Hydrodynamic forces acting on a microscopic emulsion drop growing at a capillary tip in relation to the process of membrane emulsification.

Here, we calculate the hydrodynamic ejection force acting on a microscopic emulsion drop, which is continuously growing at a capillary tip. This force could cause drop detachment in the processes of membrane and microchannel emulsification, and affect the size of the released drops. The micrometer-sized drops are not deformed by gravity and their formation happens at small Reynolds numbers despite the fact that the typical period of drop generation is of the order of 0.1 s. Under such conditions, the flow of the disperse phase through the capillary, as it inflates the droplet, engenders a hydrodynamic force, which has a predominantly viscous (rather than inertial) origin. The hydrodynamic boundary problem is solved numerically, by using appropriate curvilinear coordinates. The spatial distributions of the stream function and the velocity components are computed. The hydrodynamic force acting on the drop is expressed in terms of three universal functions of the ratio of the pore and drop radii. These functions are computed numerically. Interpolation formulas are obtained for their easier calculation. It turns out that the increase in the viscosity of each of the two liquid phases increases the total ejection force. The results could find applications for the interpretation and prediction of the effect of hydrodynamic factors on the drop size in membrane emulsification.