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A study of small-amplitude shape oscillations of a viscous compound droplet suspended in a viscous host fluid is performed. A generalized eigenvalue problem is formulated and is solved by using the spectral method. The effects of the relevant non-dimensional parameters are examined for three cases, i.e. a liquid shell in a vacuum and a compound droplet in a vacuum or in a host fluid. The fundamental mode 
$l=2$ is found to be dominant. There exist two oscillatory modes: the in phase and the out of phase. In most situations, the interfaces oscillate in phase rather than out of phase. For the in-phase mode, in the absence of the host, as the viscosity of the core or the shell increases, the damping rate increases whereas the oscillation frequency decreases; when the viscosity exceeds a critical value, the mode becomes aperiodic with the damping rate bifurcating into two branches. In addition, when the tension of the inner interface becomes smaller than some value, the in-phase mode turns aperiodic. In the presence of the unbounded host fluid, there exists a continuous spectrum. The viscosity of the host may decrease or increase the damping rate of the in-phase mode. The mechanism behind it is discussed. The density contrasts between fluids affect oscillations of the droplet in a complicated way. Particularly, sufficiently large densities of the core or the host lead to the disappearance of the out-of-phase mode. The thin shell approximation predicts well the oscillation of the compound droplet when the shell is thin.
We present an overview of the SkyMapper optical follow-up programme for gravitational-wave event triggers from the LIGO/Virgo observatories, which aims at identifying early GW170817-like kilonovae out to
$\sim200\,\mathrm{Mpc}$
distance. We describe our robotic facility for rapid transient follow-up, which can target most of the sky at
$\delta<+10\deg $
to a depth of
$i_\mathrm{AB}\approx 20\,\mathrm{mag}$
. We have implemented a new software pipeline to receive LIGO/Virgo alerts, schedule observations and examine the incoming real-time data stream for transient candidates. We adopt a real-bogus classifier using ensemble-based machine learning techniques, attaining high completeness (
$\sim98\%$
) and purity (
$\sim91\%$
) over our whole magnitude range. Applying further filtering to remove common image artefacts and known sources of transients, such as asteroids and variable stars, reduces the number of candidates by a factor of more than 10. We demonstrate the system performance with data obtained for GW190425, a binary neutron star merger detected during the LIGO/Virgo O3 observing campaign. In time for the LIGO/Virgo O4 run, we will have deeper reference images allowing transient detection to
$i_\mathrm{AB}\approx 21\,\mathrm{mag}$
.
