Forced oscillations of pendant (sessile) drops

Oscillations of supported liquid drops are the subject of wide scientific interest, with applications in areas as diverse as liquid–liquid extraction, synthesis of ceramic powders, growing of pure crystals in low gravity, and measurement of dynamic surface tension. In this study, axisymmetric forced oscillations of arbitrary amplitude of a viscous liquid drop of fixed volume which is pendant from or sessile on a rod with a fixed contact line and surrounded by an inviscid ambient gas are induced by moving the rod in the vertical direction sinusoidally in time. This nonlinear free boundary problem is solved by a method of lines using Galerkin/finite element analysis for discretization in space and an implicit, adaptive finite difference technique for discretization in time. The variation of the drop response over a wide range of the governing parameters (Reynolds number Re, gravitational Bond number G, volume, and forcing frequency and amplitude) is analyzed. The results show that as the forcing frequency is increased, a sequence of oscillation modes is observed, each with its own resonance frequency ω rn , n=1,2,…, at which drop response amplitude reaches a local maximum. While resonance frequencies depend strongly on drop size and on forcing amplitude, the effect of Reynolds number on ω rn is large when Re is small and diminishes when Re is large, in accord with observations for free oscillations. At high Re, a sharp increase in drop deformation can occur for drops forced to oscillate in the vicinity of their resonance frequencies, indicating the incipience of hysteresis. The maximum observed drop deformations increase with Re, G, and forcing amplitude, while the value of the drop deformation as a function of drop size is determined by a balance between the magnitude of the viscous shear stress imposed on the drop liquid by the solid rod relative to the capillary pressure due to surface tension acting on the fluid interface. The effects of viscous dissipation are also seen in the damping of various oscillation modes and in the creation, evolution in time, and disappearance of zones of fluid recirculation within the drop.