  The stability of a thick planar premixed flame, propagating steadily in a
direction transverse to that of unidirectional shear flow, is studied. A linear
stability analysis is carried out in the asymptotic limit of infinitely large
activation energy, yielding a dispersion relation. The relation characterises
the coupling between Taylor dispersion (or shear-enhanced diffusion) and the
flame thermo-diffusive instabilities, in terms of two main parameters, namely,
the reactant Lewis number $Le$ and the flow Peclet number $Pe$. The
implications of the dispersion relation are discussed and various flame
instabilities are identified and classified in the $Le$-$Pe$ plane. An
important original finding is the demonstration that for values of the Peclet
number exceeding a critical value, the classical cellular instability, commonly
found for $Le<1$, exists now for $Le>1$ but is absent when $Le<1$. In fact, the
cellular instability identified for $Le>1$ is shown to occur either through a
finite-wavelength stationary bifurcation (also known as type-I$_s$) or through
a longwave stationary bifurcation (also known as type-II$_s$). The latter
type-II$_s$ bifurcation leads in the weakly nonlinear regime to a
Kuramoto--Sivashinsky equation, which is determined. As for the oscillatory
instability, usually encountered in the absence of Taylor dispersion in $Le>1$
mixtures, it is found to be absent if the Peclet number is large enough. The
stability findings, which follow from the dispersion relation derived
analytically, are complemented and examined numerically for a finite value of
the Zeldovich number. The numerical study involves both computations of the
eigenvalues of a linear stability boundary-value problem and numerical
simulations of the time-dependent governing partial differential equations. The
computations are found to be in good qualitative agreement with the analytical
predictions.
