It´s clear that a moving flag is a chaotic system. But is it nontheless possible, under certain conditions and a uniform wind velocity, to make the flag look frozen in time, i.e., flat without movement?
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It is not clear what you allow to do. Is putting it on the Moon valid? Or stick it in a medium such that the Reynolds number is very small? If you only allow to tweak the air velocity then no, it will always wobble as common knowledge shows as we always are in a turbulent-chaotic regime. – DarioP Feb 03 '16 at 16:20
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Put a flag in a certain initial condition, where the flag is as thin and smooth as possible. The flag is put in it´s ideal initial position by putting it on an very thin stick and keep the flag totally flat and stretched out. Then you put on the uniform wind making machine, sending it´s wind parallel to the flag and when this machine is running in a constant mode, you let loose the flag. Will the flag stay flat, or wil their always develop some little changes wich cause the flag to flutter? And what if we make all stuff mathematical idealizations (infinitely thin flag perfectly smooth, etc.). – Deschele Schilder Feb 03 '16 at 21:13
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obviously yes : wet it , put it flat in a congelator and then maintain the temperature with an icy back and parallel wind – Feb 03 '16 at 21:25
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1@igael As always the most easy answers are staring you in the face! – Deschele Schilder Feb 03 '16 at 21:31
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What if the velocity of the homogeneous wind blowing on both sides of the flag is very high (although I just read that above a certain velocity laminar flow turn in a flow with turbulence), and a we have constructed a special pole (having for example the form of an elongated razor blade with no holes on it, made by a very strong material) to wich the flag is seamlessly attached, will you come near the desired result? I mean, if the flag shows the turbulence in the air (if it´s very light) then tere´s not much turbulence to see. – Deschele Schilder Feb 04 '16 at 11:07
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The simplest answer to your question would be: Yes, put the flag out on a day when there is no wind. If you want to be able to distinguish which country it represents, tilt it so that its pole is horizontal. But I guess that this is not the situation that you have in mind. I imagine that you are asking if it is possible that wind will be strong enough to hold the flag up and regular enough so that the flag appears motionless.
If that is your question, I would guess: 'maybe in theory, but no in practice'.
Keep in mind that it is not the dynamics flag that are chaotic, but the dynamics of the air. As long as it is light enough, the flag just follows the wind. Flowing fluids such as wind have a laminar (smooth) motion when the flow is slow, but as it speeds up, the smooth motion becomes unstable and even the slightest perturbation will produce turbulence. Usually there is a critical velocity above which the laminar flow is unstable.
If I were to try to realise the situation that you describe, I would use a flag that is very stiff (so that it holds up almost by its self) and very heavy (so that it is not affected by the wind). For example, a flag that is made out of a 1cm thick slab of metal should not move under most weather conditions. If you insist that the flag be made out of cloth then I would try a flag that is very light (so that a weak wind can lift it) and use a very thin pole (so that the laminar flow is stable at high velocities).
I have been thinking on the way to estimate what parameters would work, but this is an incredibly complex problem that I do not have the time to try to solve. Moreover, it depends on a lot of parameters that you have not specified. For example, what is the stiffness of the flag, it's mass, the radius of the pole, the height of the pole? When you say that the flag is perfectly still: Is it allowed to have small and slow movement that is imperceptible? How small? How slow?
I would say that the most efficient way to solve your problem is to go out with a flag on a windy day and do the experiment. If you flag moves, try again on a day that is less windy. If your flag falls, try another lighter one. If you can do it, then the answer to your question is 'yes'. If you can't, try a thinner pole and a different kind of flag. Keep trying until you succeed or believe the the answer is 'no'.
Steven Mathey
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But a mathematically ideal situation will do the trick? I mean, a flag of zero thickness , smooth on every level. A flagpole of zero thickness. A perfect windgenerator, and an ideal begin situation? – Deschele Schilder Feb 03 '16 at 20:32
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1Indeed, if you remove all sources of symmetry breaking from the equations, your dynamics will be symmetric. But that is not a complete description of your system. There always are small perturbations. These may be so small that you would never take them into account normally. In most cases they can (and should) be neglected. Here however the system is unstable. The tiniest perturbation is amplified by the dynamics. – Steven Mathey Feb 03 '16 at 20:58
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This is not unlike the question "can you theoretically balance a perfect pencil on its tip". The answer is always "no".
No, because the initial conditions cannot be perfectly obtained; and the equations of motion are such that a small perturbation from the initial condition will grow. You cannot generate perfectly laminar flow on a molecular scale, indefinitely: the mechanisms used to generate the wind will themselves leave small perturbations in the air, and these will be sufficient to initiate motion of the flag.
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Is this pencil standing on it's point a chaotic system? It's for certain an attractor, because no matter what the initial conditions, the pencil always in a horizontal position, just like a little ball always ends up in the hole in the middle of of a sink, no matter the initial conditions of the little ball (of course the ball must have such an energy that it staying in the sink). – Deschele Schilder Feb 03 '16 at 23:17
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Not sure if the pencil meets the definition of chaotic. Unstable, definitely. Tiny change in initial conditions will change the course of events - but as you say, with a predictable endpoint. – Floris Feb 04 '16 at 03:40
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@Floris: A pencil stending on it’s tip is not a chaotic system, but neither is a flag standing still. Nothing standing still is a chaotic dynamics. Sensitivity to intial conditions isn’t everything. You should at least have a dynamical system and some reagions in state space being revisited. – Wrzlprmft Feb 04 '16 at 23:50
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@Wrzlprmft fair point. But the flag standing still is in an unstable equilibrium, like the pencil. Any perturbation will be amplified. It was OP who was trying to map it to a chaotic system... – Floris Feb 04 '16 at 23:53
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@Floris: I wasn’t arguing against you but rather addressing your uncertainty about the definition of chaos. – Wrzlprmft Feb 05 '16 at 00:00
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@Wrzlprmft I realize that now that I re-read what you wrote. Sorry to sound defensive. – Floris Feb 05 '16 at 02:33