Migration of a surfactant-laden droplet in non-isothermal Poiseuille flow

Abstract

The motion of a surfactant-laden viscous droplet in the presence of non-isothermal Poiseuille flow is studied analytically and numerically. Specifically, the focus of the present study is on the role of interfacial Marangoni stress generated due to imposed temperature gradient and non-uniform distribution of bulk-insoluble surfactants towards dictating the velocity and direction of motion of the droplet when the background flow is Poiseuille. Assuming the thermal convection and fluid inertia to be negligible, we obtain the explicit expression for steady velocity of a non-deformable spherical droplet when the droplet is located at the centerline of the imposed unbounded Poiseuille flow and encountering a linearly varying temperature field. Under these assumptions, the interfacial transport of surfactants is governed by the surface Péclet number which represents the relative strength of the advective transport of surfactant molecules over the diffusive transport. We obtain analytical solution for small and large values of the surface Péclet number. Analytical solution is also obtained for the case in which the surface Péclet number is of order unity by considering small surfactant Marangoni number which represents the relative strength of the surfactant-induced Marangoni stress over the viscous stress. For an arbitrary surface Péclet number, a numerical solution of the surfactant transport equation is performed using an iterative method which compares well with the analytical solutions. Depending on the direction of temperature gradient with respect to the imposed Poiseuille flow, the surfactant-induced Marangoni stress affects the droplet velocity significantly. 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, the presence of surfactants may increase or decrease the magnitude of droplet velocity depending on the relevant governing parameters. Importantly, for particular values of governing parameters, we observe change in the direction of droplet motion due to the presence of surfactants, which may bear significant consequences in the design of droplet based microfluidic systems.