On the minimum size of drops composing the type of monodisperse microemulsions obtained via tip streaming

On a recent paper, Castro et al (JFM (2012), 698, 423-445, JFM12) reported the generation of concentrated monodisperse emulsions composed of drops with sizes even below 1 $\mu m$. Drops are generated from the capillary breakup of a long and thin ligament which strongly stretches downstream from the exit of an injector of radius $R_i$. The ligament is formed when a flow rate $Q_i$ of a fluid with a viscosity $\mu_i$ discharges into an immiscible liquid of viscosity $\mu_o$ flowing in parallel with the axis of the injector at a velocity $qU_0$, with $q=Q_i/(\pi R^2_iU_o)$. It was theoretically found that the diameter of the drops obtained is $D_i\propto R_iq^{1/2}$. However, experiments showed that the predicted drop size is only found when the highly stretched ligament is formed. But this occurs for values of $\lambda=\mu_i/\mu_o$ and the capillary number $Ca_o=\mu_o U_o/\sigma$, with $\sigma$ the interfacial tension coefficient, above a certain threshold which depends on the flow rate ratio $q$. In this presentation we theoretically show that the boundaries in the ($Ca_o$,$\lambda$,$q$) space in which highly stretched long ligaments are formed, corresponds to the conditions under which the jet, calculated using the slender-body description of JFM12, is \emph{globally stable}.