  In this paper, we present a theoretical, experimental, and numerical study of
the dynamics of cavitation bubbles inside a droplet suspended in another host
fluid. On the theoretical side, we provided a modified Rayleigh collapse time
and natural frequency for spherical bubbles in our particular context,
characterized by the density ratio between the two liquids and the
bubble-to-droplet size ratio. Regarding the experimental aspect, experiments
were carried out for laser-induced cavitation bubbles inside oil-in-water (O/W)
or water-in-oil (W/O) droplets. Two distinct fluid-mixing mechanisms were
unveiled in the two systems, respectively. In the case of O/W droplets, a
liquid jet emerges around the end of the bubble collapse phase, effectively
penetrating the droplet interface. We offer a detailed analysis of the criteria
governing jet penetration, involving the standoff parameter and impact velocity
of the bubble jet on the droplet surface. Conversely, in the scenario involving
W/O droplets, the bubble traverses the droplet interior, inducing global motion
and eventually leading to droplet pinch-off when the local Weber number exceeds
a critical value. This phenomenon is elucidated through the equilibrium between
interfacial and kinetic energies. Lastly, our boundary integral model
faithfully reproduces the essential physics of nonspherical bubble dynamics
observed in the experiments. We conduct a parametric study spanning a wide
parameter space to investigate bubble-droplet interactions. The insights from
this study could serve as a valuable reference for practical applications in
the field of ultrasonic emulsification, pharmacy, etc.
