A liquid drop wants to minimize its surface energy. When it is stretched into an elongated shape it will break up into several spherical droplets. The break-up process is driven by surface tension, and can involve inertia of the liquid as well as viscous dissipation in the liquid. If a liquid drop is breaking up in another liquid (such as the breaking of oil drops in vinegar when you make salad dressing), the inertia and viscous dissipation in the outside liquid can be significant as well.

Although the break-up process is, generally, complicated and depends on a variety of material parameters and ambient conditions, there is something we can say in general about the process without reference to the details. Very close to break-up, the minimum radius of the breaking drop is much smaller than all other lengthscales while the time remaining until break-up is also much briefer than all other time-scales.

As a consequence, the break-up dynamics is determined completely by local conditions, which are determined by the break-up dynamics, which are determined by the local conditions... The break-up process is self-similar.

Near break-up, profiles at different times superimpose onto each other with a suitable rescaling of length-scales. One example of such a self-similar profile is given in the figure on the right. It shows the pinch-off region for a viscous liquid drop (mixture of water and glycerin) in a viscous surrounding (silicone oil). The blue line is a calculated profile. The red circles show a profile found experimentally by Itai Cohen & Sidney Nagel (Phys. Fluids 13 2001). The drop is 10 times less viscous than the exterior fluid. The lengths are scaled in units of the minimum neck radius, so 70 corresponds to a length 70 times that of the minimum neck radius.

Notice that the profile is asymmetric. It looks like a shallow cone joined unto a steep cone. This is not an artifact due to gravity, as such structures are seen even when the experimental setup is horizontal. This asymmetry arises because, as the drop breaks, its minimum is also moving, thereby breaking the left-right symmetry. We don't know whether such asymmetry is necessary for breakup. We also don't understand why viscous drop breakup produces a shallow cone whose opening is about 6 degrees even when the viscosity of the drop is equal to the viscosity of the exterior fluid. This is a nonlinear problem with only O(1) numbers in the equations, yet its solution involved a small number (cone slope=0.03...).

Very recently, we discovered an exception to the story described above. The breakup of a water-drop in oil is not self-similar. Instead, the profile of the water drop simply collapses uniformly inward. Near breakup, the dynamics simplifies not by becoming self-similar, but by losing all dependence on the axial coordinate.