Speed of rolling droplets

We analyze the near-rolling motion of two-dimensional nonwetting drops down a gently inclined plane. Inspired by the scaling analysis of Mahadevan and Pomeau [Phys. Fluids 11, 2449 (1999)PHFLE61070-663110.1063/1.870107], we focus upon the limit of small Bond numbers, B1, where the drop shape is nearly circular and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. Our analysis reveals that the leading-order dissipation is contributed by both the flow in the flat-spot region and the correction to rigid-body rotation in the remaining liquid domain. The resulting leading-order approximation for the drop velocity U is given byμU/γ∼α/2Bln1B, wherein μ is the liquid viscosity, γ the interfacial tension, and α the inclination angle.