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

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 for a one-step Arrhenius chemistry model 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 Navier--Stokes case where they coincide in the one-step chemistry model. 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.