  Compound particles are a class of composite systems in which solid particles
encapsulated in a fluid droplet are suspended in another fluid. They are
encountered in various natural and biological processes, for e.g., nucleated
cells, hydrogels, microcapsules etc. In this work, we analyze the flow in and
around a concentric compound particle and investigate the deformation and
reorientation dynamics of the confining drop and its stability against breakup
in imposed linear flows. We obtain analytical expressions for the flow fields
upto O(Ca) (capillary number) and the deformed shape of the confining drop upto
O(Ca^2) using a domain perturbation technique. Further, we develop an O(Ca)
constitutive equation for the volume-averaged stress for a dilute dispersion of
compound particles. Compared to linear theory, O(Ca^2) calculations are found
to be important as they make qualitatively different predictions in some linear
flows. We find that the strong hydrodynamic interaction between the
encapsulated particle and the confining interface results in an increased
deformation of the confining drop compared to that of a simple drop, and
enhances the rheological quantities such as the effective shear viscosity,
extensional viscosity, and normal stress differences in a dilute dispersion.
These measures pertaining to particles, drops, and particles coated with a
fluid film are also derived as limiting cases of compound particles. Moreover,
linear viscoelastic behavior of a dilute dispersion of compound particles is
characterized in terms of complex modulus by subjecting the dispersion to a
small amplitude oscillatory shear flow.
