Parametric oscillations of the sessile drop

Waves are formed on the surface of a sessile drop driven through substrate vibrations oriented at a slanting angle from the normal. A mathematical model is derived, which leads to an infinite system of coupled Mathieu equations governing the wave dynamics that are solved using Floquet theory. The spatial structure of the waves is described by the mode number pair $[\ell,m]$ , where $\ell$ and $m$ are the polar and azimuthal mode numbers, respectively. Limiting cases corresponding to horizontal and vertical vibrations are discussed with predictions agreeing well with prior literature. We focus our results on three drop motions – (1) harmonic $[1,1]$ rocking mode, (2) harmonic $[2,0]$ pumping mode, and (3) subharmonic rocking $[1,1]$ mode – as they depend upon the slanting angle, static contact angle, and contact-line conditions, which we assume to be either pinned or freely moving with fixed contact angle. New theoretical predictions are tested through experiments over a range of parameters, showing good agreement.