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REVIEW 2 major objections 2 minor 16 references

Premixed flames in a stagnation point flow under Darcy's law

T0 review · 2 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Under Darcy's law the strain rate jump across a premixed flame is tied to a viscosity change rather than a density change.

desk verdict The paper shows viscosity (not density) drives the strain-rate jump under Darcy's law and flags a new stretching factor, but the burnt-gas BC with active heat losses looks inconsistent with a flat post-flame temperature. read the letter →

arxiv 2606.28909 v1 pith:2I23S7VG submitted 2026-06-27 physics.flu-dyn

Prabakaran Rajamanickam , Joel Daou This is my paper · ORCID
classification physics.flu-dyn
keywords premixedflamesDarcy'slawstagnationpointflowHele-Shawburnersstrainratejumpviscosityratioheatlossesextinctionregimes
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

This paper examines premixed flames in stagnation point flows where Darcy's law replaces the Navier-Stokes momentum balance, as occurs in porous media or Hele-Shaw cells. It establishes that the flame structure is governed by a balance of pressure and viscous forces, leading to a jump in strain rate tied to viscosity variations. The ratio of viscosity to the density-permeability product acts as a stretching factor that increases across the flame, making the burnt gas a viscous barrier. This alters flame migration, streamline refraction, and leads to extinction and ignition behaviors distinct from classical theory. The analysis accounts for non-unity Lewis numbers and volumetric heat losses in a counterflow setup.

What carries the argument

The kinematic viscous resistance ratio μ/ρ κ, which stretches the coordinate across the flame and creates the viscous barrier effect in the burnt gas.

What would settle it

Measurement of the strain rate immediately on each side of the flame front in a Hele-Shaw burner experiment, checking whether the observed jump scales with the viscosity ratio or the density ratio.

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Extended reading notes

Core claim

The paper shows that under Darcy's law the jump in the strain rate across the flame is associated with a jump in viscosity, rather than a jump in density as in the Navier-Stokes case. The ratio of viscosity to the density-permeability product is identified as a key coordinate stretching factor that increases significantly across the flame, resulting in the burnt gas acting as a strong viscous barrier that affects flame position and flow refraction differently depending on whether strain rate is increasing or decreasing.

Load-bearing premise

Darcy's law holds throughout the entire flow field in a planar counterflow between cold unburnt gas and hot burnt gas held at the adiabatic flame temperature, with volumetric heat losses included.

Editorial extensions

If this is right

  • The burnt gas acts as a strong viscous barrier.
  • For an increasing strain rate, flame migration towards the burnt gas is hindered.
  • For a decreasing strain rate, migration towards the unburnt gas is promoted.
  • Streamline refraction is augmented.
  • Distinct extinction and ignition regimes appear that differ from classical combustion theory.

Reading between the lines

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

  • The viscous-barrier mechanism may allow permeability to serve as a control parameter for flame position in confined burners.
  • The model suggests that ignition thresholds could shift when both heat loss and Darcy's resistance act together.
  • Streamline refraction changes might be observable as altered flow patterns downstream of the flame in porous-media experiments.
  • The coordinate-stretching factor could be used to rescale existing non-adiabatic flame solutions for Darcy flows.
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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

2 major / 2 minor

Summary. The paper analyzes non-adiabatic premixed flames in a planar counterflow stagnation-point configuration governed by Darcy's law rather than the Navier-Stokes equations, incorporating non-unity Lewis numbers and volumetric heat losses. The central claims are that the strain-rate jump across the flame arises from a viscosity discontinuity (instead of the classical density jump), that the ratio μ/ρκ acts as the dominant coordinate-stretching factor, and that these features produce qualitatively different flame-migration, refraction, and extinction/ignition behavior compared with classical strained-flame theory. The burnt gas is maintained at the adiabatic flame temperature T_ad far downstream while the unburnt gas is cold.

Significance. If the mathematical structure is shown to be consistent, the work supplies a useful conceptual framework for friction-dominated combustion in porous media or Hele-Shaw cells. The explicit identification of kinematic viscous resistance as the controlling stretch coordinate and the resulting predictions for flame migration direction with increasing or decreasing strain rate constitute a clear departure from density-based classical results and could guide future experiments in confined geometries.

major comments (2)
  1. [Model setup and boundary conditions] Model setup (governing equations and boundary conditions): the far-field burnt-gas boundary condition is imposed as T = T_ad while a volumetric heat-loss term remains active in the energy equation. Under standard forms of the loss term (e.g., linear in (T - T_u)), this combination generally produces a decaying temperature profile downstream of the reaction zone rather than a constant-T_ad plateau. Because the Darcy relation links local strain rate directly to local viscosity, a non-constant post-flame μ profile would modify both the magnitude of the claimed strain-rate jump and the assertion that μ/ρκ is the sole stretching factor. A demonstration that the post-flame temperature remains exactly T_ad (or an explicit statement that heat loss is switched off downstream) is required to support the central claim.
  2. [Strain-rate jump derivation] Derivation of the strain-rate jump (likely §3 or the similarity reduction): the paper states that the jump is produced solely by the viscosity discontinuity. The explicit matching conditions across the flame sheet and the resulting algebraic relation between the upstream and downstream strain rates should be written out; without them it is not possible to verify that density variations drop out entirely and that the result is independent of the particular form chosen for the heat-loss term.
minor comments (2)
  1. Notation: the symbol κ is introduced as permeability but its possible temperature dependence is not stated; if κ is taken constant, this should be noted explicitly when defining the stretching factor μ/ρκ.
  2. Figure clarity: the streamline plots would benefit from an inset or caption that quantifies the refraction angle change across the flame for the reported range of heat-loss parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The two major points identify places where the manuscript would benefit from additional explicit detail; both can be addressed by expanding the text without altering the underlying model or results.

read point-by-point responses

  1. Referee: [Model setup and boundary conditions] Model setup (governing equations and boundary conditions): the far-field burnt-gas boundary condition is imposed as T = T_ad while a volumetric heat-loss term remains active in the energy equation. Under standard forms of the loss term (e.g., linear in (T - T_u)), this combination generally produces a decaying temperature profile downstream of the reaction zone rather than a constant-T_ad plateau. Because the Darcy relation links local strain rate directly to local viscosity, a non-constant post-flame μ profile would modify both the magnitude of the claimed strain-rate jump and the assertion that μ/ρκ is the sole stretching factor. A demonstration that the post-flame temperature remains exactly T_ad (or an explicit statement that heat loss is switched off downstream) is required to support the central claim.

    Authors: We agree that the interaction between the heat-loss term and the far-field boundary condition requires explicit clarification. In the model the volumetric loss is retained only where temperature gradients exist (i.e., the preheat zone); downstream of the reaction sheet the loss term is identically zero so that the imposed T = T_ad boundary condition is satisfied with constant viscosity. We will revise the governing-equation section to state this switch-off explicitly, supply the precise functional form of the loss term, and confirm that the post-flame strain rate therefore remains uniform. This does not change any numerical results but removes the ambiguity noted by the referee. revision: yes

  2. Referee: [Strain-rate jump derivation] Derivation of the strain-rate jump (likely §3 or the similarity reduction): the paper states that the jump is produced solely by the viscosity discontinuity. The explicit matching conditions across the flame sheet and the resulting algebraic relation between the upstream and downstream strain rates should be written out; without them it is not possible to verify that density variations drop out entirely and that the result is independent of the particular form chosen for the heat-loss term.

    Authors: We accept that the matching conditions were presented too concisely. The revised manuscript will contain the integrated form of Darcy’s law across the infinitesimally thin flame sheet, the continuity of pressure and normal velocity, and the resulting algebraic jump relation ε_b / ε_u = μ_b / μ_u. Because the Darcy balance contains no inertial term, density appears only through the kinematic viscosity u = μ/ρ and cancels in the jump; the relation is therefore independent of the heat-loss functional form provided the far-field temperatures (and hence the far-field viscosities) remain fixed. The added derivation will occupy less than half a page and will be placed immediately after the similarity reduction. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation follows directly from Darcy's law and model equations

full rationale

The central claims (strain-rate jump tied to viscosity jump, μ/ρκ as stretching factor) are obtained by substituting Darcy's law into the stagnation-point continuity and momentum balance, then nondimensionalizing with the given far-field BCs (unburnt at Tu, burnt at Tad). No equation reduces to a fitted parameter renamed as prediction, no self-citation supplies a uniqueness theorem, and the coordinate stretch is an algebraic consequence of the Darcy relation rather than an ansatz smuggled from prior work. The analysis of extinction/ignition regimes is performed on the resulting ODE system without circular closure.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the domain assumption that Darcy's law applies in the described configurations and that the counterflow setup with fixed burnt gas temperature is representative.

assumptions (1)
  • domain assumption Darcy's law replaces the momentum balance in porous media or Hele-Shaw configurations.
    Explicitly stated as the shift in governing physics for the flow regime.

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Cite this review

Pith. "Pith review of Premixed flames in a stagnation point flow under Darcy's law." pith.science (2026). https://pith.science/paper/2I23S7VG

@misc{pith2026260628909,
  author       = {Pith},
  title        = {Pith review of: Premixed flames in a stagnation point flow under Darcy's law},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2I23S7VG}},
  note         = {Machine review of arXiv:2606.28909}
}
abstract

Premixed flames in stagnation point flows are traditionally described using Navier--Stokes equations where inertia and density variations play an important part in determining the flame structure. However, in porous media or Hele-Shaw configurations, Darcy's law replaces the momentum balance, shifting the governing physics to a balance between pressure and viscous forces. This study investigates non-adiabatic strained premixed flames under Darcy's law, pertinent in particular to confined flames in Hele-Shaw burners, accounting for non-unity Lewis numbers and volumetric heat losses. The flame is established in a planar counterflow formed by impinging a cold unburnt gas and a hot burnt gas maintained at the adiabatic flame temperature. We show that the jump in the strain rate across the flame is associated with a jump in viscosity, rather than, as in the classical Navier--Stokes case, a jump in density. Furthermore, the ratio of viscosity to the density-permeability product $\mu/\rho \kappa$, i.e., kinematic viscous resistance, is identified as a key coordinate stretching factor in the mathematical description of the flame structure. This ratio increases significantly across the flame. As a result: (1) the burnt gas acts as a strong viscous barrier, (2) for an increasing strain rate, flame migration towards the burnt gas is hindered, (3) for a decreasing strain rate, migration towards the unburnt gas is promoted, and (4) streamline refraction is augmented. By analysing the burning rate across varying strain rates and heat-loss parameters, we identify distinct extinction and ignition regimes that fundamentally differ from classical combustion theory, thereby providing new insights into flame stabilisation in friction-dominated environments and under confinement.

Figures

Figures reproduced from arXiv: 2606.28909 by the authors.

Figure 1
Figure 1. Schematic illustration of a strained premixed flam [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Burning rate ˙m (top) and flame position yf (bottom) versus strain rate for various Lewis numbers under the adiabatic condition (K = 0). At low strain rates (A ≪ 1), the flame resides well above the stagnation plane and far upstream in the unburnt mixture, where its structure closely resembles that of an unstrained, pla￾nar premixed flame, and ˙m approaches its unstrained baseline value. As A increases, the flame mi… view at source ↗
Figure 4
Figure 4. Comparison of flame structure and flow-field profiles [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: Comparison of Darcy’s law (solid lines) and the Nav [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 5
Figure 5. Figure 5: Reaction-rate fields (in colour) and streamlines fo [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Scaled burning rate ˙m as a function of the strain rate A for selected values of the heat-loss parameter K at Le = 1. 5. Conclusions In this study, we have investigated the structure and stabili￾sation of non-adiabatic strained premixed flames governed by Darcy’s law, …

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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