  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 Bellinger's 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 $t_{D} \equiv a^{2}/L\tau$ 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.
