  The critical dimension necessary for a flame to propagate in suspensions of
fuel particles in oxidizer is studied analytically and numerically. Two types
of models are considered: First, a continuum model, wherein the individual
particulate sources are not resolved and the heat release is assumed spatially
uniform, is solved via conventional finite difference techniques. Second, a
discrete source model, wherein the heat diffusion from individual sources is
modeled via superposition of the Green's function of each source, is employed
to examine the influence of the random, discrete nature of the media. Heat
transfer to cold, isothermal walls and to a layer of inert gas surrounding the
reactive medium are considered as the loss mechanisms. Both cylindrical and
rectangular (slab) geometries of the reactive medium are considered, and the
flame speed is measured as a function of the diameter and thickness of the
domains, respectively. In the continuum model with inert gas confinement, a
universal scaling of critical diameter to critical thickness near 2:1 is found.
In the discrete source model, as the time scale of heat release of the sources
is made small compared to the interparticle diffusion time, the geometric
scaling between cylinders and slabs exhibits values greater than 2:1. The
ability of the flame in the discrete regime to propagate in thinner slabs than
predicted by continuum scaling is attributed to the flame being able to exploit
local fluctuations in concentration across the slab to sustain propagation. As
the heat release time of the sources is increased, the discrete source model
reverts back to results consistent with the continuum model. Implications of
these results for experiments are discussed.
