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arxiv: 2606.07223 · v1 · pith:SS7ZM4CNnew · submitted 2026-06-05 · 🧮 math.AP

Homogenization of regularized Oldroyd-type fluids

Florian Oschmann , Jonas Sauer This is my paper

Pith reviewed 2026-06-27 21:37 UTC · model grok-4.3

classification 🧮 math.AP
keywords homogenizationOldroyd fluidperforated domainDarcy lawviscoelastic fluidregularizationscaling regimes
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The pith

In appropriate scaling regimes, the polymeric stress of regularized Oldroyd fluids vanishes from the homogenized Darcy law in perforated domains.

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

The paper analyzes the homogenization of a regularized Oldroyd-type viscoelastic fluid model in a periodically perforated domain. It establishes that the system converges to an effective Darcy law for the macroscopic flow. Under suitable relations between the perforation size and the regularization and diffusion parameters, the contribution of the polymeric extra stress disappears in the limit. The proofs rely on uniform estimates, oscillating test functions, and a relative energy method. The analysis also yields a weak-strong uniqueness result for the original viscoelastic system.

Core claim

We prove that the regularized Oldroyd-type system in a periodically perforated domain homogenizes to a Darcy law on the macroscopic scale, with the polymeric stress not contributing to the effective equation when the scaling parameters satisfy appropriate conditions. The convergence is both qualitative and quantitative, obtained via uniform estimates and a relative energy method.

What carries the argument

The relative energy method combined with oscillating test-function techniques, under scaling regimes that render the extra stress term negligible.

If this is right

  • The macroscopic velocity satisfies a Darcy-type equation without viscoelastic effects.
  • Convergence holds in suitable function spaces for the velocity and stress variables.
  • Weak-strong uniqueness is established for the viscoelastic system independent of the homogenization.

Where Pith is reading between the lines

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

  • This implies that perforation and regularization can suppress memory effects in the fluid at large scales.
  • Numerical simulations with varying perforation sizes and parameters could verify the transition to Newtonian-like macro behavior.
  • The result may extend to other constitutive laws if similar scalings are applied.

Load-bearing premise

The scaling regimes must be chosen so that the extra stress term becomes negligible compared to the viscous terms in the limit process.

What would settle it

A counterexample where the polymeric stress remains in the limit equation despite the stated scaling assumptions on perforation size and regularization parameters would falsify the claim.

read the original abstract

We study homogenization of a regularized viscoelastic Oldroyd-type model in a periodically perforated bounded domain. The system describes an incompressible non-Newtonian fluid coupled to an elastic extra stress tensor and includes both nonlinear viscosity and nonlinear stress diffusion effects. The governing model, introduced by Kreml, Pokorn\'y, and \v{S}alom (2015), covers Oldroyd-A- and Oldroyd-B-type constitutive laws. We establish qualitative and quantitative homogenization results in suitable scaling regimes and show convergence toward an effective Darcy law on the macroscopic domain. In particular, we prove that, under appropriate assumptions on the scaling parameters, the polymeric stress does not contribute to the effective limit equation. The analysis combines uniform estimates, oscillating test-function techniques, and a relative energy method, and additionally yields a weak-strong uniqueness principle for the viscoelastic system.

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

0 major / 3 minor

Summary. The manuscript studies homogenization of a regularized Oldroyd-type viscoelastic fluid (covering Oldroyd-A and Oldroyd-B laws) in a periodically perforated bounded domain. Under suitable scaling relations among perforation size, regularization parameter, and stress diffusion coefficient, the authors prove convergence to an effective Darcy law in which the polymeric extra stress vanishes from the macroscopic equation. The analysis relies on uniform a priori estimates, oscillating test functions, and a relative-energy argument; the paper also derives a weak-strong uniqueness principle for the viscoelastic system.

Significance. If the proofs are complete, the result clarifies when viscoelastic contributions become negligible in homogenized porous-media flows and extends standard homogenization tools (oscillating test functions, relative energy) to regularized non-Newtonian systems. The weak-strong uniqueness statement is a concrete additional contribution that strengthens the work.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'under appropriate assumptions on the scaling parameters' is repeated without a concise statement of the admissible relations (e.g., how the perforation size ε, regularization δ, and diffusion coefficient u scale with one another). Adding one explicit scaling example would improve readability.
  2. The abstract states that both qualitative and quantitative homogenization results are obtained, yet the quantitative part (error estimates or rates) is not mentioned again in the provided summary. Clarifying whether rates are derived and in which norm would help readers locate the quantitative contribution.
  3. Notation: ensure that the symbols for the extra-stress tensor, the regularization parameter, and the perforation size are introduced uniformly in the introduction and used consistently in all subsequent sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the homogenization of regularized Oldroyd-type fluids and for recommending minor revision. The summary accurately captures the main contributions, including the convergence to a Darcy law independent of polymeric stress and the weak-strong uniqueness result. As no specific major comments are provided in the report, we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a mathematical homogenization proof establishing convergence to a Darcy law under scaling assumptions on perforation size, regularization, and diffusion coefficients. It relies on uniform a priori estimates, oscillating test functions, and relative energy methods applied to the system introduced in independent prior work (Kreml et al. 2015). No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the vanishing of the polymeric stress term is derived from the estimates rather than assumed or renamed. The derivation is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the existence of solutions to the regularized system (from the 2015 reference) and standard functional-analytic tools for incompressible fluids; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Existence of weak solutions to the regularized viscoelastic system
    Invoked to start the homogenization analysis; referenced to Kreml, Pokorný, Šalom (2015).
  • standard math Periodic perforation geometry and incompressibility constraint
    Standard setup for homogenization in perforated domains.

pith-pipeline@v0.9.1-grok · 5660 in / 1191 out tokens · 23405 ms · 2026-06-27T21:37:15.124964+00:00 · methodology

discussion (0)

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Reference graph

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