Hydrodynamic instabilities of propagating interfaces under Darcy's law

The hydrodynamic instabilities of propagating interfaces in Hele-Shaw channels or porous media under the influence of an imposed flow and gravitational acceleration are investigated within the framework of Darcy's law. The stability analysis pertains to an interface between two fluids with different densities, viscosities, and permeabilities, which can be susceptible to Darrieus-Landau, Saffman-Taylor, and Rayleigh-Taylor instabilities. A theoretical analysis, treating the interface as a hydrodynamic discontinuity, yields a simple dispersion relation between the perturbation growth rate $s$ and its wavenumber $k$ in the form $s=(ak - bk^2)/(1+ck)$, where $a$, $b$ and $c$ are constants determined by problem parameters. The constant $a$ characterises all three hydrodynamic instabilities, which are long-wave in nature. In contrast, $b$ and $c$, which characterize the influences of local curvature and flow strain on interface propagation speed, typically provide stabilisation at short wavelengths comparable to interface's diffusive thickness. The theoretical findings for Darcy's law are compared with a generalisation of the classical work by Joulin & Sivashinsky, which is based on an Euler-Darcy model. The comparison provides a conceptual bridge between predictions based on Darcy's law and those on Euler's equation and offers valuable insights into the role of confinement on interface instabilities in Hele-Shaw channels. Numerical analyses of the instabilities are carried out for premixed flames using a simplified chemistry model and Darcy's law. The numerical results corroborate with the explicit formula with a reasonable accuracy. Time-dependent numerical simulations of unstable premixed flames are carried out to gain insights into the nonlinear development of these instabilities.