Flame dynamics and Markstein numbers in Hele-Shaw cells and porous media under Darcy's law

Abstract

The propagation of premixed flames in narrow Hele-Shaw cells and permeable porous media is governed by Darcy's law, leading to hydrodynamic behaviour distinct from conventional flames. This study investigates the role of confinement on flame dynamics, focusing on the associated Markstein numbers. A hydrodynamic model treating the flame as a discontinuity surface is presented, in which the burning rate depends on curvature and tangential flow strain, characterised by two Markstein numbers $\mathcal{M}_c$ and $\mathcal{M}_t$. A major finding is that $\mathcal{M}_c \neq \mathcal{M}_t$ under Darcy's law, as the law permits tangential velocity discontinuities at the flame front due to viscosity variations. Additionally, a third Markstein number $\mathcal{M}_g$ associated with gravity also emerges uniquely under Darcy's law. The Darcy-specific effects vanish in purely radial flows but are important for strained flames. In planar counterflows, for instance, the strain rate jump across the flame is dictated by the unburnt-to-burnt viscosity ratio $\mathfrak{m}$ rather than the density ratio $\mathfrak{r}$, a dramatic departure from conventional behaviour. The influence of confinement on the combined hydrodynamic instabilities of planar flames, namely Darrieus--Landau, Saffman--Taylor, and Rayleigh--Taylor instabilities, is discussed. Weakly nonlinear dynamics under strong confinement is found to follow a Michelson--Sivashinsky equation with modified coefficients (long-wave instability), while under moderate confinement, Ginzburg--Landau dynamics (finite-wavenumber instability) is found to apply. Strong confinement amplifies the Darrieus--Landau instability, enhancing hydrodynamic coupling in conjunction with augmented streamline refraction caused by tangential velocity discontinuities.

Figures (6)

• Figure 1: Schematic of a curved premixed flame. The flame front, defined by the level set $G(\mathbf{x},t)=0$, is assumed to have a wrinkling length scale much larger than its thickness.
• Figure 2: The six integrals $\mathcal{J}_i$, defined in \ref{['J1']}-\ref{['J6']} and appearing in formulas \ref{['marksteinc']}-\ref{['marksteing']}, are plotted versus $q$ for the case $\rho=1/(1+q\theta)$ and $\mu=\lambda=(1+q\theta)^{0.7}$.
• Figure 3: Graphical illustration of the three Markstein numbers \ref{['marksteinc']}-\ref{['marksteing']} as functions of the Lewis number $Le$ when $\beta=20$ and $q=5$ and for $\rho=1/(1+q\theta)$ and $\mu=\lambda=(1+q\theta)^{0.7}$. The critical Lewis number $Le_*=1+l_*/\beta$ below which $\mathcal{M}_t>\mathcal{M}_c$, defined in \ref{['lstar']}, is shown on the right as a function of $q$.
• Figure 4: Refraction of a streamline across a stationary oblique flame front: (a) continuous tangential velocity (usual case based on Navier-Stokes); (b) discontinuous tangential velocity with $\mathrm{g}_t=0$ (Darcy's law).
• Figure 5: Radially symmetric flame configurations: (a) stationary flame in a point-source flow of reactive gas of strength $Q$ and (b) freely propagating flame.