Does dispersed phase inertia affect the shape of sheared emulsion droplets?

Inertial effects on sheared emulsion droplets are a topic of scientific and industrial interest for several applications from processing to microfluidics. Most of the literature have addressed so far the role of inertia of the continuous phase, which is known to affect shear-induced droplet deformation and migration at values of the Reynolds number of the external fluid Re-c > 1. However, less attention has been paid to the case of inertial effects inside the droplets, corresponding to values of the Reynolds number of the droplet fluid Red > 1. Such a case is especially relevant when the viscosity ratio lambda between the droplet and the external fluid is << 1, which is typical of water-in-oil emulsions where the low values of droplet viscosity can result in Red > 1, while Re-c < 1 due to the larger oil viscosity. Here, we focus on the effect of droplet inertia under shear flow at lambda << 1 by high-speed video microscopy experiments in a microcapillary and by numerical simulations based on a front-tracking finite-difference method. The results unveil the droplet's three-dimensional shape under shear flow at low viscosity ratios and show that droplet inertia tends to increase droplet deformation and orientation along the flow direction and to form two vortices inside the droplets even at small Re-c. The latter findings are at variance with the case of external fluid inertia, where droplets become more aligned with the velocity gradient direction.