Effect of Marangoni stress on the bulk rheology of a dilute emulsion of surfactant-laden deformable droplets in linear flows

In the present study, we analytically investigate the deformation and bulk rheology of a dilute emulsion of surfactant-laden droplets suspended in linear flows. We use an asymptotic approach to determine the effect of surfactant distribution on the deformation of an isolated droplet as well as the effective shear and extensional viscosity for a dilute emulsion. The nonuniform distribution of surfactants due to the bulk flow results in the generation of a Marangoni stress, which affects both the deformation as well as the bulk rheology of the emulsion. The present analysis is done for the limiting case when the surfactant transport is dominated by the surface diffusion relative to surface convection. As an example, we have used two commonly encountered bulk flows, namely, uniaxial extensional flow and simple shear flow. With the assumption of negligible inertial forces present in either of the phases, we show that both the surfactant concentration on the droplet surface as well as the ratio of viscosity of the droplet phase with respect to the suspending fluid has a significant effect on the droplet deformation as well as the bulk rheology. It is seen that increase in the nonuniformity in surfactant distribution on the droplet surface results in a higher droplet deformation and a higher effective viscosity for either of the linear flows considered. The effect of surfactant distribution on effective viscosity is insignificant for highly viscous droplets. For the case of simple shear flow, surfactant distribution is found to have no effect on the inclination angle. However, a higher viscosity ratio predicts the droplet to be more aligned toward the direction of flow. First and second normal stress differences are present for the case of a simple shear flow, of which the former is found to increase with nonuniformity in surfactant distribution, whereas the later remains unaffected.