Slender-body theory for the generation of micrometre-sized emulsions through tip streaming

We report experiments in which a flow rate ${Q}_{i} $ of a fluid with a viscosity ${\ensuremath{\mu} }_{i} $ discharges into an immiscible liquid of viscosity ${\ensuremath{\mu} }_{o} $ that flows in parallel with the axis of the injector. When the outer capillary number verifies the condition ${\mathit{Ca}}_{o} = {\ensuremath{\mu} }_{o} {U}_{o} / \sigma ~\geqslant ~5$, where ${U}_{o} $ and $\sigma $ indicate, respectively, the outer velocity and the interfacial tension coefficient, and if the inner-to-outer velocity ratio is such that ${U}_{i} / {U}_{o} = {Q}_{i} / (\lrm{\pi} {U}_{o} { R}_{i}^{2} )\ll 1$, with ${R}_{i} $ the inner radius of the injector, a jet is formed with the same type of cone–jet geometry as predicted by the numerical results of Suryo & Basaran (Phys. Fluids, vol. 18, 2006, p. 082102). For extremely low values of the velocity ratio ${U}_{i} / {U}_{o} $, we find that the diameter of the jet emanating from the tip of the cone is so small that drops with sizes below $1~\lrm{\ensuremath{\mu}} \mathrm{m} $ can be formed. We also show that, through this simple method, concentrated emulsions composed of micrometre-sized drops with a narrow size distribution can be generated. Moreover, thanks to the information extracted from numerical simulations of boundary-integral type and using the slender-body approximation due to Taylor (Proceedings of the 11th International Congress of Applied Mechanics, Munich, 1964, pp. 790–796), we deduce a third-order, ordinary differential equation that predicts, for arbitrary values of the three dimensionless numbers that control this physical situation, namely, ${\mathit{Ca}}_{o} $, ${\ensuremath{\mu} }_{i} / {\ensuremath{\mu} }_{o} $ and ${U}_{i} / {U}_{o} $, the shape of the jet and the sizes of the drops generated. Most interestingly, the influence of the geometry of the injector system on the jet shape and drop size enters explicitly into the third-order differential equation through two functions that can be easily calculated numerically. Therefore, our theory can be used as an efficient tool for the design of new emulsification devices.