Radiation-affected hydrodynamic instability of particle-laden flames

We extend the classical stability analyses of Landau Darrieus and Markstein to flames propagating in gases that are seeded with fine inert particles, and account for the resulting radiative exchanges. Our main assumptions are that: i) the Zel'dovich number based upon the flame-speed sensitivity to reaction temperature can be considered large; ii) the two-phase mixture is a one-velocity, one-temperature continuous medium; iii) radiation follows the Eddington equation; iv) radiant exchanges are weak, yet non negligible; v) the flame front is optically very thin but local curvature effects can be accounted for via a Markstein length that is proportional to the actual front thickness. Using asymptotic techniques and the normal-mode method we show that radiative exchanges modify the classical dispersion relation in several, possibly antagonistic, ways: 1) radiation-enhanced propagation speed tends to strengthen the Landau-Darrieus mechanism and to weaken the (stabilizing or destabilizing) influence of gravity and that of curvature. 2) transverse radiant exchanges bring about a stabilizing nonlocal curvature effect which is akin to Markstein's for long wavelengths of wrinkling but saturates for short waves (transverse optically-thin limit). The combined effects may result, e.g., in two disjoint ranges of unstable wavelengths. As the evolution equation equivalent to the dispersion relation is of fourth-order in time, the parametric destabilization/stabilization of particle-laden flames might possibly differ from the classical case.