What causes water droplets to drop in periodic, but not uniform time intervals?

Question

A little while ago I noticed water droplets forming from a slightly overflowing reserve in my sink. They dropped in a special periodic time pattern, which was not uniform. Instead two droplets would be created and fall right after each other followed by a longer pause than the distance in time between the two droplets. Then the pattern would repeat, resembling something like "oo…oo…oo…" and so on, if "o" represents a single droplet and "…" a pause longer than the distance in time between two droplets "oo".

I have recorded this with my camera and made it available here: https://nextcloud03.webo.hosting/s/eY5cAsb3XsxDy57

Is there any theory capable of predicting such a behaviour (ideally qualitatively and quantitatively)?

(I am grateful for any suggestion for improving the question and its associated tags.)

Answer 1

The dripping faucet is a classic example of chaos in everyday life.

What the OP describes is possibly the first step in a complex bifurcation route to chaos, where

As the flow speeds up, the dripping pattern changes abruptly at certain thresholds, their mathematics predicts. For slow speeds, there is regular, metronome-like dripping. Then the sequence splits, or 'bifurcates': instead of repeating every drip, it repeats every other drip. For instance, the researchers saw this bifurcation experimentally too, as an alternation between two drops of different size.

The system has been studied since at least the 1980s and there's quite some literature on it so, yes, there are quantitative models which can predict its behavior - though I'm not aware of any for the OP's specific configuration of an overflowing container.

Its details are pretty complicated, because fluid dynamics plays such a decisive role in it. For instance, the specific shape of the drop (e-print), as well as its vibrations (e-print) are also important.

Also, the system is not only chaotic, it also displays hysteresis, i.e.,

in some ranges of flow speed, the behaviour differs according to whether the speed is increasing or decreasing -- whether the 'tap' is being turned on or off. The tap, in other words, remembers its own 'history'.

Answer 2

The droplet formation process depends on gravity, surface tension, the nozzle diameter, and any velocity that the water has before it begins forming up into a droplet at the nozzle tip. When one droplet breaks off the nozzle tip, it tends to leave behind a velocity field in the water right next to the nozzle tip which perturbs the formation of the next droplet. This makes the behavior of droplet #2 dependent on the behavior of droplet #1, since droplet #1 establishes the initial conditions for the formation of the next drop.

Note also that because the growing droplet has mass and the surface of the droplet acts like a springy membrane, there will be a natural frequency at which the system likes to oscillate. Note further that the mass goes like ~(cube root of radius) and the compliance of the surface goes like ~(some godawful nonlinear function of the radius). This means that the natural frequency is a strong function of the displacement of the droplet, meaning in turn that the system is strongly nonlinear.

The presence of strong nonlinearity, feedback effects between consecutive droplets, and sensitive dependence on initial conditions add up to something called quasiperiodic behavior, which is related to (but different from) chaotic dynamics. Water droplets breaking off a nozzle are a good example of quasiperiodicity.

Answer 3

Water molecules have a specific force of attraction between them and also between the molecules of other materials. That is why they form droplets instead of being perfectly flat and uniform. Gravity pulls water molecules down, but they don't immediately fall because of this attraction. However gravity combined with the force of attraction causes more water molecules to gradually buildup in each droplet, making it heavier and also elongating it in height. When the molecular attraction is no longer able to counteract the combined weight of a certain portion of the droplet, it falls. The mechanism is similar to a string holding multiple weights. If you keep adding weights, at one point the string will no longer be able to hold the weights and it will snap, depending on the strength of the string. The interval is not uniform because the distribution of water surrounding each droplet is not perfectly uniform, so the rate at which the droplets buildup is also not uniform. And also the frequency of the falling droplets should gradually decrease over time, as there will less and less 'free' water molecules to add to the droplets, provided the water is not constantly being replenished. In order to quantitatively describe it exactly, you would need to take into account how the water is distributed surrounding each droplet, the force of attraction between the molecules, the gravitational force, and also the temperature of the water, which affects its viscosity.

Answer 4

When cohesion properties cause a water droplet to surpass the adhesion abilities of the surface to which it is adhered, gravity can then cause the drop of water to fall.