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Bernoulli theorem paradox

  1. Dec 18, 2011 #1
    Here is a paradox that came to my mind during my fluid mechanics course last term. I don't know the solution to it:

    We have this experiment in which we hold two sheets of paper parallel to each other and blow between them. They are brought closer to each other:

    http://sepulki.net/loro/benoulli1.bmp [Broken]

    The air outside is stationary, and the air between the sheets moves, so from the Bernoulli theorem it follows that the pressure is higher outside and lower inside - this implies that the forces on the sheets point inwards and bring them closer to each other.

    Now let's consider this experiment in the reference frame of the air moving between the sheets of paper:

    http://sepulki.net/loro/benoulli2.bmp [Broken]

    Now the air in between is stationary, and the air outside is moving, so it would mean that the pressure is higher inside, and lower outside and so now the forces on the sheets point outwards and draw them apart.

    What is wrong with this reasoning in the moving frame? To be honest I'm not exactly sure if this situation can really be treated as an irrotational flow (and if the Bernoulli theorem is applicable).
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Dec 18, 2011 #2
    One problem is you're going from velocity = v to velocity = 0 discontinuously - there has to be a gradient as you get near the walls of the tube, pipe, paper, channel, whatever you want to call it.

    This is known as a "boundary layer" and its a small region where the fluid goes from the uniform velocity to zero at the edges of the boundary in a very rapid but continuous way.

    My guess is that this is where the error lies.
  4. Dec 18, 2011 #3


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    Actually, the error lies in the fact that your thought experiment changes frames but fails to consider that your paper is now moving. Static pressure is frame independent, and that is what gives rise to the forces involved in your problem.
  5. Dec 18, 2011 #4
    The thing is that this simple explaination by Bernoulli theorem doesn't take the bounadry layer into account. It would still apply if we used Euler's equations (the ones that neglect viscosity) instead of Navier-Stokes equations.

    But if we neglect viscosity - it doesn't matter whether or not the paper is moving - if there's no viscosity, it doesn't interact with air.

    In other words the Bernoulli theorem follows from Euler's equations, that are Galilean invariant, and so the pressure calculated from them should also be Galilean invariant, which it isn't...
  6. Dec 18, 2011 #5
    I was thinking that maybe it is that the Bernoulli theorem doesn't apply here?

    Because what I think we have in the case of no viscosity is a flow which is piecewise irrotational - but if we think of a point on the boundary (paper) there is vorticity around it.

    And the Bernoulli theorem holds only in irrotational flows.

    But these are just my thoughts.
  7. Dec 18, 2011 #6
    Surely it is a simple case of two distinct bodies of fluid to which you can apply Bernoulli separately.

    You cannot pick one variable value from one body and apply it to the other.
  8. Dec 18, 2011 #7


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    Depending on the source of the air stream, that's not what will happen. Hold the end of a hair dryer (in cold mode) between two sheets of paper and facing downwards. When you turn the blow dryer on (in cold mode), the sheets will not be drawn together.

    The Bernoulli principle that relates faster moving air to lower pressure only holds when there are no external forces involved and therefore no change in the total energy. In the case of a blow dryer, the total energy of the air is increased, both pressure and velocity are increased at the nozzle of the blow dryer.

    If you change the experiement to blow air between two suspended balloons or empty soda cans, the combination of coanda and venturi effects will cause the balloons or cans to converge. The coanda effect is only indirectly related to Bernoulli.

    If you blow air into a venturi pipe, then at the narrowest part of the venturi, the velocity will be highest and the pressure the lowest. Ignoring the effects of friction with the walls of the pipe and viscosity, this follows Bernoulli principle. (For a real pipe of constant diameter, pressure decreases with distance moved along the pipe due to friction and viscosity).
    Last edited: Dec 18, 2011
  9. Dec 18, 2011 #8


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    Doesn't look like you read this one:
    You transformed from one frame to another incorrectly.
  10. Dec 19, 2011 #9
    Yes you can apply Bernoulli, but you have to do it correctly and your analysis does not have enough information.

    The simplest form of Bernoulli is

    [tex]{p_1} + 1/2\rho v_1^2 = {p_2} + 1/2\rho v_2^2[/tex]

    Now the point of this is that condition 1 (LHS) and condition 2 (RHS) have to be taken for different points in the same body of fluid.

    The fluid between the sheets and the fluid external to the sheets are different fluid bodies.

    A more rigorous statement would be that you have Bernoulli applies along streamlines (1D) stream strips (2D) or stream tubes (3D), but not across them.

    What you are proposing tries to apply Bernoulli across the stream tubes.

    You can only use Bernoulli to compare when you can trace the streamlines to infinity or a common point. Otherwise it would be equivalent to trying to compare the forces exerted on a pipe by the air outside and some fluid flowing inside by using Bernoulli. You can only do this if you can find some point where the fluid enters or exits at atmospheric pressure.
  11. Dec 19, 2011 #10
    Thanks for the answers! I read them all :)

    rcgldr, thanks this is interesting:
    Does it mean, that its energy is increased, because the motor is doing work on it? Isn't it similar to us doing work on the air when we blow it out?

    That's exactly what I was thinking

    So I guess the main problem is that I don't know how to change reference frames

    Ok so now when I think of that, is it just that the Bernoulli potentials for the two stream tubes are different:

    In the frame of the paper the air inside has got a high Bernoulli potential and outside - a low Bernoulli potential.

    In the moving frame the air outside has got a low Bernoulli potential and inside - a high one.

    But the potentials change when I change frames, so in order to change frames correctly I should assume that pressures are invariant, velocities transform as usually, and from that I can calculate the new Bernoulli potentials, which change differently because we're considering different bodies of fluid.

    So in fact what happens when I go to the moving frame, is that the Bernoulli potential of the air inside goes up, because its velocity goes up ; the Bernoulli potential of the air outside goes down, because its velocity goes down - but the pressures remain the same and give rise to the same forces?

    And does the actual motion of paper really matter in the sense other than just distinguishing the two frames?
  12. Dec 19, 2011 #11
    The information you provide is not enough to solve the problem.

    Look at it like this:

    Suppose you supply air from a compressed air source to a bicycle tyre.

    Air enters the tyre (has a velocity) and the tyre walls expand against the external atmosphere. No one is suprised because the (static) pressure inside the tyre is greater than the external.

    Now let the air move the other way - the tyre walls deflate. Again no one is suprised.

    In both cases the calculations are a balance of forces due to prevailing static pressures. Bernoulli is not invoked.

    There are two bodies of fluid.

    Your paper experiment is similar.

    Now consider a aerofoil or sphere or other shape moving in an atmosphere.

    There is now only one body of fluid viz the atmosphere.

    Streamlines start at equal pressure a long way in front of the moving object, flow over and past it in some way, to finally reconnect and return to the same pressure a long way behind the object.

    In these circumstances you can apply Bernoulli (along a streamline) to gain an insight into the exchange of energy between static (potential) and velocity energies throughout the process.
    Depending upon the shape and other factors this insight can be pretty accurate.

    go well
  13. Dec 19, 2011 #12


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    Yes, but the motor in the blow dryer does much more work than a person blowing on the air. The other thing to consider is that if the static pressure inside the blow dryer's nozzle or a persons mouth wasn't higher than the ambient static pressure outside, then the air wouldn't flow (accelerate) outwards.

    Static pressure isn't changed by reference frames. Dynamic pressure is relative to a reference frame, just like kinetic energy, but that's not a factor unless the air is going to be accelerated or decelerated to the speed of the reference frame.

    I'm not sure what you mean by potential (dynamic pressure?), but the static pressure of the air inside and outside of the two sheets of paper is independent of the frame of reference. Getting back to the point I made above, the pressure of the air being blown is higher than the ambient pressure, so the air accelerates as it's pressure decreases, until momentum of the air ahead of the stream results in an adverse pressure gradient (increasing with distance), slowing the stream down. There is a term called "exit velocity" which refers to the speed of the affected air at the moment it's pressure returns to ambient. This NASA article on propellers explains this.


    Except that a wing also increases the energy of the air which violates Bernoulli, in the same manner as mentioned in the NASA propeller article linked to above, but on a smaller scale (a smaller pressure jump, with a smaller exit velocity). Also as mentioned in that NASA article outside of the zone near the wing where the energy of the air is increased, then Bernoulli applies (ignoring issues like turbulent flow).

    According to this article:

    http://home.comcast.net/~clipper-108/Lift_AAPT.pdf [Broken]

    A Cessna 172 weighing about 2300 lbs, traveling in level flight at about 140 mph, diverts about 5 tons of air per second downwards at 11.5 mph right at the wing (that 11.5 mph might actually be the exit velocity). This increases the energy of the air by about 44,000 lb-ft per second, which translates into about 80 horsepower. It's engine can produce 180 hp or more, so it can deliver the 80 hp consumed by lift and the power consumed by drag. Bernoulli principle is violated in the zone where the 80 hp is added to the air related to lift, plus the power added to the air related to drag, but Bernoulli would apply outside that zone (again ignoring issues like turbulence).

    So the Cessna 172 isn't all that efficient. At the other extreme, a high end glider like a Nimbus 4 (87 foot wingspan), weighing about 1500 lbs with pilot, might have a 60:1 glide ratio at 60 mph forward speed, 1 mph descent rate. This translates into a gravitational power input of only 4 hp, all of which goes into the air, most of it in the form of kinetic energy, some of it into temperature.
    Last edited by a moderator: May 5, 2017
  14. Dec 19, 2011 #13
    Perhaps if you read my text more closely you would understand why I wrote

    rather than stating that Bernoulli or some other equation is followed exactly.
  15. Dec 19, 2011 #14
    This makes sense, because if we consider a more realistic model for my original experiment - that the sheets of paper are of finite size, still no viscosity, and the air between the sheets comes from our mouth - it wouldn't make sense to me to apply Bernoulli in the simplest form across the paper now, if we couldn't do it in the simple model - because we still have two separate bodies of fluid.

    Oh by Bernoulli potential I meant this quantity: (I guess it's not really called that...)

    [itex]H = p + \frac{1}{2} \rho v^2[/itex]

    So taking this into account does my reasoning make sense?
  16. Dec 19, 2011 #15


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    If the sheets have finite size, then you can make a statement about which part is moving and which part is not; also, the bodies of air meet at the end, right?
  17. Dec 19, 2011 #16
    Yes, but they have different total energies, so my quantity H is still different for the air inside and outside.
  18. Dec 19, 2011 #17


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    Yes, but there is no longer a paradox since there's clearly a difference between the two frames you originally described if the sheets have finite extent.

    You would probably also be assuming that the streamlines going between the sheets of paper eventually meet up with streamlines which passed outside the paper.
  19. Dec 19, 2011 #18


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    Which why I mentioned that Bernoulli applies outsize the zone where energy is added to the air.

    This is the total energy per unit volume. H is normally used for the energy head in a different form of Bernoulli equation. Bernoulli can have a potential term, but that term is gravitational potential energy per unit volume:

    p + (1/2) ρ v2 + ρ g h

    Link to wiki article:

    They have have different total energies, but the force perpendicular to the sheets of paper is due to the differences in static pressure, not total energies. The difference in speed between the air and paper is the same in any frame, so any friction / viscosity effects between paper and air will be the same in any frame. The difference in speed between the two streams of air is also the same in any frame, so any interaction between those streams is also the same in any frame. Only speed dependent quantities such as kinetic or dynamic energy are frame dependent, but since there's no direct interaction related to the frame of reference it doesn't matter.
    Last edited: Dec 19, 2011
  20. Dec 19, 2011 #19
    Could you describe the streamlines which exhibit this behaviour, both before and after the sheets?

    Hear! hear!

    Several responders have now stated this (or equivalent) in different ways, Loro, please take note - you are getting there, fluid mechanics is a complicated subject.

    And yes there is a link between the first law of thermodynamics and Bernoulli, but it is not applicable here.
    Last edited: Dec 19, 2011
  21. Dec 19, 2011 #20


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    I imagine there would be a streamline along the inside surface of the channel and one along the outside surface (just like streamlines over a wing), and they would have to meet up at the end of the sheet. Am I confused?
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