Two-layer viscous drops on inclined surfaces

We present a theoretical and experimental study of the dynamics of two-layer viscous drops driven by gravity on inclined surfaces, motivated by natural, biological and industrial phenomena involving the interactions between two fluid layers, e.g. lubrication. A model describing the evolution of two fluids on an inclined substrate is developed and explored to reveal a variety of flow regimes for different modes of release. The asymptotic reduction of this problem due to the dominance of the along-slope component of gravity is shown to yield significant analytical inroads compared to previous studies of multi-layer flow configurations, which have focused exclusively on the case of horizontal beds. For the canonical example in which two fluids are introduced at a constant flux, the flow forms two regions: an upstream region containing both fluids, and a downstream region comprised purely of the lighter fluid, with a sharp intervening jump in thicknesses between the two. By constructing similarity solutions, we establish a full regime diagram of the possible configurations over all asymptotic limits of the viscosity, flux and density ratios. For the release of two fixed volumes of fluid, the layers separate completely into two disjoint but connected drops, contrasting in essential structure from the constant-flux case. Even a small volume of the heavier drop is able to significantly accelerate the propagation of the lighter drop in front of it. Excellent agreement is found between our theoretical predictions and the results of a series of laboratory experiments.