Viscous exchange flows

Gravitationally driven exchange flows of viscous fluids with different densities are analysed theoretically and investigated experimentally within a horizontal channel. Following initiation from rest when there is a vertical boundary dividing the two fluids, the denser fluid slumps under the less dense along the underlying boundary, while the less dense fluid intrudes along the upper boundary. The motion is driven by the pressure gradients associated with the density differences between the two fluids, resisted by viscous stresses, and mathematically modelled by a similarity solution that depends on the ratio of the viscosities of the two fluids. When the viscosity of the less dense fluid is much smaller than the viscosity of the denser fluid, the shape of the interface between the fluids varies rapidly close to the upper boundary and depends weakly on the viscosity ratio within the interior of the flow. Matched asymptotic expansions are employed in this regime to determine the shape of the interface and the rates of its propagation along the boundaries. The similarity solutions are shown to be linearly stable and thus are expected to represent the intermediate asymptotics of the flow. Experiments confirm the similarity form of solutions and demonstrate close agreement with the theoretical predictions when the viscosities of the fluids are comparable, but exhibit some discrepancies when the viscosities differ more substantially. It is suggested that these discrepancies may be due to mixing between the fluids close to the boundaries, which is induced by the no-slip boundary condition. Exchange flows within porous domains are also investigated to determine the shape of the interface as a function of the ratio of the viscosities of the two fluids and using asymptotic analysis, this shape is determined when this ratio is much larger, or smaller, than unity. (c) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3685723]