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arxiv: 2606.26953 · v1 · pith:I6XSHZJLnew · submitted 2026-06-25 · ⚛️ physics.flu-dyn

Hydrodynamic theory of premixed flames under Darcy's law: Interfacial conditions and effects of nonunity Lewis number and heat loss

Prabakaran Rajamanickam , Joel Daou This is my paper

Pith reviewed 2026-06-26 03:02 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords premixed flamesDarcy's lawMarkstein numbershydrodynamic theorydispersion relationporous mediaHele-Shaw channelsLewis number
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The pith

Premixed flames under Darcy's law acquire three distinct Markstein numbers that shape their burning rate and stability.

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

This paper establishes a first-principles hydrodynamic theory for premixed flames propagating under Darcy's law in porous media or Hele-Shaw channels. It derives interfacial jump conditions that account for the finite thickness of the flame, incorporating effects of nonunity Lewis number and heat loss through large activation-energy asymptotics. The theory shows that the adiabatic burning rate depends on three separate Markstein numbers for curvature, tangential strain, and gravity-induced strain. These lead to a dispersion relation for small perturbations that differs from the classical Navier-Stokes version, providing a basis for analyzing flame dynamics in confined environments.

Core claim

The paper derives interfacial conditions for premixed flames under Darcy's law from large activation-energy asymptotics and multiple-scale analysis. The conventional continuity of mass flux and pressure receives corrections due to finite flame thickness. The adiabatic burning rate involves three distinct Markstein numbers corresponding to curvature, tangential flow strain, and gravity-induced strain, with explicit formulas given. The resulting dispersion relation is s = (a|k| - bk^2 - d|k|^3) / (1 + c|k|), which is to be compared with the Clavin-Garcia relation from Navier-Stokes equations.

What carries the argument

Interfacial jump conditions derived via large activation-energy asymptotics and systematic multiple-scale analysis, which introduce corrections to mass flux and pressure continuity and yield three Markstein numbers under Darcy's law.

Load-bearing premise

Large activation-energy asymptotics and multiple-scale analysis provide accurate interfacial corrections for the finite-thickness flame under Darcy's law.

What would settle it

Measurement of the linear growth rate of perturbations to a planar flame front in a Hele-Shaw apparatus as a function of wavenumber, to check agreement with the derived dispersion relation including the gravity term.

Figures

Figures reproduced from arXiv: 2606.26953 by Joel Daou, Prabakaran Rajamanickam.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of a curved premixed flame of t [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Leading-order burning rate ˙m [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The plot of various integrals as functions of the heat [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

Premixed flames propagating in porous media or Hele-Shaw channels are governed by Darcy's law, which accounts for the strong frictional forces imposed by the solid matrix or confining walls. Prior theoretical studies of such flames have typically employed phenomenological Markstein-type corrections and have assumed unity Lewis numbers and adiabatic conditions. In this work, we develop a rigorous hydrodynamic theory for premixed flames under Darcy's law that incorporates nonunity Lewis numbers and heat losses. Using large activation-energy asymptotics and a systematic multiple-scale analysis, we derive the interfacial jump conditions across the flame from first principles. The conventional continuity requirements of mass flux and pressure at an interface under Darcy's law acquire corrections to the finite thickness of the flame. The adiabatic burning rate is shown to involve three distinct Markstein numbers, corresponding to curvature, tangential flow strain, and gravity-induced strain. The gravity term is unique to Darcy's law and has no counterpart in classical Navier--Stokes formulations. Moreover, the curvature Markstein number and the tangential strain Markstein number are found to be unequal, in contrast to the classical case where they coincide under constant transport properties. Explicit formulas for the Markstein numbers are provided, and the resulting new dispersion relation, linking the perturbation wave number $k$ to the growth rate $s$, takes the form $s = (a|k| - bk^2 - d|k|^3) / (1 + c|k|)$. This relation, applicable under Darcy's law, is to be compared to the classical Clavin--Garcia dispersion relation derived from the Navier--Stokes equations. The theory provides a rigorous foundation for flame dynamics in strongly confined environments, with direct applications to porous media combustion and Hele-Shaw cell experiments.

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 / 1 minor

Summary. The paper develops a hydrodynamic theory for premixed flames under Darcy's law (porous media or Hele-Shaw channels) that incorporates nonunity Lewis numbers and heat losses. Using large-activation-energy asymptotics combined with multiple-scale analysis, it derives corrected interfacial jump conditions on mass flux and pressure from first principles, shows that the adiabatic burning rate involves three distinct Markstein numbers (curvature, tangential flow strain, and gravity-induced strain), provides explicit formulas for these numbers, and obtains the dispersion relation s = (a|k| − b k² − d |k|³)/(1 + c |k|).

Significance. If the asymptotic analysis is accurate, the work supplies a first-principles derivation of flame stability under Darcy's law that is independent of phenomenological Markstein corrections, isolates a gravity-induced Markstein number absent from Navier–Stokes formulations, and demonstrates that curvature and tangential-strain Markstein numbers are unequal. The resulting dispersion relation offers concrete, testable predictions for confined combustion that can be compared directly with the classical Clavin–Garcia relation.

major comments (2)
  1. [Derivation of interfacial conditions (multiple-scale analysis)] The central derivation rests on the β → ∞ limit plus multiple-scale matching to obtain the three Markstein numbers and the coefficients a–d in the dispersion relation. No finite-β benchmarks, numerical simulations at realistic β ≈ 10, or error estimates for O(1/β) corrections to the preheat-zone structure are provided; if secular terms from the Darcy drag or buoyancy are missed at this order, the claimed first-principles status of the Markstein numbers and the explicit dispersion relation is compromised.
  2. [Dispersion relation and Markstein-number formulas] The abstract states that explicit formulas for the three Markstein numbers are given and that the dispersion relation follows, yet the relation between the Markstein numbers and the coefficients a, b, c, d is not shown in a single equation or table; without this reduction the claim that the dispersion relation is fully determined by the derived Markstein numbers cannot be verified.
minor comments (1)
  1. [Abstract] The abstract refers to a comparison with the Clavin–Garcia dispersion relation but does not state the precise differences in functional form or coefficient structure that arise under Darcy's law.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight important aspects of the asymptotic derivation and presentation. We address each major comment below and will revise the manuscript accordingly where the points are valid.

read point-by-point responses

  1. Referee: The central derivation rests on the β → ∞ limit plus multiple-scale matching to obtain the three Markstein numbers and the coefficients a–d in the dispersion relation. No finite-β benchmarks, numerical simulations at realistic β ≈ 10, or error estimates for O(1/β) corrections to the preheat-zone structure are provided; if secular terms from the Darcy drag or buoyancy are missed at this order, the claimed first-principles status of the Markstein numbers and the explicit dispersion relation is compromised.

    Authors: The multiple-scale analysis is constructed precisely to remove secular terms order by order; the Darcy drag and buoyancy contributions have been checked at each stage and do not generate additional secular growth at the orders retained. We agree, however, that explicit error estimates or finite-β comparisons would strengthen the presentation. As the work is purely asymptotic, we will add a paragraph in the conclusions discussing the expected O(1/β) corrections and the regime of validity, but we cannot supply new numerical benchmarks within the scope of this theoretical study. revision: partial

  2. Referee: The abstract states that explicit formulas for the three Markstein numbers are given and that the dispersion relation follows, yet the relation between the Markstein numbers and the coefficients a, b, c, d is not shown in a single equation or table; without this reduction the claim that the dispersion relation is fully determined by the derived Markstein numbers cannot be verified.

    Authors: We thank the referee for pointing out this presentational gap. In the revised manuscript we will insert a new equation (or compact table) that explicitly maps each of the three Markstein numbers onto the coefficients a, b, c, d of the dispersion relation, thereby making the reduction transparent and verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from governing equations via asymptotics is self-contained

full rationale

The paper derives interfacial jump conditions and three distinct Markstein numbers (curvature, tangential strain, gravity-induced) explicitly from the Darcy-law governing equations using large-activation-energy asymptotics plus multiple-scale analysis. The resulting dispersion relation s = (a|k| - b k² - d |k|³)/(1 + c |k|) is obtained directly from these derived coefficients rather than from any fitted input, self-citation chain, or renaming of prior results. No load-bearing step reduces by construction to quantities defined in the authors' own earlier work; the abstract and description present the Markstein numbers as first-principles outputs with explicit formulas. This is the normal case of an independent asymptotic analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of Darcy's law for the bulk flow, the large-activation-energy limit, and the multiple-scale separation between flame thickness and hydrodynamic scales. No free parameters are introduced beyond the physical inputs (Lewis number, heat-loss parameter). No new entities are postulated.

axioms (2)
  • domain assumption Large activation-energy asymptotics combined with multiple-scale analysis yields the correct interfacial jump conditions
    Invoked to derive the corrections to mass-flux and pressure continuity across the flame front
  • domain assumption Darcy's law governs the flow outside the thin reaction zone
    Stated as the governing equation for premixed flames in porous media or Hele-Shaw channels

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discussion (0)

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Reference graph

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