  In the present study we analytically investigate the deformation and bulk
rheology of a dilute emulsion of surfactant-laden droplets suspended in a
linear flow. We use an asymptotic approach to predict the effect of surfactant
distribution on the deformation of a single droplet as well as the effective
shear and extensional viscosity for the dilute emulsion. The non-uniform
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 suspension. 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
are able to 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 non-uniformity in surfactant
distribution on the droplet surface results in a higher droplet deformation and
a higher effective viscosity for either of linear flows considered. 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 towards the direction of flow.
