Effect of Bond number on the longitudinal and lateral oscillation of drops supported by flat surfaces

Accurate prediction of the natural frequency for the oscillation of a liquid drop supported by a planar substrate is important to many drop applications. Similar to the oscillation of a free drop, the oscillation mode for a drop supported by a planar surface can be characterized by spherical harmonics, with n and m representing the longitudinal and azimuthal wavenumbers. Different from free drops, the two n=1 modes corresponding to the centroid translation in the directions normal (n=1,m=0) and tangential (n=1,m=1) to the surface trigger shape oscillations for supported drops. These two modes typically dominate the oscillations and the natural oscillation frequencies, normalized by the capillary frequency, are functions of the equilibrium contact angle, the Bond number (Bo), the contact line mobility, and the surface orientation. Parametric numerical and experimental studies have been performed to establish a comprehensive understanding of oscillation dynamics. In particular, for drops pinned on a vertical surface, an inviscid model has been developed to predict the oscillation frequency for wide ranges of Bo and contact angles. The model reveals the scaling relation between the normalized frequency and Bo. For a given equilibrium contact angle, the lateral oscillation frequency decreases quadratically with Bo.