  The three-dimensional diffusive-thermal stability of a two-dimensional flame
propagating in a Poiseuille flow is examined. The study explores the effect of
three non-dimensional parameters, namely the Lewis number $Le$, the Damk\"ohler
number $Da$, and the flow Peclet number $Pe$. Wide ranges of the Lewis number
and the flow amplitude are covered, as well as conditions corresponding to
small-scale narrow ($Da \ll 1$) to large-scale wide ($Da \gg 1$) channels. The
instability experienced by the flame appears as a combination of the
traditional diffusive-thermal instability of planar flames and the recently
identified instability corresponding to a transition from symmetric to
asymmetric flame. The instability regions are identified in the $Le$-$Pe$ plane
for selected values of $Da$ by computing the eigenvalues of a linear stability
problem. These are complemented by two- and three-dimensional time-dependent
simulations describing the full evolution of unstable flames into the
non-linear regime. In narrow channels, flames are found to be always symmetric
about the mid-plane of the channel. Additionally, in these situations, shear
flow-induced Taylor dispersion enhances the cellular instability in $Le<1$
mixtures and suppresses the oscillatory instability in $Le>1$ mixtures. In
large-scale channels, however, both the cellular and the oscillatory
instabilities are expected to persist. Here, the flame has a stronger
propensity to become asymmetric when the mean flow opposes its propagation and
when $Le<1$; if the mean flow facilitates the flame propagation, then the flame
is likely to remain symmetric about the channel mid-plane. For $Le>1$, both
symmetric and asymmetric flames are encountered and are accompanied by temporal
oscillations.
