  The motion of a surfactant-laden viscous droplet in the presence of
background non-isothermal Poiseuille flow is studied analytically and
numerically. Specifically, the effect of interfacial Marangoni stress due to
non-uniform distribution of surfactants and temperature at the droplet
interface on the velocity and direction of motion of the droplet along the
centerline of imposed Poiseuille flow is investigated in the presence of
linearly varying temperature field. In the absence of thermal convection, fluid
inertia and shape deformation, the interfacial transport of bulk-insoluble
surfactants is governed by the surface Peclet number which represents the
relative strength of the advective transport of surfactant over the diffusive
transport. We obtain analytical solution for small and large values of the
surface Peclet number. Numerical solution is obtained for arbitrary surface
Peclet number, which compares well with the analytical solution. Depending on
the direction of temperature gradient with respect to the imposed Poiseuille
flow, the surfactant-induced Marangoni stress affects the droplet velocity
differently. When the imposed temperature increases in the direction of imposed
Poiseuille flow, surfactants retard the droplet motion as compared with a
surfactant-free droplet. However, when the imposed temperature decreases in the
direction of imposed Poiseuille flow, presence of surfactants may increase or
decrease the magnitude of droplet velocity depending on the relevant governing
parameters. Further, for particular values of governing parameters, we observe
change in direction of droplet motion due to presence of surfactants, which may
bear significant consequences in the design of droplet based microfluidic
systems.
