  We present the analytical solution for the two-dimensional velocity and
density fields within an approximation for laminar stratified inclined duct
(SID) flows where diffusion dominates over inertia in the along-channel
momentum equation but it is negligible in the density transport equation. We
refer to this approximation as the hydrostatic/gravitational/viscous in
momentum and advective in density (HGV-A) approximation due to the leading
balances in the governing equations. The analytical solution is valid for
laminar flows in a two-layer configuration in the limit of long ducts. Under
such conditions, the non-dimensional volume flux is given by the Froude number
$Fr^* =Re_g/(A\,K)$ with $Re_g$ the gravitational Reynolds number, $A$ the
aspect ratio of the duct, and $K$ a geometrical parameter that depends on the
tilt of the duct and is obtained from the analytical solution. The analytical
solution in the HGV-A approximation is validated against results from
laboratory experiments, and allows us to gain new insight into the dynamics and
properties of SID flows. Most importantly, constant values of $Fr^*$ describe,
in both horizontal and inclined ducts, the transitions between increasingly
turbulent flow regimes: from laminar flow, to interfacial waves, to
intermittent turbulence and sustained turbulence.
