  Direct interactions between the flow field and the chemical reaction in
premixed flames occur when the reaction zone thickness is comparable to, or
greater than flow length scales. To study such interactions, a laminar model is
considered that has direct bearings to steadily propagating deflagrations in a
Hele-Shaw channel with a background plane Poiseuille flow. The study employs
asymptotic analyses, pertaining to large activation energy and lubrication
theories and considers a distinguished limit where the channel width is
comparable to the reaction zone thickness, with account being taken of
thermal-expansion and heat-loss effects. The reaction zone structure and
burning rates depend on three parameters, namely, the Peclet number,
$\mathcal{P}$, the Lewis number, $Le$ and the ratio of channel half-width to
reaction zone thickness, $\lambda_*$. When the parameter $\lambda_*$ is small,
transport processes are controlled by Taylor's dispersion mechanism and an
explicit formula for the effective burning speed $S_T$ is obtained. The formula
indicates that $S_T/S_L \propto 1/Le$ for $\mathcal{P}\gg 1$, which
interestingly coincides with a recent experimental prediction of the flame
speed in a highly turbulent jet flame. The results suggest that the role played
by differential diffusion effects is significant both in laminar and turbulent
cases. The reason for the peculiar $1/Le$ dependence can be attributed, in our
laminar model, to Taylor dispersion. Presumably, this dependence may be
attributed to a similar but more general mechanism in the turbulent case,
rather than to diffusive-thermal curvature effects. The latter effects play
however an important role in determining the flame speed when $\lambda_*$ is
large. The magnitude of heat losses at extinction, is multiplied by a factor
$1/Le^2$ in comparison with those corresponding to the no-flow case in narrow
channels.
