Weak-Stretch Premixed-Flame Response¶
A convention-specific first-order change in premixed-flame propagation speed under weak surface stretch relative to a planar reference, with curvature handled separately where needed.
Core Idea¶
A weakly stretched premixed flame can have a propagation speed different from a comparable planar, unstretched flame. Once its tracked surface, speed, gas-side reference, and stretch sign are specified, a local first-order speed difference can be described by a Markstein sensitivity. Flame stretch is the fractional area-change rate of a moving flame-surface element; tangential strain and propagation of a curved front can both contribute. The resulting coefficient is tied to these conventions and the weak regime, not a universal material constant.[ref-fe7313ce6fac][ref-d8da46a32970][^ref-169954bc9dd7]
The narrow relation applies in unlike settings: an outward spherical flame has curvature-bearing area growth, while a near-planar counterflow flame is mainly strained. Zhang and colleagues test spherical zero-stretch extrapolation with synthetic data; Davis and colleagues report numerical counterflow results. These establish two geometrical routes to a convention-specific response, without making their coefficients interchangeable.[ref-3339f61a16cd][ref-d8da46a32970][ref-169954bc9dd7][ref-fe7313ce6fac]
Scope of Application¶
Use this entry when a premixed flame has a defined tracked surface and speed, a comparable planar unstretched reference, and a weak, approximately quasi-steady stretch interval supporting a local slope. Specify displacement versus consumption speed, burned versus unburned gas, and any separate curvature term. Equal total stretch need not produce equal speed corrections when curvature and strain differ. Strongly nonlinear or rapidly transient regimes need additional modeling.[ref-fe7313ce6fac][ref-d8da46a32970]
The cited spherical study tests inference on synthetic flame data, and the cited Davis counterflow papers describe simulations rather than new direct measurements in those papers. A confined flame may also change speed through pressure rise; that is a limit on complete prediction from one local stretch coefficient, not an exclusion of all local stretch response.[ref-3339f61a16cd][ref-d8da46a32970][ref-169954bc9dd7][ref-86759b81ccc9]
Clarity¶
“Positive Markstein length” is incomplete without the flame surface, speed, gas side, sign convention, and fitting interval. Davis and colleagues distinguish burned- and unburned-gas values, while Thiesset and colleagues show the influence of a tracked iso-surface and the distinction between displacement and consumption speeds. Zero total stretch also need not mean a planar flame: compensating strain and curvature can leave a speed effect.[ref-d8da46a32970][ref-169954bc9dd7][^ref-fe7313ce6fac]
Manages Complexity¶
Ask four questions: Which surface and speed are tracked? How do strain and curvature enter stretch? What is the planar reference? What local response slope is supported over the stated interval? This lets spherical extrapolations and counterflow simulations be compared without silently equating their conventions. Model choice, radius range, and image noise can materially affect an inferred spherical Markstein length.[ref-fe7313ce6fac][ref-d8da46a32970][^ref-3339f61a16cd]
Abstract Reasoning¶
Hold mixture and conventions fixed, subtract the planar reference speed, and inspect the difference ΔS over a weak stretch interval. A retained first-order term is locally homogeneous and additive in its weak perturbation inputs; test whether curvature needs a separate contribution. This approximate Linearity is an internal constituent of the admitted first-order relation, not a claim that raw speed or the entire flame is globally linear. Report a slope only with the conventions and range that give it meaning.[ref-fe7313ce6fac][ref-d8da46a32970]
Knowledge Transfer¶
Transfer the role test from spherical expansion to near-planar counterflow: identify the surface, speed, stretch contributions, reference, and local correction anew. The geometry and gas-side conventions change, so the numerical coefficient must be checked again. A generic small-signal response elsewhere is only an analogy unless a premixed reaction front and its area-rate stretch are present.[ref-fe7313ce6fac][ref-169954bc9dd7]
Example¶
Zhang, Coronel, and Mével compare extrapolation models using synthetic spherically expanding flame data. Curvature-bearing growth supplies stretch; a selected radius and speed convention support a fitted zero-stretch intercept and Markstein slope. Their result tests the inference method, including its sensitivity to model and noise, rather than reporting a new direct flame measurement.[ref-3339f61a16cd][ref-fe7313ce6fac]
Davis, Quinard, and Searby numerically study near-planar counterflow flames. Flow strain supplies much of the stretch with little curvature; they distinguish burned- and unburned-gas speed conventions and compare their Markstein results with spherical literature. This is the second, geometrically unlike realization of the same bounded role pattern.[ref-d8da46a32970][ref-169954bc9dd7][^ref-fe7313ce6fac]
Relationships to Other Abstractions¶
Current abstraction Weak-Stretch Premixed-Flame Response Domain-specific
Parents (1) — more general patterns this builds on
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Weak-Stretch Premixed-Flame Response is part of Linearity Prime
A small-signal linear speed correction is an identity-bearing constituent of the weak-stretch response.
Hierarchy path (1) — routes to 1 parentless root
- Weak-Stretch Premixed-Flame Response → Linearity
Neighborhood in Abstraction Space¶
Weak-Stretch Premixed-Flame Response sits in a sparse region of the domain-specific corpus (96th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Diffusive–Thermal Instability — 0.81
- Mechanical Strain — 0.77
- Nusselt number — 0.76
- Equilibrium Climate Sensitivity — 0.76
- Pressure — 0.76
Computed from structural-signature embeddings · 2026-10-17
Not to Be Confused With¶
Flame stretch is the area-rate input, not the speed response; planar laminar flame speed is the reference; Markstein length is a convention-dependent coefficient. The historical Matalon–Matkowsky–Clavin–Joulin theory is source lineage rather than an independent positive case. High-strain extinction, cellular instability, and confined-vessel pressure rise require further conditions or mechanisms and do not follow from one local slope. The Linearity parent applies to ΔS's retained first-order correction, not to the full flame.[ref-fe7313ce6fac][ref-d8da46a32970][^ref-86759b81ccc9]
References¶
[^ref-fe7313ce6fac]: F. Thiesset et al., “Isolating strain and curvature effects in premixed flame/vortex interactions,” Journal of Fluid Mechanics 831 (2017), 618–654, https://doi.org/10.1017/jfm.2017.641. Full authored manuscript consulted at https://caprysses.fr/IMG/publis/2017_Thiesset_JFM.pdf, especially equations (1.3), (1.5), (1.7) and discussion of surface and regime limits. [^ref-d8da46a32970]: S. G. Davis, J. Quinard, and G. Searby, “Determination of Markstein numbers in counterflow premixed flames,” Combustion and Flame 130 (2002), 112–122, https://doi.org/10.1016/S0010-2180(02)00369-3. Author-hosted abstract consulted at https://www.irphe.fr/~searby/GSPapers.html, especially the burned/unburned reference distinction; full article was inaccessible. [^ref-169954bc9dd7]: S. G. Davis, J. Quinard, and G. Searby, “Markstein numbers in counterflow, methane- and propane-air flames: a computational study,” Combustion and Flame 130 (2002), 123–136, https://doi.org/10.1016/S0010-2180(02)00368-1. Author-hosted original abstract consulted at https://www.irphe.fr/~searby/GSPapers.html; full article was inaccessible. [^ref-3339f61a16cd]: Y. Zhang, S. A. Coronel, and R. Mével, “Numerical study of synthetic spherically expanding flames for optimization of laminar flame speed experiments,” Fuel 310 (2022), article 122367, https://doi.org/10.1016/j.fuel.2021.122367. Institutional original-article abstract consulted at https://authors.library.caltech.edu/records/3b0sq-9dg56; the study uses synthetic data. [^ref-86759b81ccc9]: J. K. Bechtold, G. Krishnan, and M. Matalon, “Hydrodynamic theory of premixed flames propagating in closed vessels: flame speed and Markstein lengths,” Journal of Fluid Mechanics 998 (2024), A59, https://doi.org/10.1017/jfm.2024.919. Full publisher-hosted article consulted for confined pressure-rise effects and convention limits.