Shape and rheology of droplets with viscous surface moduli

We develop perturbation theories to describe the flow dynamics of a droplet with a thin layer of insoluble surfactant whose mechanics are described by interfacial viscosity, i.e. a Boussinesq–Scriven constitutive law. The theories quantify droplet deformation in the limit of small capillary number, large viscosity ratio, or large shear Boussinesq number, to a sufficient level of approximation where one can extract information about nonlinear rheology and droplet breakup. In the first part of this manuscript, we quantify the Taylor deformation parameter and inclination angle in shear and extensional flows, developing expressions that resolve discrepancies between current analytical theories and boundary element simulations. Interestingly, the theories we develop appear to accurately describe the inclination angle of a clean droplet over a wider range of viscosity ratios and capillary numbers than previous works. In the second part of the manuscript, we calculate how interfacial viscosity alters the extra stress of a dilute suspension of droplets, in particular the shear stress, normal stress differences, shear thinning and extensional thickening. The normal stresses are intimately related to the lateral migration of droplets in wall-bound shear flow, and we explore the influence of interfacial viscosity on this phenomenon. We conclude by discussing how one can use these theories to describe droplet breakup, and how one can incorporate additional effects into the perturbation theories such as viscoelastic membranes and/or Marangoni flows.