Transient buoyancy-driven front dynamics in nearly horizontal tubes

The interpenetration of light and heavy liquids has been studied in a long tube inclined at small angles $α$ to the horizontal. For angles greater than a critical angle $αc$ (whose value decreases when the density contrast measured by the Atwood number $At$ increases), the velocity of the interpenetration front is controlled by inertia and takes the steady value $Vf=ki(Atgd)1∕2$⁠, with $ki≃0.7$⁠. At lower angles, the front is initially controlled by inertia, but later limited by viscous effects. The transition occurs at a distance $Xfc$⁠, which increases indefinitely as $α$ increases to $αc$⁠. Once the viscous effects act, the velocity of the front decreases in time to a steady value $Vf∞$ which is proportional to $sinα$⁠. For a horizontal tube in the viscous regime, the velocity of the front decreases to zero as $t−1∕2$⁠. At the same time, the profile of the interface $h(x,t)$ only depends on the reduced variable $x∕t1∕2$⁠. A quasi-unidirectional model reproduces well the variation of the velocity of the front and the profiles of the interface, both in inclined and horizontal tubes. In the inclined tube, the velocity of the front is determined by matching rarefaction waves to a shock wave.