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The oscillation of droplets supported by solid surfaces is important for a wide variety of applications such as dropwise condensation. In the present study, the axisymmetric natural oscillations of a liquid drop supported by a flat surface are investigated by direct numerical simulation. The liquid–gas interface is captured using a geometric volume-of-fluid method. A parametric study is carried out by varying the equilibrium contact angle and the gravitational Bond number (Bo). Both positive and negative gravities are considered, and thus the results cover both pendant and sessile drops. To incorporate the effect of contact line mobility, the two asymptotic limits, namely, the pinned contact line (PCL) and free contact line (FCL) conditions, are considered and their effects on the drop oscillation features are characterized. The predicted oscillation frequencies for PCL and FCL serve as the upper and lower bounds for general situations. The drop oscillation is initiated by increasing the gravity magnitude for a short time. The first mode due to the drop centroid translation dominates the excited oscillation. The oscillation frequency scales with the capillary frequency, and the normalized frequency monotonically decreases with the equilibrium contact angle. For zero gravity, the computed frequencies for all contact angles agree remarkably well with the inviscid theory for both the PCL and FCL conditions. The kinetic energy correction factor is introduced to account for the additional contribution of the oscillation-induced internal flow to the overall kinetic energy of the drop. Both the frequency and the kinetic energy correction factor increase with Bo, decrease with the contact angle, and increase when the contact line condition changed from FCL to PCL. The variation of oscillation frequency due to the change of Bo is particularly significant when the contact angle is large, suggesting that the gravity effect must be incorporated to accurately predict the oscillation frequency for drops supported by hydrophobic or superhydrophobic surfaces.

Accurate prediction of the natural frequency for the lateral oscillation of a liquid drop pinned on a vertical planar surface is important to many drop applications. The natural oscillation frequency, normalized by the capillary frequency, is mainly a function of the equilibrium contact angle and the Bond number ( $Bo$ ), when the contact lines remain pinned. Parametric numerical and experimental studies have been performed to establish a comprehensive understanding of the oscillation dynamics. An inviscid model has been developed to predict the oscillation frequency for wide ranges of $Bo$ and the contact angle. The model reveals the scaling relation between the normalized frequency and $Bo$ , which is validated by the numerical simulation results. For a given equilibrium contact angle, the lateral oscillation frequency decreases with $Bo$ , implying that resonance frequencies will be magnified if the drop oscillations occur in a reduced gravity environment.

While drop oscillation dynamics has been widely studied for many decades, the influence of a moving contact line on the oscillation modes of drops remains underexplored. Herein, we report the oscillation dynamics of drops on thin liquid films with different viscosities where lower viscosities provide a slipping surface and higher viscosities immobilize the contact line. A gently deposited drop onto an oil film undergoes shape oscillations due to capillarity, where the frequency, amplitude, and apparent contact angle are tracked via a high-speed camera. This study demonstrates that restraining the mobility of the drop contact line by increasing the viscosity of a thin oil film underneath the drop increases the extent of the drop oscillation time as well as affecting the natural frequency of the drop oscillation. The drop oscillation time was defined by the time at which the changes in the drop height dropped to values less than 1% of the equilibrium height. The experimental results for the first longitudinal mode oscillation frequencies as a function of the equilibrium contact angles for the pinning and slipping contact lines were in good agreement with previously reported numerical simulations and model predictions.

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