# Satellite droplets in a breaking liquid jet

The famous example of a dripping faucet is an example of a Rayleigh-Plateau instability in which there is a certain jet radius below which perturbations on the surface will grow to break the jet into droplets. The drop radius, according to this theory, can be predicted by calculating the fastest growing unstable wavelength.

In practice you often have small satellite droplets after the 'main' droplet, see below

What I wonder is how it is possible that these satellite droplets occur in a breaking jet? Something which, as far as I understand, is not predicted by Rayleigh-Plateau theory.

Does this have to do with the fact that Rayleigh-Plateau theory is only applicable to the onset of breakup? Or is it for example caused by inertial or viscous effects?

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It's caused by surface tension of the air/water interface. As the jet starts to pinch off it creates a neck between the jet and the developing drop, and this neck stretches the surface. When the drop breaks free, the stretched region rebounds. This creates a splash that ejects the small droplets from the trailing edge of the drop and the leading edge of the jet it's just broken away from.

I don't think there is any easy way to model the formation of the small drops. I think you'd have to reach for your finite element package.

Later:

Having done some thinking about it I suspect the above is at best only a partial answer. As the drop separates from the water in the tap a neck does form, but I'm not sure retraction of the neck is the only mechanism for forming the satellite drops. If you have a look at this video it seems fairly clear that the stretched out neck develops an instability of its own and it breaks up to form the satellite droplets. As those droplets separate they may develop necks between them and the process can repeat to form still smaller droplets, though at some point presumably the pressure in the drops (which is inversely proportional to their radius) prevents any further breakup.

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Ok, so you're saying that the rebound causes the small droplets. That sounds like a good explanation. However, that would mean that inertia should play a role as well, right? Purely on the basis of surface tension I would expect the droplet to stay intact because this will minimize its surface area per volume. – Michiel Feb 10 '13 at 13:32
@michielm: See my revision: I think my first answer is at best only a partial answer. – John Rennie Feb 10 '13 at 14:56
So then the satellites come from a series of Rayleigh-Plateau instabilities?! That would certainly explain why every next satellite is smaller than the previous one. – Michiel Feb 12 '13 at 6:26
@michielm: Yes, that certainly seems to be the case in the Youtube video I found, and your picture looks like this as well. – John Rennie Feb 12 '13 at 7:29