Rolling of non-wetting droplets down a gently inclined plane

In their pioneering 1999 paper, Mahadevan & Pomeau argued that small non-wetting drops roll, rather than slide, down gently inclined surfaces. They showed that the rolling speed possesses an anomalous scaling with an inverse dependence upon drop size, in contrast with conventional modes of drop mobility. The Mahadevan–Pomeau scaling was corroborated by the experiments of Quéré and co-workers using superhydrophobic surfaces (1999) and liquid marbles (2001). We here go beyond scaling arguments, carrying out an asymptotic analysis of the well-posed hydrodynamic problem governing a perfectly non-wetting drop moving down an inclined plane under gravity, in the rolling regime where the drop size and the inclination angle are suitably small. The analysis reveals that the Mahadevan–Pomeau scaling should be multiplied by the prefactor (3π/16)√(3/2)≈0.72, in good agreement with the experiments. The analysis also illuminates intriguing characteristics of the flow field inside the drop and a unique "peeling" mechanism by which the contact line propagates along the surface.