  Planar flames are intrinsically unstable in open domains due to the thermal
expansion across the burning front--the Landau-Darrieus instability. This
instability leads to wrinkling and growth of the flame surface, and
corresponding acceleration of the flame, until it is stabilized by cusp
formation. We look at the Landau-Darrieus instability for C/O thermonuclear
flames at conditions relevant to the late stages of a Type Ia supernova
explosion. Two-dimensional direct numerical simulations of both single-mode and
multi-mode perturbations using a low Mach number hydrodynamics code are
presented. We show the effect of the instability on the flame speed as a
function of both the density and domain size, demonstrate the existence of the
small scale cutoff to the growth of the instability, and look for the proposed
breakdown of the non-linear stabilization at low densities. The effects of
curvature on the flame as quantified through measurements of the growth rate
and computation of the corresponding Markstein number. While accelerations of a
few percent are observed, they are too small to have any direct outcome on the
supernova explosion.
