Flow around a droplet suspended in a yield-stress fluid

We investigate the minimal yield-stress required in order to hold static an ellipsoidal Newtonian droplet inside a yield-stress liquid. This critical limit (Y-c) is influenced by the droplet aspect ratio (chi), the interfacial tension (gamma), and the viscosity ratio (M) between the droplet and the surrounding liquid, as well as the ratio of the yield-stress to the buoyancy stress (Y). The droplet will remain trapped by the liquid yield-stress for Y > Y c. Our study bridges the gap in the published results between those calculated for bubbles ( M -> 0) and the solid rigid particles ( M -> infinity), being of practical use for those estimating the design of stable yield-stress emulsions. In general, the critical yield number increases with the interfacial tension and the droplet aspect ratio and will decrease with the droplet viscosity. For spherical droplets, our results computed for yield numbers below Y-c suggest that the spherical shaped droplet may propagate in steady motion.