Numerical study of natural oscillations of supported drops with free and pinned contact lines

Abstract

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.

Oscillation Dynamics of Drops on Immiscible Thin Liquid Films

“…Through this study, we describe the relationship among the viscous stresses in the oil film, natural frequency of the drop, and wettability of the liquid oil film. We find that the oil film thickness and viscosity determine viscous stresses that provide a spectrum of cases of the slipping and pinning contact lines for the first mode oscillation frequency, which are well predicted by previously established simulation results and corroborated by the experiments herein.…”

“…Sakakeeny and Ling showed that the frequencies normalized by the capillary frequency decrease with contact angle and increase with the Bond number, and this trend occurs in the first and high-order modes of oscillation . Sakakeeny and Ling also performed simulations for SCL and PCL situations to calculate the natural frequency of the drop, which was in agreement with Bostwick and Steen’s theory . Recently, Kern et al widened the prediction on natural frequencies by studying the PCL case on different solid substrates and with different liquids to clarify the independence of the postpinning energy from the preimpact energyprior to substrate contact …”

“…We assume that the deformation of the liquid film can be ignored because the liquid film thickness is small compared to the contact radius of the drop ( ). To further understand the influence of the liquid film on the drop oscillation dynamics, the normalized frequency of the drop on the liquid film is plotted with the SCL and PCL simulation results from Sakakeeny and Ling, which serve as the two limiting cases for our experimental data, one for perfectly pinned and one for perfectly slipping contact lines (Figure a). It was shown that the simulation results for the two limiting contact line conditions agree well with the inviscid theoretical models of Bostwick and Steen .…”

“…(a) Experimental normalized frequencies at different apparent contact angles with SCL and PCL simulations (the simulation data is on a solid surface) and (b) ratio of the experimental normalized frequencies to the SCL and PCL frequency models. Each data point is an average of five experiments.…”

“…Each data point is an average of five experiments. Reproduced with permission from ref . Copyright 2021 the American Institute of Physics.…”