Thermally modulated cross-stream migration of a surfactant-laden deformable drop in a Poiseuille flow

Abstract

A theoretical model is developed to study the cross-stream migration of a deformable surfactant-laden droplet suspended in a nonisothermal Poiseuille flow. In addition to the thermocapillary migration of the droplet, presence of shape deformation due to the imposed flow redistributes the surfactants along the interface that has a significant effect on its dynamics, which has yet remained unexplored. Owing to the nonlinearity present in the system of governing equations, an asymptotic approach is adopted to capture the intricate and nontrivial coupling between the various influencing parameters. Assuming negligible fluid inertia and thermal convection, the droplet migration velocity is obtained through small-deformation analysis in the limits of convective and diffusive surfactant transport. For the former limiting case, the droplet migrates towards the flow centerline when the temperature of the suspending medium increases in the direction of the imposed flow; however, the direction of its cross-stream migration reverses for a system with a high viscosity ratio of droplet to carrier phase. Under the same limiting scenario and for systems with low viscosity ratio, when the temperature linearly decreases in the direction of the imposed flow, the cross-stream migration velocity reduces with the increase in the applied temperature gradient till a critical point is reached at which there occurs no cross-stream migration. Beyond the critical point, there is a gradual increase in the magnitude of the cross-stream velocity with further rise in the imposed temperature gradient. The droplet, below the critical temperature gradient, migrates towards the flow centerline; however, above it the droplet moves away from the centerline. For the other limiting case where surfactant transport is dominated by surface convection, the magnitude of the cross-stream velocity is found to be significantly larger and at the same time independent of the droplet-carrier phase viscosity ratio.