Why Water Decreases in Cross-Sxn Area: Conservation of Mass

[quasi426]

Why does water bottleneck or decrease in cross sectional area when it falls from the faucet. The book says it is conservation of mass stuff...
(Cross-Sxn Area)* (velocity) = constant.

How can that be if all the water is accelerating at the same pace. What I mean is that all the water in the cross sectional area at the start should be increasing in velocity at the same rate due to gravity.

 

[Crosson]

What I mean is that all the water in the cross sectional area at the start should be increasing in velocity at the same rate due to gravity.

This is absolutely true. But the water further away from the faucet has been falling for a longer duration and has a higher velocity.

 

[Antimatter]

Bear with me; per second, the amount (volume) of water that passes through the hole is equal to the amount that falls onto the sink, right? Otherwise there would be some kind of water buildup in between. If the 'water beam' were to have constant cross-section, there would be less water exiting the faucet than what fell on the ground: imagine the velocity of the water is 1m/s at the faucet, and 10m/s just before hitting the surface underneath. Then, in 1 second, one meter of 'beam' will have past through the faucet, and, in that same second, 10 meters of beam will have hit the surface. Because the volume of either is the same, the cross section must have decreased. Hope that clears things up.

 

[quasi426]

Would the cross section of the water decrease towards the bottom if it was in a vacuum?

 

[FredGarvin]

Would the cross section of the water decrease towards the bottom if it was in a vacuum?

Yes it would. A vacuum is not the issue here, it is gravity. Gravity accelerates the stream of water.

 

[arildno]

quasi:
Consider the inverted case, with a fountain:
Do you agree that the cross-section of the spurt INCREASES until you get to the top where the water starts falling down again?

 

[El Hombre Invisible]

Just a couple of points I thought I'd throw in. First off, the pertinent law is the conservation of energy, not mass. As gravitational potential energy is converted into kinetic energy, and the longer any given water molecule has been accelerated the faster it must be falling, then on average a molecule nearer the sink than the tap will be traveling faster than one nearer the tap than the sink. If you take any given area, say a cubic centimetre, the molecules at the top of that area also must be moving slower than those at the bottom, so the water is kind of being stretched out vertically. This in itself does not necessitate that the cross-sectional area must decrease. If this were the only concern, the cross-sectional areas may be the same at the top as the bottom, but the density would be greater at the top. What makes the cross-sectional area decrease, rather than the density, is (on a fundamental level) the intermolecular interactions between the water molecules. For the density to decrease, the average distance between molecules has to to increase, meaning an increase in potential energy. This requires an increase in temperature, but no heat is being transferred to the water (bar the negligible friction with this air, and not even that in a vacuum). Therefore the average distance between molecules has to be constant, and so it is the cross-sectional area that has to decrease. This is explained macroscopically by the laws of fluid mechanics, involving field lines and viscosity and other things I can't remember much about.

 

[Doc Al]

Would the cross section of the water decrease towards the bottom if it was in a vacuum?

Yes. Surface tension holds the stream together. (As Fred says, vacuum is not the issue.)