Nonlinear oscillations of pendant drops

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Abstract
Whereas oscillations of free drops have been scrutinized for over a century, oscillations of supported (pendant or sessile) drops have only received limited attention to date. Here, the focus is on the axisymmetric, free oscillations of arbitrary amplitude of a viscous liquid drop of fixed volume V that is pendant from a solid rod of radius R and is surrounded by a dynamically inactive ambient gas. 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 dynamics of such nonlinear oscillations are governed by four dimensionless groups: (1) a Reynolds number Re, (2) a gravitational Bond number G, (3) dimensionless drop volume V/R3 or some other measure of drop size, and (4) a measure of initial drop deformation a/b. In contrast to free drops whose frequencies of oscillation ω decrease as the amplitudes of their initial deformations increase, the change in frequency Δω of pendant drops with increasing initial deformation is drop size dependent. As the average linear size of pendant drops characterized by V1/3 becomes large compared to the rod radius, V1/3/R≫1, Δω falls as a/b rises, in accordance with results for free drops. The dynamics of very small drops, i.e., ones for which V1/3/R≪1, however, are profoundly affected by the presence of the solid rod. For such small drops, Δω rises as a/b rises, a remarkable fact. The results show that for drops of a given size, the frequency is insignificantly affected by viscosity over a wide of range of Reynolds numbers. However, when Re falls below a critical value, the nature of drop motion changes from underdamped oscillations to an aperiodic return to the rest state. Detailed examination of flow fields inside oscillating drops and decomposition of drop shapes into their linear modes supply further insights into the underlying physics. The effect of finite G in modifying the frequencies of oscillations and the rate at which they are damped is also investigated.