When a vertical stream of liquid is high enough, usually the stream starts to fragment into drops. The standard explanation for that that I believe is that the falling liquid is sped up, so the stream narrows, and at some point the stream is narrow enough that droplets are a lower energy state from a surface tension POV.
I today noticed that streams of my shower gel either never fragments into drops, or they require a height that I cannot provide before that happens.
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I think you are referring to tide forces, or what is known as spaghettification when it happens around very strong gravitational gradients like black holes.
I'm not sure if that is what happens to a falling stream of water near the Earth because of the effects of aerodynamic drag. The water will reach terminal velocity very quickly.
It's an interesting problem, though.
@Pat @freemo "Very quickly" still leaves an observable portion of the stream where this hasn't happened yet:
If you open your tap just a little bit, you'll notice that the stream narrows downward, and at some point breaks into droplets.
The narrowing obviously is caused by the water still accelerating: the total volume-based flow of water is conserved, so the crosssectional area of the stream is inversely proportional to the speed.
Also note that terminal velocity of a stream should be much higher than that of a droplet, because only viscous friction slows it down (as opposed to having to move the air "sideways" too).
Yes, aerodynamic forces wouldn't be very much at velocities obtained after falling only a few centimeters.
If the stream of water was to suddenly become "disconnected" at the top (at the faucet) then the stretching doesn't continue (I think), and eventually, if it falls far enough, it would form into a sphere. So the connection to the fixed faucet has something to do with it, I think.
FYI I forgot to mention it at the time but the phenomenon of water breaking up in the stream is called Plateau–Rayleigh instability
https://en.wikipedia.org/wiki/Plateau%E2%80%93Rayleigh_instability
@freemo So your hypothesis is just that the heights are scaled up due to high viscosity, not that the effect will never happen? (I can see how viscosity can change the time required, but can't see how it can affect whether the whole thing happens at all.)
Actually, I might be able to predict how viscosity, vertical air speed, initial radius, and maybe something else (density?) too affect the distance just by dimensional analysis. Hm~ figuring out how those parameters and liquid-air surface tension affect the height sounds like a nice problem, solutions to which can be verified: I can adjust many of these parameters by e.g. mixing up various concentrations of soap in water, and should be able to measure them independently. If by some amount of luck I find enough time and will to do it, will be sure to write it up.