Natural oscillations of a sessile drop on flat surfaces with mobile contact lines

A model to predict the oscillation frequency for drops pinned on a vertical planar surface

“…The B-S predictions generally agree well with simulation results, see figure 4. Small discrepancies are observed for θ 0 90 • , which may be due to the simplified contact line boundary conditions used in the B-S model (Sakakeeny & Ling 2020). The C-K model predicts the frequency by a different approach.…”

“…The height-function method is used to calculate the interface curvature (Popinet 2009) and to specify the contact angle at the surface (Afkhami & Bussmann 2009). Details of the numerical methods and validation studies for the Basilisk solver can be found in previous studies (Sakakeeny & Ling 2020, 2021 and the code website.…”

“…The contact with the solid surface has a significant impact on the oscillation dynamics. The natural frequency of the drop oscillation depends on the equilibrium contact angle (θ 0 ), contact line mobility and Bo (Bostwick & Steen 2014;Sakakeeny & Ling 2020, 2021. The key dimensionless parameters and the ranges of values considered are summarized in table 1.…”

“…Different from free drops, the two n = 1 modes, corresponding to the centroid translation in the directions normal (n = 1, m = 0) and tangential (n = 1, m = 1) to the surface, trigger shape oscillations for supported drops. Since the natural frequency and damping rate increase with n (Lamb 1932), the n = 1 modes typically dominate the oscillations of supported drops (Strani & Sabetta 1984;Noblin et al 2004;Sakakeeny & Ling 2020, 2021. While the longitudinal mode (n = 1, m = 0) is axisymmetric, see figure 1(a), and has been extensively studied (Strani & Sabetta 1984, 1988Noblin et al 2004;Sakakeeny & Ling 2020, 2021, less attention has been paid to the lateral mode (n = 1, m = 1), see figure 1(b), which is the focus of the present study.…”

“…A simple capillary-gravity wave model has been suggested by Noblin et al (2004) to include the effect of gravity on the oscillation frequency, but significant discrepancies with experiment was found for surfaces with large contact angles (Yao et al 2017). Numerical simulations have been recently performed by Sakakeeny & Ling (2020, 2021 to investigate the effect of Bond number on the axisymmetric zonal oscillation modes for supported drops. Their results indicated that the drop oscillation frequency varies significantly with Bo, even when the drop radius is smaller than the capillary length (Bo < 1).…”