We present numerical results for the breakup of a pair of colloidal particles enveloped by a droplet under shear flow. The smoothed profile method is used to accurately account for the hydrodynamic interactions between particles due to the host fluid
. We observe that the critical capillary number, $Ca_{rm B}$, at which droplets breakup depends on a velocity ratio, $E$, defined as the ratio of the boundary shift velocity (that restores the droplet shape to a sphere) to the diffusive flux velocity in units of the particle radius $a$. For $E < 10$, $Ca_{B}$ is independent of $E$, as is consistent with the regime studied by Taylor. When $E > 10$, $Ca_{B}$ behaves as $Ca_{rm B} = 2E^{-1}$, which confirms Karam and Bellingers hypothesis. As a consequence, droplet break up will occur when the time scale of droplet deformation $dot{gamma}^{-1}$ is smaller than the diffusive time scale $t_{D} equiv a^{2}/Ltau$ in units of $a$, where $L$ is the diffusion constant and $tau$ is the 2nd order coefficient of the Ginzburg-Landau type free energy of the binary mixture. We emphasize that the breakup of droplet dispersed particles is not only governed by a balance of forces. We find that velocity competition is one of the important contributing factor.
