Faucet Physics Question

[Matt Flynn]

oh bother.

a = acceleration of 1G

yes viscosity affects a but it is negligible. if you want to get that accurate you should just measure it empirically or get somebody who took physics to help you.

so v2 = v1 + (a x t)

...conveniently allowing you to adjust a for gravities on other worlds.

(v1 + (at)) / v1 = (r1 squared)/(r2 squared)

r2 squared = (r1^2) x v1 / (v1 +at)

So diameter at time t =

2r1 x the square root of ( v1 / (v1 + at) )


hope that's what you were looking for. on to superatoms...

matt

[daryn]

yes, once it ceases to be laminar flow, all bets are off so to speak.

[Patrick del Poker Grande]

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the pressure terms in bernoulli's equation cancel so it doesn't matter if it's in a vacuum or not. bernoulli's equation just shows that the velocity increases as the water moves down. then mass conservation shows that because the water is moving faster the diameter must decrease.

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Everyone agrees that for a steady stream of water or sand or anything else, the mass / unit time passing through any plane of the stream (the flux) has to be equal - this is the conservation of mass part, and is essential. The flux for any cross-section of the stream is equal to Area x Velocity x Density. Since the velocity increases further down the stream, the area and/or the density must decrease to compensate. Since the stream is a unbroken column of water, the density cannot decrease (because its a liquid) and so the cross-sectional area must decrease. In the sand case, the individual grains of sand act sort of like a gas, so the density of sand grains will go down and the stream will not narrow.

So the part about water molecules being attracted to each other is just saying that the bonds are strong enough to make it a liquid, not a gas.


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i just want to take this opportunity to say damn i'm good. patrick, don't let your employer see this!

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Well I'm glad this makes you feel good about yourself having apparently stuck it to me or something. I'm not worried about my job over a silly fluid dynamics question, as I'm sure they're more concerned with my structural analysis on things like weapon systems, satellites, and my analysis on the space shuttle starting next week.

Not only that, but I'm still not going along with the mass-conservation as the reason the stream narrows. Yes, you may have mass conservation, but it's not the reason it happens. First, there's nothing saying that the water/air mixture must keep the same density. There's nothing saying that air doesn't enter or exit the stream. In fact, my position is that it does exit the stream and that's what allows it to narrow. You could probably say that if you make it a closed system inside some set cylinder of diameter greater than or equal to the original faucet diameter. Then you could say that at each cross-section of this cylinder, the air has just moved towards the outside of the cylinder and then you have the same average density or flux through any given section. Fine, that matches conservation of mass. That doesn't explain why the air moves to the ouside of the column or the water moves towards the inside. The water could keep increasing speed and have less composite density through any given section and the stream would still have the same outer diameter and it'd still be within the constraints of mass conservation.

In short, mass conservation is most likely preserved, but it's not the cause of the phenomenon we're looking at here. There's nothing about mass conservation that says the water stream has to narrow instead of spraying out wider. The cause we're looking for is the effect of the air leaving the stream, thus narrowing it, instead of entering it and diffusing it into a spray.

I'm sorry this all wasn't very well worded, but I'm a bit busy writing reports and proposals for work right now, so I can't exactly give it a ton of attention.

[Patrick del Poker Grande]

I guess the problem is that most people here are already just assuming that the water attracts to itself, thus it's easy to just apply conservation of mass and then say that the diameter of the stream is getting smaller because the velocity is increasing. You never say why it must be getting smaller instead of the water just being more diffuse. I'm saying that the hydrophilic effect you've already assumed... is the answer, or at least the beginning of it, that you've jumped past half the answer and applied it to get the other half. In the end, it's this effect in combination with conservation of mass that leads to the narrowing stream phenomenon. The disagreement here all stems from differing assumptions. I didn't assume as much as others did here.

[daryn]

i hear what you are saying, but some things just aren't necessary to answer a question. for instance, if i ask you what is 2+2, do you answer 4, or do you start in with elementary number theory, to define the terms first?

[Patrick del Poker Grande]

Yeah, that's fine, but the problem here is that you can't necessarily assume these things. It's not 2+2. In fact, there was a post some time earlier where one guy observed the stream breaking up again after a longer fall. Your assumption doesn't allow for this, yet it does happen and mass is still being conserved. There definitely is a significant other part to this phenomenon that you're skipping completely over. There are two parts to the solution here - one is mass conservation, but the other is whatever is making the water attract to itself. You just assumed the latter of these two when I don't think it can just be assumed. It clearly cannot, as it falls apart when the stream reaches higher speeds. The question that needs to be answered is what makes the water attract itself? Is it just something about surface tension and water's natural affinity toward itself? Is it aerodynamics? This is the interesting part of the question if you ask me. The equations that were listed before are the part that I thought was elementary and didn't need to be talked about.

[wacki]

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The question that needs to be answered is what makes the water attract itself? Is it just something about surface tension and water's natural affinity toward itself? Is it aerodynamics? This is the interesting part of the question if you ask me.

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It's called hydrogen bonds which can get to about 10% of the the strength of covalent bonds. I must say that Patrick is the first person to view this problem in the proper manner IMO.

[mostsmooth]

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Why does a stream of water coming out of a faucet narrow as it falls?

Are there any equations which model the diameter of the stream as a function of the distance below the faucet?
The initial speed of the water and the diameter of the faucet are given constants.

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i know not much about physics, but im gonna take a stab at this before i read any replies.
the water coming out of the faucet is spread out at the tap by the screen or whatever. as it falls it contracts to form the smallest package. i imagine the air pressure is what forces the water together. (possibly something to do with surface tension as well??)
i wonder how close i am?

[mostsmooth]

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p.s. I'm right, it has to do with the attraction of water molecules to each other.

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sorry buddy, you're not. what he described will happen even if the water molecules are packed as closely as possible at the top.

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not if there was no attraction to each other though, no?
if the attraction wasnt there, the water would fall like the previously mentioned sand, straight down and no diameter tightening, correct?

[styleish]

for this faucet problem, by using bernoulli's equation we are assuming that the viscosity is 0. the viscosity, i believe, along with surface phenomena are what cause the instabilities in the flow at higher speeds among other things. however, for the flow prior to this i believe the solution posted by wacki is a good enough approximation of the actual phenomena for our purposes. i completely agree with you, however, in that we are taking an extremely elementary look at this problem. but doesn't every solution to a physical problem include certain simplifying assumptions?