Hydrodynamic theory of premixed flames under Darcy's law

TL;DR

This paper develops a hydrodynamic model of premixed flames under Darcy's law, deriving a simplified pressure equation and analyzing flame stability and propagation in porous media and confined geometries.

Contribution

It introduces a novel theoretical framework applying Darcy's law to premixed flames, including explicit expressions for Markstein numbers and new insights into flame speed contributions.

Findings

Derived a Laplace equation model for pressure across the flame front.

Identified two Markstein numbers relevant for confined flame stability.

Discovered new effects on flame speed from tangential velocity and gravity discontinuities.

Abstract

This paper investigates the theoretical implications of applying Darcy's law to premixed flames, a topic of growing interest in research on flame propagation in porous media and confined geometries. A multiple-scale analysis is carried out treating the flame as a hydrodynamic discontinuity in density, viscosity and permeability. The analysis accounts in particular for the inner structure of the flame. A simple model is derived allowing the original conservation equations to be replaced by Laplace's equation for pressure, applicable on both sides of the flame front, subject to specific conditions across the front. Such model is useful for investigating general problems under confinement including flame instabilities in porous media or Hele-Shaw channels. In this context, two Markstein numbers are identified, for which explicit expressions are provided. In particular, our analysis reveals novel contributions to the local propagation speed arising from discontinuities in the tangential components of velocity and gravitational force, which are permissible in Darcy's flows to leading order, but not in flows obeying Euler or Navier-Stokes equations.