Dynamics of sessile drops. Part 3. Theory of forced oscillations
  A partially-wetting sessile drop is driven by a sinusoidal pressure field that produces capillary waves on the liquid/gas interface. The analysis presented in Part 1 of this series (Bostwick & Steen 2014) is extended by computing response diagrams and phase shifts for the viscous droplet, whose three phase contact-line moves with contact-angle that is a smooth function of the contact line speed. Viscous dissipation is incorporated through the viscous potential flow approximation and the critical Ohnesorge number bounding regions beyond which a given mode becomes over-damped is computed. Davis dissipation originating from the contact-line speed condition leads to damped oscillations for drops with finite contact-line mobility, even for inviscid fluids. The critical mobility and associated driving frequency to generate the largest Davis dissipation is computed. Lastly, regions of modal coexistence where two modes can be simultaneously excited by a single forcing frequency are identified. Predictions compare favorably to related experiments on vibrated drops. 
Joshua B. Bostwick and Paul H. Steen
physics.flu-dyn