension, which removes the rotational component of shear.

If this is right

  • Steady planar extensional viscosity of an unknown fluid can be obtained solely from standard shear rheometry data.
  • The equivalence applies equally to phenomenological constitutive models and to microscopically motivated frame-invariant models.
  • Experimental tests with viscoelastic polymer solutions validate that the reconstructed extensional viscosity agrees with direct measurements.
  • The two deformation histories are shown to be rheologically equivalent once the rotational component is accounted for.

Where Pith is reading between the lines

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

  • Standard shear rheometers could become sufficient for estimating extensional properties in many lab settings.
  • The same kinematic mapping approach might be tested on other pairs of flow histories that differ only by rotation.
  • Industrial flow simulations involving mixed shear and extension could incorporate shear-only data more directly.

Load-bearing premise

The kinematic equivalence between steady shearing and planar extensional flows can be used to define an effective extension rate that removes the rotational component of the shearing flow.

What would settle it

Independent planar extensional viscosity measurements on the same fluid sample that fail to match the values reconstructed from its shear viscosity and normal stress differences using the derived effective extension rate.

Figures

Figures reproduced from arXiv: 2604.11678 by Gareth H. McKinley, Nicholas King.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The different stresses [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Transient shear and planar extensional viscosity [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Steady planar extensional viscosity curves recreated [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Experimental steady shear [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Steady shearing and planar extension are commonly viewed as two distinct types of flow field, especially in the context of probing the rheology of complex fluids. By leveraging the kinematic equivalence between the two flows, we derive an effective extension rate experienced by a material element which removes the rotational component of the shearing flow. This enables reconstruction of the steady planar extensional viscosity of an unknown fluid using only material functions measured in a steady shearing flow, revealing a deep rheological equivalence between the two deformation histories. We demonstrate this equivalency through phenomenological and microscopically motivated frame-invariant constitutive models as well as experiments with a viscoelastic polymer solution.

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 / 2 minor

Summary. The manuscript claims that steady simple shear and planar extensional flows are kinematically equivalent once an effective extension rate is defined by removing the rotational (vorticity) component from the shear velocity gradient. This equivalence is asserted to permit direct reconstruction of the steady planar extensional viscosity of an arbitrary frame-invariant fluid using only the shear viscosity and first normal-stress coefficient measured in steady shear. The claim is supported by explicit calculations for several phenomenological and microscopically motivated constitutive models together with experimental data on one viscoelastic polymer solution.

Significance. If the mapping is shown to be independent of the constitutive equation, the result would be significant for soft-matter rheology: it would allow extensional properties, which are experimentally challenging, to be obtained from routine shear measurements. The work earns credit for testing the idea across both phenomenological and microscopic models and for including experimental validation rather than remaining purely theoretical.

major comments (2)
  1. [§2] §2 (derivation of effective extension rate): the central reconstruction procedure assumes that replacing the shear rate with twice the effective extension rate in the shear material functions yields the exact steady extensional viscosity for any frame-invariant fluid. The manuscript demonstrates this for selected models but does not provide a general argument that residual history effects arising from the continuously rotating principal axes of the shear flow vanish identically in the steady state; this assumption is load-bearing for the claim that the procedure works for an unknown fluid.
  2. [§4] §4 (experimental validation): the equivalence is shown for a single polymer solution. To substantiate the claim that the reconstruction works for arbitrary unknown fluids, the manuscript must clarify whether any model-specific adjustments were required to match the data and must report quantitative uncertainty or goodness-of-fit metrics for the reconstructed versus measured extensional viscosity; without these, the experimental support remains illustrative rather than confirmatory of generality.
minor comments (2)
  1. [Introduction] The introduction would benefit from a brief comparison to prior kinematic mappings between shear and extension that have appeared in the rheology literature, to clarify the precise novelty of the effective-rate construction.
  2. [Figures] Figure captions should explicitly state the constitutive models used in each panel so that readers can immediately connect the plotted curves to the derivations in §3.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the constructive major comments. We address each point below and have made revisions to improve the manuscript's clarity and rigor.

read point-by-point responses

  1. Referee: [§2] §2 (derivation of effective extension rate): the central reconstruction procedure assumes that replacing the shear rate with twice the effective extension rate in the shear material functions yields the exact steady extensional viscosity for any frame-invariant fluid. The manuscript demonstrates this for selected models but does not provide a general argument that residual history effects arising from the continuously rotating principal axes of the shear flow vanish identically in the steady state; this assumption is load-bearing for the claim that the procedure works for an unknown fluid.

    Authors: We agree that a fully general proof for arbitrary frame-invariant fluids would strengthen the central claim. The current manuscript verifies the reconstruction explicitly across a range of phenomenological and microscopically motivated constitutive models, which span different levels of complexity and history dependence. In the revised version we will expand §2 with an explicit discussion of the steady-state limit: because the effective extension rate is constructed from the symmetric part of the velocity gradient (removing vorticity), and because frame-invariant constitutive equations depend only on objective deformation measures, the accumulated strain history in steady shear maps directly onto that of steady planar extension once the rate is rescaled. We will also state the assumptions under which this mapping holds and note that counter-examples, if they exist, would require non-objective or explicitly time-dependent constitutive behavior outside the scope of the paper. revision: yes

  2. Referee: [§4] §4 (experimental validation): the equivalence is shown for a single polymer solution. To substantiate the claim that the reconstruction works for arbitrary unknown fluids, the manuscript must clarify whether any model-specific adjustments were required to match the data and must report quantitative uncertainty or goodness-of-fit metrics for the reconstructed versus measured extensional viscosity; without these, the experimental support remains illustrative rather than confirmatory of generality.

    Authors: We accept this criticism. No model-specific adjustments or parameter fitting to the extensional data were performed; the reconstruction used only the independently measured steady shear viscosity and first normal-stress coefficient. In the revised manuscript we will add quantitative metrics: the root-mean-square relative deviation between reconstructed and measured extensional viscosity, together with propagated experimental uncertainties from the shear rheometer data. We will also state explicitly that the comparison is for one well-characterized viscoelastic solution and that broader experimental confirmation across additional fluids would be valuable future work. revision: yes

Circularity Check

0 steps flagged

No significant circularity; kinematic derivation is independent and equivalence is demonstrated rather than assumed

full rationale

The paper's core step defines an effective extension rate by decomposing the simple-shear velocity gradient into its symmetric strain-rate tensor (whose eigenvalues supply the extension rate) and antisymmetric vorticity tensor. This decomposition is a standard, external kinematic identity, not derived from or fitted to the paper's rheological results. The subsequent claim that steady stresses (hence viscosities) match under this mapping is presented as a testable equivalence, verified explicitly on several frame-invariant constitutive models plus one polymer solution experiment. No parameters are fitted to extensional data, no self-citations supply load-bearing uniqueness theorems, and no ansatz is smuggled in via prior work. The reconstruction procedure therefore does not reduce to its inputs by construction; it remains a non-tautological prediction whose validity for arbitrary fluids is left open by the demonstrations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard kinematic decomposition of the velocity gradient into extensional and rotational parts, a domain assumption in continuum mechanics that requires no new free parameters or invented entities.

axioms (1)
  • domain assumption Kinematic equivalence between steady simple shear and planar extensional flows via removal of the antisymmetric (rotational) component of the velocity gradient
    Invoked to define the effective extension rate experienced by a material element.

pith-pipeline@v0.9.0 · 5392 in / 1268 out tokens · 57902 ms · 2026-05-10T15:36:26.430637+00:00 · methodology

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Reference graph

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