As water is pulled into an opening by gravity, it begins to spin. Why does it spin?
Because angular momentum from the initial state of the water is preserved. It's the same thing that a skater uses to start in an open, slow spin and pull their arms in to go into a closed tight spin.
Actually, there are two regimes of spinning with two different speeds of spinning: immediately after opening a hole in the bath and about a minute later.
Immediately after opening the hole, conservation of angular momentum already works and one may see very slow spinning far from hole and faster spinning close to hole.
A minute later, the spinning becomes many times faster than in the very beginning. So, what is the reason of the fast spinning? Or what is the reason of increasing of the speed of spinning a minute later?
Angular momentum for a spinning object is mass times velocity times radius. As momentum is preserved and as the radius decreases (because of the water going out), the velocity must increase.
That's corect, but that was an answer to a different question.
Consider the numerical exapmle.
R = 30 cm, r = 3 cm.
Immediately after opening the hole in the bath, we have:
v(R) = 1 cm/sec, v(r) = 10 cm/sec
A minute later, BOTH of the speeds, the speed far from the funnel and the speed close to funnel becomes much larger,
v(R) = 12 cm/sec, v(r) = 120 cm/sec
The question is:
Why a minute later the speed at the distance 30 cm from funnel increased from 1 cm/sec to 12 cm/sec? Why a minute later the speed at the distance 3 cm from funnel increased from 10 cm/sec to 120 cm/sec?
i probably think that in this case what is decreased is the fluid mass, so it start to spinn faster... L=mvr. thats it.
[nitpick]In the case of water down a drain angular momentum is not precisely conserved. The tub (and maybe gravity?) does exert torque on the water.[/nitpick]Accounting for that amount of torque the rest of what has been said about angular momentum is correct.
In addition to the conservation of angular momentum there is also conservation of energy. As the water moves down into the drain there is some loss of PE. By conservation of energy you can also get an overall increase in KE in the tub depending on the KE of the water going down the drain. [nitpick]Of course, accounting for energy lost to viscous heating etc.[/nitpick]
There is a large bath, about 100 gallons of water and a small hole, about 1 inch diameter. A minute later there is still about 95 galons of water. Decrease of the mass of water is only 5%, but increase of rotation speed of the whole funnel is about 1200%.
I feel like I've been hustled. Your OP belied the depth of your knowledge on the subject.
But I don't think the whole volume participates at that point. Due to inertia and friction I imagine you can consider the dynamics of a smaller volume of only a few gallons surrounding the drain.
I am not sure about gravity, but the tub, actually bottom of it near the hole, exert friction. So it should reduce the angular momentum. But actually, the angular momentum increases a minute after beginning of the process.
Yes. There are two mechanisms of the speed increase as the water approaching the hole. The first one is that water goes closer to vertical ax, momentum conservation and so on... The second one is that water goes to a lower level, PE => KE and so on... But the question was not about speed increase as the water approaching the hole, but about increase of the speed of the funnel as whole a minute after beginning the process.
I am not satisfied with my knowledge of the subject... what I actually want is to find any effective measures against tornadoes and tropical storms that are too annoying in my lovely Florida. But in order to find something, I need deep understanding of rotation phenomena.
So, I am not satisfied with the hurricanes in Florida and not satisfied with the present knowledge of the subject...
You cannot consider part of the water in isolation to the rest. Viscous forces "connect" the water approaching the hole to the rest of the water in the funnel. The viscous forces are small, but not negligible. That is why, as you observed, it takes a rather large amount of time.
Consider only FIVE gallons of water surrounding the drain.
At t = 0 (or t = 10 sec), the funnel spins slowly.
At t = 1 min, the first five gallons are gone. There are another five gallons of water surrounding the drain. The funnel spins quickly. Why behavior of the next five gallons, which forms quickly spinning funnel is different from behavior of the first five gallons, which formed slowly spinning funnel?
That is exactly what I was thinking about, but I needed an independent opinion... thanks
The five gallons of water surrounding the drain is a very poor system to choose. It is not an isolated system and the boundaries and interactions are very difficult to define. You are much better off considering all of the water in the tub. That makes the boundaries much easier to define as well as the interactions with the surroundings.
Different initial conditions.
You are absolutely right!
And different boundary conditions!
i might be wrong but i think as the rotation progresses, the viscous resistance decreases, so as letting the velocity increases. just might be
but one thing that kicks me is the fact, it rotates. why does it rotate at all?? i have a large tank, full of water, i punch a hole in it, water, a lil after, drops below forming a vortex. why does it happen?? i asked this question, all through my course, but didnt get any answer OR i am ultra stupid;))
What exactly do you mean? The coefficient of viscosity decreases or the viscous resistance as a global phenomenon decreases at constant coefficient of viscosity?
I believe that answer to the question why does it rotate at all?, is the same as the answer to the question "why it rotates faster and faster as the rotation progresses?".
So, there is a mechanism exists that accelerates spinning the funnel as whole. In such a situation an initial fluctuations of angular momentum are enough to develop global spinning until nonlinearity restricts it at some reasonable level.
This question has got me thinking. The Wikipedia page (http://en.wikipedia.org/wiki/Vortex) mentions that for a free vortex "The tangential velocity v varies inversely as the distance r from the center of rotation, so the angular momentum, rv, is constant". I believe that this is constant as a function of r, not as a function of t.
As you indicate the whole thing can start spinning faster, so something must be exerting a net torque on the fluid in the same direction as the angular momentum. The viscous shear forces should exert a net torque in the opposite direction, the normal forces in a symmetric vessel should not exert a net torque, and I can't see how gravity would exert a torque about a vertical axis.
Where's the torque?
EDIT: I cannot reproduce the "points far away start spinning faster" thing in my sink even though anecdotally I think I have seen such occurences. The drain plug may be interfering. The situation you described above, was that just hypothetical, or have you done such an experiment?
I don’t think that precise measurements are possible in hydrodynamics. Deviations about 1% from the law 1/r may 'save' the model. It takes about 1 minute to develop stationary fast spinning funnel. For that time liquid makes several hundreds turns around center of vortex. So, the process of acceleration of the whole thing is comparatively slow. The torque required for such slow acceleration may cause deviations of about 1% or less from exact 1/r law.
I think that wikpedia describes a stationary, developed vortex. But during the process of slow acceleration the law may be a little bit different from 1/r and again the model works.
I believe the torque is within 1% of experimentally measured 1/r law.
The question is why the torque accelerates the whole thing. Friction due to walls and bottom must decelerate the whole thing.
First, you should remove drain plug at all and close the hole by your business card. Then wait 5 minutes until the water is in rest. After that remove business card using piece of wire (a long needle would be the best), moving it ALONG the bottom of the bath. After such non-disturbing opening of the hole, you should get the funnel WITHOUT spinning for a minute or more.
Well, if there is no external torque in the right direction then the only way possible for the "far away" fluid to gain angular momentum is if the fluid going down the drain has less angular momentum per unit mass than the rest of the fluid. In the ideal irrotational vortex the angular momentum is uniform throughout the fluid, so you would get no such effect.
However, I don't know the derivation of the irrotational vortex equations, it could be that they are assuming no viscosity. If so then it would make sense that the innermost fluid would have the highest shear rates and therefore rotate slightly slower than the inviscid limit and therefore have less angular momentum than the bulk of the fluid.
Yes, that is like Cheshire Cat smile...
Try Focault's pendulum - coriolis effect.
The effect is due to the rotation of the earth on its axis (of rotation...). We always subconciously assume Earth is staionary when it simply isn't.
And it's caused by moving in toward that axis (ever so slightly with Focault pendulum as it falls), and likewise opposite effect when it swings away away from the Earth's axis. It's like trying to keep a straight line as you walk inwards towards the centre of a roundabout or carousel. The fact you're already rotating throws you to one side of the line that you're trying keep.
You could also demonstrate the effect by dropping something verticlly from 100m at the equator. I think it should hit the ground about 1mm to one side.
So basically I think there wouldn't be any torque if you were on a planet that doesn't spin.
The reason the vortex speeds up is probably the positive feedback effect that somebody earlier mentioned. ie it's because all the particles in the water are connected by intermolecular forces :-) and obviously a larger effect from surface tension. -So what happens at the center has a knock on effect on the water further out. Especially true for the water on the surface where moleculr forces are stronger..
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