Miscible displacement flows in near-horizontal ducts at low Atwood number

We study buoyant displacement flows with two miscible fluids of equal viscosity in the regime of low Atwood number and in ducts that are inclined close to horizontal. Using a combination of experimental, computational and analytical methods, we characterize the transitions in the flow regimes between inertial- and viscous-dominated regimes, and as the displacement flow rate is gradually increased. Three dimensionless groups largely describe these flows: densimetric Froude number $\mathit{Fr}$, Reynolds number $\mathit{Re}$ and duct inclination $\ensuremath{\beta} $. Our results show that the flow regimes collapse into regions in a two-dimensional $(\mathit{Fr}, \mathit{Re}\cos \ensuremath{\beta} / \mathit{Fr})$ plane. These regions are qualitatively similar between pipes and plane channels, although viscous effects are more extensive in pipes. In each regime, we are able to give a leading-order estimate for the velocity of the leading displacement front, which is effectively a measure of displacement efficiency.