  Compound droplets can be used in substance encapsulation and material
compartmentalization to achieve a precise control over the relevant processes
in many applications, such as bioanalysis, pharmaceutical manufacturing, and
material synthesis. The flow fields in compound droplets directly affect the
performance of these applications, but it is challenging to measure them
experimentally. In this study, the flow in compound droplets in axisymmetric
microchannels is simulated using the Finite Volume Method, and the interface is
captured using the Level Set Method with surface tension accounted for via the
Ghost Fluid Method. The combination of the Level Set Method and the Ghost Fluid
Method reduces spurious currents that are produced unphysically near the
interface, and achieves a precise simulation of the complex flow field within
compound droplets. The shape of compound droplets, the vortical patterns, the
velocity fields, and the eccentricity are investigated and the effects of the
key dimensionless parameters, including the size of the compound droplet, the
size of the core droplet, the capillary number, and the viscosity ratio, are
analyzed. The flow structures in multi-layered compound droplets are also
studied. This study not only unveils the complex flow structure within compound
droplets moving in microchannels, but can also be used to achieve a precise
control over the relevant processes in a wide range of applications of compound
droplets.
