Nonlinear rheology of a dilute emulsion of surfactant-covered spherical drops in time-dependent flows

The effect of an insoluble surfactant on the rheological behaviour of a dilute emulsion is theoretically studied under low-capillary-number conditions. The dynamics depends on three dimensionless time-scale parameters that characterize the strength of the mechanisms that control the magnitude of the distortion of the surfactant distribution on the drop interface. These mechanisms include Marangoni relaxation, drop rotation by the imposed flow, and oscillations of the imposed flow. The interaction of the time scales gives rise to a complex rheological behaviour. The evolution of the system is described by a nonlinear matrix equation derived by expanding the fluid velocity and surfactant distribution in spherical harmonics. Analytical expansions are developed for conditions where the surfactant distribution is only slightly perturbed, which occurs when one of the time-scale parameters is small.