Influence of complex interfacial rheology on the thermocapillary migration of a surfactant-laden droplet in Poiseuille flow

Abstract

The effect of surface viscosity on the motion of a surfactant-laden droplet in the presence of a non-isothermal Poiseuille flow is studied, both analytically and numerically. The presence of bulk-insoluble surfactants along the droplet surface results in interfacial shear and dilatational viscosities. This, in turn, is responsible for the generation of surface-excess viscous stresses that obey the Boussinesq-Scriven constitutive law for constant values of surface shear and dilatational viscosities. The present study is primarily focused on finding out how this confluence can be used to modulate droplet dynamics in the presence of Marangoni stress induced by nonuniform distribution of surfactants and temperature along the droplet surface, by exploiting an intricate interplay of the respective forcing parameters influencing the interfacial stresses. Under the assumption of negligible fluid inertia and thermal convection, the steady-state migration velocity of a non-deformable spherical droplet, placed at the centerline of an imposed unbounded Poiseuille flow, is obtained for the limiting case when the surfactant transport along the interface is dominated by surface diffusion. Our analysis proves that the droplet migration velocity is unaffected by the shear viscosity whereas the dilatational viscosity has a significant effect on the same. The surface viscous effects always retard the migration of a surfactant-laden droplet when the temperature in the far-field increases in the direction of the imposed flow although the droplet always migrates towards the hotter region. On the contrary, if a large temperature gradient is applied in a direction opposite to that of the imposed flow, the direction of droplet migration gets reversed. However, for a sufficiently high value of dilatational surface viscosity, the direction of droplet migration reverses. For the limiting case in which the surfactant transport along the droplet surface is dominated by surface convection, on the other hand, surface viscosities do not have any effect on the motion of the droplet. These results are likely to have far-reaching consequences in designing an optimal migration path in droplet-based microfluidic technology.