5 ms·

> When I was in high school, I'd see water running out of a faucet growing narrower, and wonder if I could figure out what determines that curve. I found it was

by bradrn 6y ago

> When I was in high school, I'd see water running out of a faucet growing narrower, and wonder if I could figure out what determines that curve. I found it was rather easy to do. Out of curiosity, does anyone know the solution to this problem? I wouldn’t even know how to approach it. (I’m an undergraduate physics major; I know nothing about fluid dynamics, but from Feynmann’s description I suspect that isn’t necessary…)

Anon84 6y ago
It’s not. It’s just conservation of volume. Imagine a thing disk of water just as it comes out of the faucet (pi r^2 v*dt) as it falls, the speed increases so the same volume of what is now thinner and longer (r decreases to account for the increase in v).
- fastasucan 6y ago
  *volumetric flow
- robochat 6y ago
  I think that you mean pi.r*2.v.dt ?
  - robochat 6y ago
    pi.r².v.dt
  - Anon84 6y ago
    Yes, thank you for noticing it :)
phonon 6y ago
The water is accelerating downward due to gravity...but the amount/s flowing out of the faucet is equal to the amount/s any distance below it...so the stream has to get thinner as it gets "faster".
knolan 6y ago
Surface tension plays a role. Water wants to reduce its surface area because forces between molecules at the surface pull them together. So a droplet of water will want to form a sphere. A column of water will first form into a cylinder and then break up into spherical droplets. It’s called the Plateau–Rayleigh instability. https://en.m.wikipedia.org/wiki/Plateau–Rayleigh_instability https://en.m.wikipedia.org/wiki/Plateau–Rayleigh_instability
- eigenket 6y ago
  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.
  - robocat 6y ago
    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!
    - knolan 6y ago
      > 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.
EVa5I7bHFq9mnYK 6y ago
I think it's a power of -1/4: velocity of a falling object is proportional to square root of height, and the stream diameter is inverse proportional to square root of velocity.
deleted 6y ago
[deleted]
jackdawed 6y ago
I imagined it has to do with how the water coming out of a faucet is affected by gravity, and gains velocity. If you imagine two cross section slices, one near the faucet, and the other near the sink, the bottom slice is going faster than the top due to gravity. Another insight is that the amount of water (flow rate) passing through these slices remain constant. To keep the flow rate the same, it has to be narrower. If you let the water fall far enough, it would hit terminal velocity and it would not narrower any further. I looked it up and found it was called the continuity equation.
- benrbray 6y ago
  I like this way of looking at the problem, I wouldn't have thought of this! I think this relies implicitly, too, on incompressibility of the fluid, no? As well as surface tension? Otherwise, we might imagine the fluid having less density at the bottom, while occupying a circular cross-section of the same area.
  - leephillips 6y ago
    It does rely on incompressibility, but, to this order, does not include surface tension. The water faucet puzzle that interested me as a student was after the water had reached the sink and begun to spread out radially from the stream. Have you noticed that there is a circular boundary where the layer of water suddenly changes depth?
    - bradrn 6y ago
      > Have you noticed that there is a circular boundary where the layer of water suddenly changes depth? I stumbled upon an explanation for this a while ago: https://en.wikipedia.org/wiki/Hydraulic_jump https://en.wikipedia.org/wiki/Hydraulic_jump
    - brilee 6y ago
      google "hydraulic jump" Separately, when you model surface tension, you get an equation for when a stream breaks up into droplets - https://en.wikipedia.org/wiki/Plateau%E2%80%93Rayleigh_instability https://en.wikipedia.org/wiki/Plateau%E2%80%93Rayleigh_insta...
MaxBarraclough 6y ago
Bernoulli's principle seems relevant here, I'm surprised no one has mentioned it. https://en.wikipedia.org/wiki/Bernoulli%27s_principle#Incompressible_flow_equation https://en.wikipedia.org/wiki/Bernoulli%27s_principle#Incomp...

by bradrn 6y ago

Anon84 6y ago

It’s not. It’s just conservation of volume. Imagine a thing disk of water just as it comes out of the faucet (pi r^2 v*dt) as it falls, the speed increases so the same volume of what is now thinner and longer (r decreases to account for the increase in v).

fastasucan 6y ago
*volumetric flow
robochat 6y ago
I think that you mean pi.r*2.v.dt ?
- robochat 6y ago
  pi.r².v.dt
- Anon84 6y ago
  Yes, thank you for noticing it :)

phonon 6y ago

The water is accelerating downward due to gravity...but the amount/s flowing out of the faucet is equal to the amount/s any distance below it...so the stream has to get thinner as it gets "faster".

knolan 6y ago

Surface tension plays a role. Water wants to reduce its surface area because forces between molecules at the surface pull them together. So a droplet of water will want to form a sphere. A column of water will first form into a cylinder and then break up into spherical droplets. It’s called the Plateau–Rayleigh instability. https://en.m.wikipedia.org/wiki/Plateau–Rayleigh_instability https://en.m.wikipedia.org/wiki/Plateau–Rayleigh_instability

eigenket 6y ago
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.
- robocat 6y ago
  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!
  - knolan 6y ago
    > 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.

EVa5I7bHFq9mnYK 6y ago

I think it's a power of -1/4: velocity of a falling object is proportional to square root of height, and the stream diameter is inverse proportional to square root of velocity.

deleted 6y ago

[deleted]

jackdawed 6y ago

I imagined it has to do with how the water coming out of a faucet is affected by gravity, and gains velocity. If you imagine two cross section slices, one near the faucet, and the other near the sink, the bottom slice is going faster than the top due to gravity. Another insight is that the amount of water (flow rate) passing through these slices remain constant. To keep the flow rate the same, it has to be narrower. If you let the water fall far enough, it would hit terminal velocity and it would not narrower any further. I looked it up and found it was called the continuity equation.

benrbray 6y ago
I like this way of looking at the problem, I wouldn't have thought of this! I think this relies implicitly, too, on incompressibility of the fluid, no? As well as surface tension? Otherwise, we might imagine the fluid having less density at the bottom, while occupying a circular cross-section of the same area.
- leephillips 6y ago
  It does rely on incompressibility, but, to this order, does not include surface tension. The water faucet puzzle that interested me as a student was after the water had reached the sink and begun to spread out radially from the stream. Have you noticed that there is a circular boundary where the layer of water suddenly changes depth?
  - bradrn 6y ago
    > Have you noticed that there is a circular boundary where the layer of water suddenly changes depth? I stumbled upon an explanation for this a while ago: https://en.wikipedia.org/wiki/Hydraulic_jump https://en.wikipedia.org/wiki/Hydraulic_jump
  - brilee 6y ago
    google "hydraulic jump" Separately, when you model surface tension, you get an equation for when a stream breaks up into droplets - https://en.wikipedia.org/wiki/Plateau%E2%80%93Rayleigh_instability https://en.wikipedia.org/wiki/Plateau%E2%80%93Rayleigh_insta...

MaxBarraclough 6y ago

Bernoulli's principle seems relevant here, I'm surprised no one has mentioned it. https://en.wikipedia.org/wiki/Bernoulli%27s_principle#Incompressible_flow_equation https://en.wikipedia.org/wiki/Bernoulli%27s_principle#Incomp...