Deformation of a surfactant-covered drop in a linear flow

Abstract

We study the effect of adsorbed surfactant on drop deformation in linear flows by means of analytical solutions for small perturbations of the drop shape and surfactant distribution, and by numerical simulations for large distortions. We consider a drop with the same viscosity as the suspending fluid. Under these conditions, the problem simplifies because the disturbance flow field results solely from the interfacial stresses that oppose the distortion of shape and surfactant distribution induced by the incident flow. A general form of perturbation analysis valid for any flow is presented. The analysis can be carried out to arbitrary order given its recursive structure; a third-order perturbation solution is explicitly presented. The expansions are compared to results from boundary integral simulations for drops in axisymmetric extensional and simple shear flows. Our results indicate that under weak-flow conditions, deformation is enhanced by the presence of surfactant, but the leading-order perturbation of the drop shape is independent of the (nonzero) surfactant elasticity. In strong flows, drop deformation depends nonmonotonically on surfactant elasticity. The non-Newtonian rheology in a dilute emulsion that results from drop deformation and surfactant redistribution is predicted. Shear thinning is most pronounced for low values of the surfactant elasticity. In the weak-flow limit with finite surfactant elasticity, the emulsion behaves as a suspension of rigid spheres. In strong flows, the stresses can approach the behavior for surfactant-free drops.