The only role surface tension plays in the Feynman problem is to ensure that the cross section of the stream is a circle. The rest you can work out based on classical mechanics and the fact that the amount of water is conserved.
You actually get exactly the same "thinning" behavior if somehow the stream is square shaped or anything else in cross-section.
Air pressure probably dominates over surface tension.
Imagine a water column large enough that surface tension is very weak, say a tap with an outlet 1 metre in diameter.
At the exit of the tap the pressure of the water is at air pressure ( = 1 bar: any less and the water would be pushed back into the tap).
As the column speeds up, the pressure inside the column "wants" to decrease, however the air pressure outside forces the column to be narrower to maintain 1 bar of pressure.
Also the forces within the column depend on the hydrogen bonds between the water molecules, which will "pull" the column together. Effects at the actual surface will have little overall influence (the term surface tension seems misleading to me).
Disclaimer: I have only done undergrad physics, and although I do try to think things through, I am regularly surprised at how wrong I am about basic physics!
> Also the forces within the column depend on the hydrogen bonds between the water molecules, which will "pull" the column together. Effects at the actual surface will have little overall influence (the term surface tension seems misleading to me).
Water is a polar molecule with positive and negative charges corresponding to the hydrogen and oxygen atoms. This makes it a very ‘sticky’ molecule and so water coalesces into droplets and reaches an equilibrium internally. It’s the forces between molecules at the surface that are not at equilibrium, they experience a net attraction which causes the pressure in the droplet to increase — the Laplace pressure. The smaller the droplet the higher the Laplace pressure, lots of interesting stuff to read about there for you.
Now, consider the water tap, a very small flow rate of water will form a droplet which increases in size held in place by adhesion to the metal of the tap. Surface tension and the wet ability of the metal determines the shape of the droplet until it’s weight due to gravity overcomes the surface tension forces and it falls. The ambient pressure has no effect, the pressure of the water supply and the air are pretty much balanced, the pressure in the droplet is slightly higher due to the surface tension.
Increase the flow rate a bit and you will get a thin laminar column of water. Here the thinning effect due to acceleration is observed. However, the column quickly breaks up as per the Plateau-Rayleigh instability; the water is pulled into a chain of droplets. The air pressure is invariant so you can continue to ignore it. You can even work in a vacuum.
Increase the flow rate and the diameter increases due to the greater supply pressure needed to drive the flow. The supply can have zero velocity but higher pressure. Now you get a longer stream of laminar flow before the breakup occurs.
Increase the flow rate further and you start to see turbulent flow.
"For the water droplets of 1 micron and 4 mm, the capillary pressures are of the order of 10e5 Pa and 10² Pa, respectively (σ=0.072 N/m for water)."[∆]
Atmospheric pressure is approximately 10e5 Pa at ground level. Capillary pressure is another name for Laplace pressure from a quick Google.
So as your water column increases diameter, air pressure matters more and.
For my example of 1m diameter, I would expect Laplace pressure and surface tension to be negligible. And the rayleigh instability will still occur at a small column diameter, when the fluid is going very fast.
I guess there are two thought experiments:
1. what happens when you use a liquid with an extremely low surface tension in air? (Or maybe even a column of heavy gas like uranium hexafluoride thick enough that diffusion doesn't dominate?).
2. How could you model a system to avoid the dynamic effects of air circulation (donut flow as occurs with a helicopter) and the dynamic effects of having water laden air where the column starts to break up. Does a thick downwards jet of water at speed (over the terminal velocity of rain) still show narrowing? Or do other surface interface effects cause problems? What happens to a thick downwards jet that is over the speed of sound?
Edit: The water exiting the tap is at approximately 1bar, I need to be clearer about pressure differentials rather than absolute pressure...
Edit: what I was trying to point out was that "surface tension" is misleading, and that air friction and gravity are not the only other major forces acting on a stream of water from a tap.
Edit: Also thinking of a dam with holes down the side more than 10m. I am guessing air pressure matters the most when the water is at low velocity exiting a wide nozzle, when the water is accelerating the most due to gravity relative to its velocity.
The Laplace pressure is in addition to the static pressure. Working in gauge pressure you can ignore the air pressure and the pressure within the droplet.
Don’t confuse gauge and absolute pressure.
A 1m column of water is going to be a turbulent waterfall.
Remove surface tension and the liquid will form more complex shapes as it is less constrained to form a sphere. Instead of forming a droplet the stream will simply fall and break up as other influences dominate.
You actually get exactly the same "thinning" behavior if somehow the stream is square shaped or anything else in cross-section.