  The motion of a viscous deformable droplet suspended in an unbounded
Poiseuille flow in the presence of bulk-insoluble surfactants is studied
analytically. Assuming the convective transport of fluid and heat to be
negligible, we perform a small-deformation perturbation analysis to obtain the
droplet migration velocity. The droplet dynamics strongly depends on the
distribution of surfactants along the droplet interface, which is governed by
the relative strength of convective transport of surfactants as compared with
the diffusive transport of surfactants. The present study is focused on the
following two limits: (i) when the surfactant transport is dominated by surface
diffusion, and (ii) when the surfactant transport is dominated by surface
convection. In the first limiting case, it is seen that the axial velocity of
the droplet decreases with increase in the advection of the surfactants along
the surface. The variation of cross-stream migration velocity, on the other
hand, is analyzed over three different regimes based on the ratio of the
viscosity of the droplet phase to that of the carrier phase. In the first
regime the migration velocity decreases with increase in surface advection of
the surfactants although there is no change in direction of droplet migration.
For the second regime, the direction of the cross-stream migration of the
droplet changes depending on different parameters. In the third regime, the
migration velocity is merely affected by any change in the surfactant
distribution. For the other limit of higher surface advection in comparison to
surface diffusion of the surfactants, the axial velocity of the droplet is
found to be independent of the surfactant distribution. However, the
cross-stream velocity is found to decrease with increase in non-uniformity in
surfactant distribution.
