Is the Bernoulli eqn. incompatible with relativity?

Question

The following thought experiment is driving me mad, but I can't seem to reconcile relativity with the Bernoulli eqn.

In a classic textbook example you are asked to imagine that you are a smoker sitting on a fast moving train travelling through a narrow tunnel with the window open. The fast flowing air outside of the carriage means a reduced pressure, and so the air (and thus the smoke) from within the carriage is pushed out through the window.

However, now imagine you are a smoker sitting in the tunnel by the track watching the train go past. You see the air in the carriage moving faster, and so the air inside the carriage is at a lower pressue. Your smoke is pushed into the carriage.

Now these can't both be true.

We can even forget the specifics of the train example, and say that this must be true of any fluid flow with differential speed. It seems that Benoulli requires an absolute velocity, but relativity says we can't have one. I must be missing something fundamental, but what?

Answer 1

You have missed a fundamental principle involved in Bernoulli's equation, and that is that it is for a fixed mass of gas, i.e. a change in pressure within a streamtube that has been accelerated (i.e. the outside air at rest before the passing of the train compared with whilst it is being disturbed), not for directly comparing two separate masses of air (the air inside the carriage to that outside).

For your "classic textbook example" the assumption is that the air within the carriage is at local static pressure the same as the air outside the train. The air that is accelerated over the train therefore sees a reduction in pressure. It does not matter whether the air is blown over the train, or the train moved through the air. This is why wind-tunnels are a vital tool in aerodynamics research. Bernoulli's equation does not need absolute velocity.

Answer 2

This is a very clever and well thought out question and I don't agree with what has previously been said. I think much of what you have been said is correct

The idea that 'it is perfectly obvious what is moving With Wespect To [WRT] something else' is careless. The whole point of relativity is that when things are at constant velocity it matters not one jot one way or the other.
The clarification of this is that the train starts and finishes at rest WRT with the air. This means that the slowing of time effects of relativity and all that stuff occur to the train and not the platform
It's the person who changes their reference frame WRT and then returns to that frame of reference later that suffers the effects of relativity. This notion is easier to accept than using General Relativity [accelerating reference frames] is to prove.

In reference to the 'they can't both be true' comment, I insist on asking 'why not?'. If only stuff was removed from the train and nothing put back in, the people in the train would eventually die of asphyxiation. As a [shame] smoker, I know that the smoke goes out the window as long as the car is moving and so far I've never run out of air in the car despite the bernoulli effect. The air leaving has to be replaced by air coming in. Even relativity must agree with this.
Have you ever noticed that if a farmer has been muck spreading his fields as you drive past, the smell gets if you open the windows? How does that work if the bernoulli effect says due to pressure difference, no air can come in? As air goes out, some comes in.

Answer 3

Special relativity isn't even involved in this seeming paradox. The basic "paradox" is that, given 2 identical tubes containing identical incompressible fluids rushing in opposite directions, and there is a hole connecting the two, how can it be said that it's true for both that the "pressure is lower in the other"? In the simple case of incompressible fluids, the following is a constant:

1/2 mv² + pV = constant

where m is mass of fluid in volume V, v is fluid speed and p is pressure

But carefully note that the v is the speed relative to the SAMPLING device. That is, instead of an interconnecting tube between the main tubes, I can with equal validity use instrumentation to make these measurements, and compare the difference, if any. If you think in this way, then the paradox starts to clear up, without any reference to Special Relativity. In the example given above, stationary probe A for tube A would see a fluid speed of v, and stationary probe B for tube B would see a fluid speed of v also, so that there would be no net difference of pressure, and symmetry is upheld. If we to move probes A and B at speed v, so that probe A measures v = 0 in tube A and probe B measures v = 2v in tube B, then it WOULD measure a difference of pressure, tube B having the lower pressure, but then symmetry is lost, because in tube A, probe A appears to be stationary, but in tube B, probe B will NOT appear to be stationary, but will be seen to be moving at 2v. So, going back to the train analogy, the open window is stationary relative to the smoker and his smoke, but to the smoker outside of the train, the window is moving relative to him, so there is no symmetry. You HAVE to take into consideration of the relative speeds of the "point of measurement" with the fluid.

The more complex explanation is that if a given volume of fluid has a constant total energy, if it is at rest, the spatial distribution of kinetic energy (momentum vectors, actually) is isotropic. When the fluid is in motion, this distributionis no longer isotropic, which is why we see a drop of the internal pressure, because some of the energy has been converted into motion kinetic energy. Now, how this distribution APPEARS to a sampling instrument, such as a window or a probe, does radically depend on relative velocities. As a very crude analogy, when you are standing in a straight rain downpour, it looks like the source of the rain is straight overhead. But if you run in the rain, the source no longer looks like it's straight overhead, but coming from an angle from vertical.

Answer 4

To put it simply, the important thing is the speed of the air relative to the window.

If the train was standing still, in still air, but you put a great big fan inside the train [A], smoke would be drawn IN because the air INSIDE the train would be moving relative to the gap.

If the train is moving and the air inside it is still (relative to the train and therefore the window) [B], smoke will be drawn OUT. Equally, if the train is stationary (relative to the tracks) but there's a gale blowing outside [C], smoke will be drawn OUT.

It's completely compatible with relativity and indeed backs it up, because in case [B] above there's a PERCEIVED pressure drop, from the Frame of Reference of the window, in the air outside - even though that air is stationary relative to the tracks and general environment, and a land-based barometer would show it at normal pressure.

Answer 5

I think this confuses what Einstien said, This isn't whats relative to you the smoker. Its the difference in air pressures at the openinng of the window. The air is still rushing past the window on the outside and the air inside is moving with the train. You can argue the point, but the math works

Answer 6

Bernoulli equation applies only to situations where the
flow of gas or fluid is stationary, because it is based on
conservation of energy. Therefore the only frame of
reference, where Bernoulli equation is valid, is the
frame of reference where the boundaries (the wall of
a pipe, the wing of a bird, the wall of bulding, etc) are
at rest.

In your example such frame of reference is train.
True, the walls of the tunnel move backward in this
frame of reference, but since they move parellel
to themselves, they do not violate the condition of
stationary flow (i.e. the force of pressure of air on the
walls of the tunnel does not perform mechanical work,
and conservation of energy stands).

So, the answer to your 'relativity' question is:
The Bernoulli equation is valid only in the frame of
reference, where the flow is stationary. In all other
frames of reference it is rewritten with work performed
by movig boundaries taken into account.

Answer 7

Relativity does not apply, because unlike for relativistic effects, it's perfectly obvious what is moving WITH RESPECT TO THE SURROUNDING AIR.

The air moves a bit as the train rushes past, but the motion of the train is much greater. Thus, you can unambiguously state the pressure will be lower outside the open window.

Answer 8

particular relativity and quantum mechanics have been very effectively itegrated into quantum electrodynamics. This thought is the foundation of lots of our information of the universe, and is the foundation for info even the semiconductors that make up your laptop. typical relativity is extra complicated. the reason being amazingly elementary. typical relativity is a tender thought - it predicts a similar behaviour spectacular all the way down to the very smallest scales. Quantum mechanics is on no account tender - it predicts granularity of mass, potential, rather lots each actual sources on the smallest point. generally, this only would not rely, via fact gravity is a very negligible stress over small distances. even nonetheless, while the effect of gravity will become very large over very small distances this motives a extensive subject. it rather is exactly what happened on the beginning place of the universe, and of direction interior of black holes.

Answer 9

Einstein had some very simple thoughts. The type of thing you mention was exactly what he went in for. In fact, a lot of what he proposed was based on what some of us would call optical illusions. For instance, you might see a man walk on water - but we would know there was something there under the surface to walk on. HTH LOL