Dynamics of sessile drops. Part 2. Experiment

Abstract

High-speed images of driven sessile water drops recorded under frequency scans are analysed for resonance peaks, resonance bands and hysteresis of characteristic modes. Visual mode recognition using back-lit surface distortion enables modes to be associated with frequencies, aided by the identifications in Part 1 (Bostwick & Steen, J. Fluid Mech., vol. 760, 2014, pp. 5–38). Part 1 is a linear stability analysis that predicts how inviscid drop spectra depend on base state geometry. Theoretically, the base states are spherical caps characterized by their ‘flatness’ or fraction of the full sphere. Experimentally, quiescent shapes are controlled by pinning the drop at a circular contact line on the flat substrate and varying the drop volume. The response frequencies of the resonating drop are compared with Part 1 predictions. Agreement with theory is generally good but does deteriorate for flatter drops and higher modes. The measured frequency bands agree better with an extended model, introduced here, that accounts for forcing and weak viscous effects using viscous potential flow. As the flatness varies, regions are predicted where modal frequencies cross and where the spectra crowd. Frequency crossings and spectral crowding favour interaction of modes. Modal interactions of two kinds are documented, called ‘mixing’ and ‘competing’. Mixed modes are two pure modes superposed with little evidence of hysteresis. In contrast, modal competition involves hysteresis whereby one or the other mode disappears depending on the scan direction. Perhaps surprisingly, a linear inviscid irrotational theory provides a useful framework for understanding observations of forced sessile drop oscillations.

Footprint geometry and sessile drop resonance

“…Square and circular drops are tested with the same approach as in our previous works [21,22]. can be clearly resolved on the platform.…”

“…For circular drops, our prior work has identified the first 37 modes according to their numbers of layers and sectors [22]. As figure 3 shows, layers and sectors are most recognizable in side and top views, or a cross.…”

“…Modes with lower frequencies are also preferred in these experiments. Low frequency modes exhibit less spectral crowding [22] which means that coexistence and competition of modes are less likely. This favours isolated pure modes and unambiguous modal identification.…”

“…For both, typical observables are mode shapes [11,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], resonance frequencies [11,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], and the evolution of surface waves with forcing amplitude [18,23,[29][30][31][33][34][35]. The free surface waves oscillate at the forcing frequency (harmonically) when the forcing acceleration is low.…”

“…(For a planar rest state, the Laplace-Beltrami operator reduces to the classical Laplace operator.) This model has proved successful in capturing resonant frequencies for driven sessile drops that are not too flat [12,21,22]. These standing waves, pinned along circular footprints, are therefore classified using the spherical harmonic Square Drop Resonance February 21, 2017 Chang, Daniel, Steen symmetries inherited from free spherical drop oscillations [38,39].…”