Stationary residual layers in buoyant Newtonian displacement flows

We study buoyant displacement flows with two miscible fluids of equal viscosity in ducts that are inclined at angles close to horizontal $(β≈90°)$⁠. As the imposed velocity $(V̂0)$ is increased from zero, we change from an exchange flow dominated regime to a regime in which the front velocity $(V̂f)$ increases linearly with $V̂0$⁠. During this transition, we observed an interesting phenomenon in which the layer of displaced fluid remained at the top of the pipe (diameter $D̂$⁠) during the entire duration of the experiment, apparently stationary for times $t̂≳103D̂/V̂0$ (the stationary residual layer). Our investigation revealed that this flow marks the transition between flows with a back flow, counter to the imposed flow, and those that displace instantaneously. The same phenomena are observed in pipes (experiments) and in plane channels (two-dimensional numerical computations). A lubrication/thin-film model of the flows also shows the transition from back flow to instantaneous displacement. At long times, this model has a stationary residual layer solution of the type observed, which is found at a unique ratio $χ$ of the axial viscous velocity to the imposed velocity. The prediction of the stationary residual layer from the lubrication model is compared with the transition in observed behavior in our pipe flow experiments and our 2D numerical displacements in the channel. Reasonable agreement is found for the pipe and excellent agreement for the channel. Physically, in either geometry at transition, the upper layer of the fluid is in a countercurrent motion with zero net volumetric flux; the velocity at the interface is positive, but the velocity of the interface is zero. This results from a delicate balance between buoyancy forces against the mean flow and viscous forces in the direction of the mean flow.