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James Fingas
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The Crazy Sprinkler
« on: Dec 4th, 2002, 1:28pm » |
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The Crazy Sprinkler
A little while ago, Willy Wutang was thinking about the sprinkler puzzle, and reading the sprinkler thread in the forum:
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_med ium;action=display;num=1028067360
Willy, having a creative mind (or at least a well-developed sense of mischief) had a flash of inspiration, and came up with a very novel sprinkler design. However, before he can patent his new sprinkler, Willy must figure out what it does--his lawyer says it's a necessary part of the patent process.
1) First of all, does the sprinkler turn? If so, in which direction does it turn?
2) Second, what do the jets of water do? In the picture above, the sprinkler is shown at the instant just before the jets of water collide. However, in the picture the sprinkler is being held still, so that even if it wants to turn, it cannot.
3) Third, does the behaviour of the jets of water depend on whether or not the sprinkler is allowed to rotate? Does it maybe depend on how fast the sprinkler rotates?
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Speaker
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Re: The Crazy Sprinkler
« Reply #1 on: Dec 5th, 2002, 8:09pm » |
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1. It turns clockwise.
2. The jets leave their respective nozzles and are deflected to a degree equal to some combination of the angle at which the jet leaves the nozzle and the angle to which it is deflected. (Remember that the deflecting jet is itself being deflected, so the diagram here will change substantially when the jets finally collide) Thinking becomes a little circular when you assume that the jets will be deflected so much that they won't collide with the other jets, meaning that because the jets won't collide that they won't be deflected...
3. The jets of water will change if the sprinkler is rotating. They will be pushed away from the center, until they finally no longer collide with the each other. The faster the rotation, the greater the degree to which the jets veer from center.
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James Fingas
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Re: The Crazy Sprinkler
« Reply #2 on: Dec 6th, 2002, 1:52pm » |
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Speaker,
I think I understand what you're trying to say about the jets being deflected. Essentially there is a point of equilibrium where each jet is deflected just a little, so that it almost misses the next jet, such that the next jet is just barely deflected (exactly the same as the first jet), and so on.
I might agree that that's an equilibrium, but it seems like an unstable equilibrium point. Any small change in the jet from A would cause a larger change in the jet from C, causing an even larger change in the jet from B, causing an even larger change in the jet from A. Add in a time delay, and I think the whole thing would run away from that particular equilibrium.
As for the behaviour of the jets under rotation, keep thinking 
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SWF
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Re: The Crazy Sprinkler
« Reply #3 on: Dec 7th, 2002, 6:08pm » |
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Certain idealizing assumptions will be made, such as flow from all nozzles is identical, surface tension does not cause the fluid streams to break up into drops, and flow velocity is high enough that gravity does not cause the flows streams to miss one another.
If the nozzles are fixed so they cannot rotate, each fluid stream will be periodically blocked by another stream. Say it takes time T for a particle in stream A to travel from where stream B can intersect stream A and the point where stream A can intersect with stream C. Say time=0 when the streams first pass the first potential intersection points. When time=T the fluid in A first reaches and blocks the flow of C. Simultaneously flow from B begins to block A, and C begins to block B. However there remains a portion of stream A between where B blocked A and where A blocks C. It takes time T for this portion of the stream to finish flowing against stream C. Similarly for B blocking A and C blocking B. When the blocking flow runs out, the situation is like at when time=0 and all streams can flow again. So streams flow for time T, then are blocked for time T, then flow for time T and so on.
When I say flow is blocked, I mean the streams impinge on one another such that they do not continue in the same straight line. When this occurs, the colliding streams spread into a thin sheet of fluid in the plane which bisects the angle between the two incoming lines of flow.
If free to rotate, nozzles will start to rotate clockwise. All that pulsing described above does not affect the angular acceleration- it will increase steadily instead of in pulses. As the angular velocity increases, the flows pick up a component of velocity that causes them to be directed more toward the axis of rotation. This causes the intersection points to move closer together and the period of the pulses, T, to decrease. Also the angular acceleration decreases since moment arms and the angular momentum of the streams decrease. Eventually angular velocity is such that all streams deflect and meet on axis of rotation. Since all flows go toward the center, the moment arm is zero, and the rotation speed will remain constant.
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Speaker
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Re: The Crazy Sprinkler
« Reply #4 on: Dec 8th, 2002, 11:40pm » |
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It looks like SWF has the hydro-dynamics well in hand. And, I do not have any expertise in that area, but to me it is counter-intuitive that the streams of water bend towards the axis of rotation.
What causes this? How does this happen? What is going on here? Are there any examples in nature that might shed light on this phenomona?
If anybody is still monitoring this topic. I don't need the answers in any particular order, but simple single syllable words would be appreciated.
Okay, I went back and read the answer again. The above still applies.
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James Fingas
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Re: The Crazy Sprinkler
« Reply #5 on: Dec 9th, 2002, 9:55am » |
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Speaker,
I agree that it's counter-intuitive that the water streams point to the middle (in the frictionless steady-state). It took me a while to figure that out myself, but intuition is not helpful here.
When you look at the regular sprinkler case, what you find is that a stationary sprinkler sends jets of water out with both a radial and a tangential velocity component. The tangential component imparts a force on the sprinkler head to make it turn. As the sprinkler starts turning, then the water leaving the sprinkler head starts to get a component of velocity in the direction that the sprinkler head is turning. Because the sprinkler head is turning in the opposite direction to the way the water is coming out, the tangential velocity of the exiting water effectively decreases.
Assuming that the sprinkler head is frictionless, then we can see that the steady state is when the water is going perfectly radially. That is to say, the speed of rotation of the sprinkler head perfectly matches the tangential velocity of the water leaving the nozzle. They cancel out, and only the radial component remains.
This is easier to see on the normal outwards-pointing sprinkler heads, but it still holds true in this one. The important thing to remember is that the water drops move in perfectly straight lines. There is no centrifugal force on the water once it leaves the nozzle. The velocity of the water is just the speed it comes out the nozzle in the direction of the nozzle, plus the speed of rotation of the nozzle, in a tangential direction. Since the head is spinning clockwise, then the nozzle movement cancels out the tangential component of velocity that the water would usually have, directing it more towards the center of the sprinkler head.
SWF, I agree with your analysis of the water jets cutting each other off. Is this a stable operating equilibrium for the system? If the behaviour of the water jets were to drift over time, what sort of solution would you expect after a long time?
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SWF
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Re: The Crazy Sprinkler
« Reply #6 on: Dec 11th, 2002, 7:33pm » |
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As to the behavior when prevented from rotating and allowed to sit for a long time, I can see the pulsing of the various streams varying in a chaotic manner rather than reaching an equilibrium. This happens if I use a fairly simplistic set of equations to model the flows. With a little bit of "damping" the flows settle into a state where there is no longer any pulsing of the flows. In that state a fraction of each flow misses the target stream, and the amount the does reach the target, blocks exactly the right amount so that the fraction of A hitting C equals the fraction of C hitting B equals fraction of B hitting A.
The chaos comes in if the flow from the nozzles is not perfectly steady. A small change propagates through the system because the flow of a stream depends on its own flow rate at an earlier time. I am not sure what would happen in an actual device, since I modeled it by some simplfying assumptions but I am guessing chaos.
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D Guy
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Speaker,
I couldn't quite follow SWF either (the level of physics is a bit beyond my current knowledge), but maybe I can help explain why the water streams would head towards the axis of the sprinker, rather than away from it:
Imagine driving in a car at 20 MPH, and then tossing a ball *straight* out the window (perpendicular to the direction the car is moving). The ball will then be moving away from the car, *and* in the same direction as the car.
With the Crazy Sprinkler, the water will be moving away from the nozzle (imagine that as being analogous to tossing the ball out the window of the car) and the movement of the sprinkler (clockwise) will give the water stream a velocity/momentum in the direction it's moving (imagine that as being analogous to the movement of the car, when the ball is tossed).
Right?
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James Fingas
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Re: The Crazy Sprinkler
« Reply #8 on: Dec 13th, 2002, 10:11am » |
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D Guy,
Good analogy!
Since a few people have said they don't follow SWF's explanation of the interaction of the jets, I'm going to try to explain it here. My apologies if this doesn't help.
The basis for the argument is that when two water jets collide straight on, neither will continue in its original direction. Instead, they will hit each other produce a spray in a third direction. The third direction is the average of the other two directions. In the "diagram" below, jets from A and B collide, sending off a stream of water towards C.
A B
\ /
\ /
*
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C
As SWF argues (and I agree) the stream of water will be more of a spray rather than a cohesive jet, and near the collision point will be nearly planar. This doesn't really affect the question, however.
Applying this to the sprinkler, we can consider what happens when the jet from B initially reaches the jet from A (immediately after what is shown in the original diagram). The two jets will collide, sending off a spray in a third direction. This spray won't be pointed at jet C, so while the jets from A and B are colliding, jet A is effectively directed away from jet C.
Keep in mind that this happens to all jets simultaneously. Therefore, all three jets are diverted away from where they were originally pointing. That is to say, jet B is diverted so it doesn't point at jet A any more. This would seem logically inconsistent! We assume that jet B points at jet A, then go on to prove that jet B doesn't point at jet A.
The key here is that the change doesn't happen immediately. Even though jet B has been diverted, there is still some water on the way to jet A. During the time that there is still some water on the way, then jet A will continue to be diverted (and so will jets B and C). When the jets become un-diverted, then they will repeat the whole process. Ignoring the rotation of the sprinkler, I think that sums up SWF's argument.
My question to SWF is essentially: is this the fundamental mode of operation of the sprinkler, or is there a simpler pattern that the sprinkler might settle into eventually?
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D Guy
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James,
I did follow that -- thanks.
A few other points:
1. I would assume that when/if the streams collided, then that would have no effect on the rotation of the sprinkler (i.e., the RPMs would remain the same).
2. Like you said, none of that takes into account the rotation of the sprinkler, which would draw the streams towards the axis (and beyond?). So at some point, the streams might actually collide at the axis, in which case that would be a persistent state.
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D Guy
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... and, unless I miss my guess, the speed of rotation would be affected by two factors:
1. Friction, wind resistance, and other inefficiencies.
2. The vector (mass x velocity x direction?) of the water coming out of the nozzles.
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mike1102
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Re: The Crazy Sprinkler
« Reply #11 on: May 30th, 2003, 11:22am » |
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Obviously we have too many cs people and not enough physicists that visit this website.....
Let's review the basics: 1. Without a net force (or torque) acting upon an object, it will not accelerate. 2. Fluids, by definition, can not support (transmit) a shearing force. So when one stream intersects another, no force is transmitted to the sprinkler via the stream. The drawing indicates the fluid streams do not hit the sprinkler body directly. So the fact that the fluid streams intersect is of no consequence to the dynamics of this problem. Now, lets deal with the forces and torques.
Force (F) is the first time derivitave of momentum (P). Momentum is mass (M) times velocity (V). Thus, writing Newton's law, F = d/dt(P) = V*dM/dt + M*dV/dt. So..... is F non-zero? Certainly dV/dt = 0 i.e., there is no "mechanical" force on the sprinkler. But dM/dt (the thrust term) is certainly non-zero i.e., the fluid has mass and it is exiting via the nozzel, thus the thrust is non-zero and the force it generates is in the opposite direction of the flow.
Looking at the sprinkler geometry, the sprinkler experiences thrusting forces at A, B & C that produce a net torque on the sprikler body that cause it to rotate clockwise. The higher the flow rate (thrust), the higher the torque.
Moving on to Question 3 - The intersecting water streams have nothing to do with sprinkler rotation - fluids can't impart a sheering force. However, once the sprinkler rotates quickly enough such that the streams directly hit the sprinkler, the net force changes - it looks as though this will cause the sprinkler to slow down - somewhat like the classic flyball governer for a steam engine.
Kind of a cute idea - But tell Willy to talk to the market research people before the patent attorney.
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SWF
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Re: The Crazy Sprinkler
« Reply #12 on: Jun 2nd, 2003, 10:26pm » |
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on May 30th, 2003, 11:22am, mike1102 wrote:Obviously we have too many cs people and not enough physicists that visit this website.....
Let's review the basics: 1. Without a net force (or torque) acting upon an object, it will not accelerate. 2. Fluids, by definition, can not support (transmit) a shearing force. |
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Quote:fluids can't impart a sheering force.
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All the CS people I know think fluids can support a shear force. Please explain why physicists think fluids cannot. How does a physicist define viscosity?
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Chronos
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Re: The Crazy Sprinkler
« Reply #13 on: Jun 4th, 2003, 11:51pm » |
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It depends on the physicist's specialization. A theoretical physicist is apt to define a "fluid" as something which cannot support shear forces. When confronted with water, he'll say that it's approximately a fluid, because it can only support really small shear forces. Once you admit that something with viscosity can be a fluid, you have to ask just what is a fluid. Is ketchup a fluid? How about Silly Putty? Or glass?
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mike1102
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Standard textbook definitions.........
"Fluid – A substance that deforms continuously when subjected to a sheer stress.
Viscosity – A fluid property that relates the shear stress in a fluid to the angular rate of deformation." (Direct quotes from a fluid dynamics textbook).
By "support" I mean a substance that will not deform continuously under a shear stress . i.e., a solid. Granted, there are fluids that deform very slowly, e.g., glass - but I asume you're not going to use glass in your sprinkler; nor does anyone I know (cs, physicist, engineer, whatever) sprinkle glass on their lawn.
Also granted is the fact that a solid will deform under a load but unlike a fluid, a solid will continue to offer a restoring force (support) on that load indefinately - a fluid will not.
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SWF
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Re: The Crazy Sprinkler
« Reply #15 on: Jun 11th, 2003, 5:51pm » |
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Those textbook definitions make sense, mike1102, but we have a different interpretation as to what "support" means. I'd say that when one thing supports another it does not matter what the thing doing the supporting needs to do (ie. flow or deform) as long as it applies the required force. Although the water would apply a shear force to the sprinkler, I do not think shear it is very relevant here. Normal force or pressure is more significant.
I disagree your explanation for where the thrust comes from: Quote:| Certainly dV/dt = 0 i.e., there is no "mechanical" force on the sprinkler. But dM/dt (the thrust term) is certainly non-zero i.e., the fluid has mass and it is exiting via the nozzel |
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and also do not agree with Quote:| ... once the sprinkler rotates quickly enough such that the streams directly hit the sprinkler, |
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What the streams will do is already described in the posts above.
Also, I hope comparisons of glass to a very viscious fluid are not references to the common misconception that old windows are distorted because the glass flows over time.
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mike1102
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Ok SWF..... we're making some progress. I'd like you to think about a few things some undergrad physics students wrestle with:
1. What force acts on a rocket ship that cause it to lift-off its launching pad or accelerate in space (a vacuum) and where does that force come from? Hint: What actually makes a "regular" sprinkler turn in the first place? Another hint: Why bother to ignite the rocket fuel?
3.What's the fundamental difference between a solid and a fluid?
4. If you lean up against a brick wall (a solid) and nothing moves, what forces are involved?
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Chronos
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Re: The Crazy Sprinkler
« Reply #17 on: Jun 14th, 2003, 12:18am » |
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Quote:| Also, I hope comparisons of glass to a very viscious fluid are not references to the common misconception that old windows are distorted because the glass flows over time. |
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Sorry, I didn't mean to perpetuate a misconception like that. Glass is a fluid in the same sense that rubber is a conductor. While it's true that glass has a non-zero fluidity, it's also true that rubber has a non-zero conductivity. In both cases, though, it's very very close to zero. My point was just that you have to assign some cut-off, and say that anything with a viscosity above X is considered a solid, and anything with a viscosity below X is a fluid. By any sensible choice of cutoff, glass is a solid. But theoretical physicists don't like arbitrary cutoffs, even if they are sensible.
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LZJ
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Re: The Crazy Sprinkler
« Reply #18 on: Jun 14th, 2003, 2:04am » |
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mike1102:
I think you should re-read the previous posts. For example, here's something from a previous post of SWF's:
Quote:| If free to rotate, nozzles will start to rotate clockwise. All that pulsing described above does not affect the angular acceleration- it will increase steadily instead of in pulses. |
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You see? They already know that
Quote:| Moving on to Question 3 - The intersecting water streams have nothing to do with sprinkler rotation |
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What they were talking about was simply the deflection of the water jets and its behaviour.
Lastly...
Quote:I'd like you to think about a few things some undergrad physics students wrestle with:
1. What force acts on a rocket ship that cause it to lift-off its launching pad or accelerate in space (a vacuum) and where does that force come from? Hint: What actually makes a "regular" sprinkler turn in the first place? Another hint: Why bother to ignite the rocket fuel?
3.What's the fundamental difference between a solid and a fluid?
4. If you lean up against a brick wall (a solid) and nothing moves, what forces are involved? |
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(...rolls eyes...get the hint?)
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