  Liquid drops and vibrations are ubiquitous in both everyday life and
technology, and their combination can often result in fascinating physical
phenomena opening up intriguing opportunities for practical applications in
biology, medicine, chemistry and photonics. Here we study, theoretically and
experimentally, the response of pancake-shaped liquid drops supported by a
solid plate that vertically vibrates at a single, low acoustic range frequency.
When the vibration amplitudes are small, the primary response of the drop is
harmonic at the frequency of the vibration. However, as the amplitude
increases, the half-frequency subharmonic Faraday waves are excited
parametrically on the drop surface. We develop a simple hydrodynamic model of a
one-dimensional liquid drop to analytically determine the amplitudes of the
harmonic and the first superharmonic components of the linear response of the
drop. In the nonlinear regime, our numerical analysis reveals an intriguing
cascade of instabilities leading to the onset of subharmonic Faraday waves,
their modulation instability and chaotic regimes with broadband power spectra.
We show that the nonlinear response is highly sensitive to the ratio of the
drop size and Faraday wavelength. The primary bifurcation of the harmonic waves
is shown to be dominated by a period-doubling bifurcation, when the drop height
is comparable with the width of the viscous boundary layer. Experimental
results conducted using low-viscosity ethanol and high-viscocity canola oil
drops vibrated at 70 Hz are in qualitative agreement with the predictions of
our modelling.
