Paradox of Bernoulli's Theorem and Moving Frames

[Loro]

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

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

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).

 

[pergradus]

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.

 

[boneh3ad]

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.

 

[Loro]

The thing is that this simple explanation 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...

 

[Loro]

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.

 

[Studiot]

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.

 

[rcgldr]

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.

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).

 

[russ_watters]

Doesn't look like you read this one:

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.

You transformed from one frame to another incorrectly.

 

[Studiot]

I was thinking that maybe it is that the Bernoulli theorem doesn't apply here?

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

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

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.

 

[Loro]

Thanks for the answers! I read them all :)

rcgldr, thanks this is interesting:

In the case of a blow dryer, the total energy of the air is increased

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?

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

That's exactly what I was thinking

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

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.

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?

 

[Studiot]

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?

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

 

[rcgldr]

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.

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?

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.

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

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.

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.

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.

http://www.grc.nasa.gov/WWW/K-12/airplane/propanl.html

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.

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

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.

 

[Studiot]

Except that a wing...etc

Perhaps if you read my text more closely you would understand why I wrote

gain an insight

rather than stating that Bernoulli or some other equation is followed exactly.

 

[Loro]

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.

I'm not sure what you mean by potential

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

H = p + \frac{1}{2} \rho v^2

So taking this into account does my reasoning make sense?

 

[olivermsun]

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.

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?

 

[Loro]

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?

Yes, but they have different total energies, so my quantity H is still different for the air inside and outside.

 

[olivermsun]

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.

 

[rcgldr]

Except that a wing also increases the energy of the air ... outside of the zone near the wing where the energy of the air is increased, then Bernoulli applies ...

Perhaps if you read my text more closely you would understand why I wrote "gain an insight" rather than stating that Bernoulli or some other equation is followed exactly.

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

Oh by Bernoulli potential I meant this quantity: (I guess it's not really called that...)
H = p + (1/2) ρ v²

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:

http://en.wikipedia.org/wiki/Bernoulli's_principle

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?

Yes, but they have different total energies, so my quantity H is still different for the air inside and outside.

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.

 

[Studiot]

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.

Could you describe the streamlines which exhibit this behaviour, both before and after the sheets?

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.

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.

 

[olivermsun]

Could you describe the streamlines which exhibit this behaviour, both before and after the sheets?

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?

 

[Studiot]

Hello, oliver,

The point of a streamline is that the quantity referred to in posts 9, 14 and 18 is constant along a streamline.

Furthermore if two streamlines meet at any point they have the same pressure (and velocity), since you can't have two different pressures (and velocities) at one point. Streamlines don't cross, of course.

Now given that information, try to draw streamlines that meet at the end of the sheet, but have different pressures (and velocities) elsewhere given that the line in the channel can have any pressure and velocity impressed upon it by the blower. Where would these streamlines from the blower start?

 

[olivermsun]

Studiot,

If the sheets end then the flow has to sort itself out at some point (and I assume the pressure and velocity will have to join up with ambient eventually). So, there is a big difference between the finite sheet problem and the infinite sheets, as posed by the OP, where the inner flow never meets the outside, never has to match with it, pretty much can be anything you want (including any static pressure you desire).

Anyhow, I'm not the OP, why does it sound like you're trying to assign me an exercise?

 

[rcgldr]

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.

At the end of the sheet, the streamlines will interact due to viscosity (friction within the air), most likely resulting in turbulent flow. I'm not sure at what point you could consider the streamlines "merged" into a new streamline. This interaction will have some effect on the streamlines somewhat before the end of the sheets, and friction with the sheets of paper and viscosity is also affecting the flow (converting pressure energy into heat, which reduces static pressure).

 

[boneh3ad]

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?

Hello, oliver,

The point of a streamline is that the quantity referred to in posts 9, 14 and 18 is constant along a streamline.

Furthermore if two streamlines meet at any point they have the same pressure (and velocity), since you can't have two different pressures (and velocities) at one point. Streamlines don't cross, of course.

Actually, if two streamlines meet at any point, you have broken physics. The core concept behind streamlines stipulates that they can never meet or cross. They can only get asymptotically closer to one another.

At the end of the sheet, the streamlines will interact due to viscosity (friction within the air), most likely resulting in turbulent flow.

They may, but there is nothing to say for certain that this will happen. However, saying that streamlines will interact is somewhat incorrect and doesn't make a whole lot of sense.

 

[olivermsun]

Actually, if two streamlines meet at any point, you have broken physics. The core concept behind streamlines stipulates that they can never meet or cross. They can only get asymptotically closer to one another.

What is a stagnation point?

 

[rcgldr]

At the end of the sheet, the streamlines will interact due to viscosity (friction within the air), most likely resulting in turbulent flow.

Actually, if two streamlines meet at any point, you have broken physics. They may (interact), but there is nothing to say for certain that this will happen. However, saying that streamlines will interact is somewhat incorrect and doesn't make a whole lot of sense.

The physics isn't "broken", but technically the streamlines stop being streamlines once they interact. Then again technically, the flow between the sheets is affected by friction and viscosity (it's pressure is reduced towards ambient with distance along the sheets), so it's not really a streamline either. What you end up in a real world situation with air, are approximations to streamlines.

 

[boneh3ad]

What is a stagnation point?

That would be the end of a streamline considering that the definition of a streamline is that its tangent is in the direction of the local velocity vector and at a stagnation point, the velocity is zero.

The physics isn't "broken", but technically the streamlines stop being streamlines once they interact. Then again technically, the flow between the sheets is affected by friction and viscosity (it's pressure is reduced towards ambient with distance along the sheets), so it's not really a streamline either. What you end up in a real world situation with air, are approximations to streamlines.

Excuse the colloquialism. I am aware that physics isn't "broken", but the definition of a streamline does not allow them to cross or meet. Streamlines still make sense in the world of viscous flows... to a point. The concept of a streamline itself is perfectly valid. Even the Bernoulli equation is valid along a single streamline in a viscous flow so long as it is not turbulent. After that, streamlines are little more than a tool for visualizing the flow direction.

 

[rcgldr]

The concept of a streamline itself is perfectly valid. Even the Bernoulli equation is valid along a single streamline in a viscous flow so long as it is not turbulent. After that, streamlines are little more than a tool for visualizing the flow direction.

So to avoid using the term streamline, there was a question as to what happens when the flow goes past the end of the sheets into the ambient air, and the effect on the surrounding air outside but near the end of the sheets. Since the flows are at different pressures and speeds (similar to what happens at the trailing edge of a wing), there's an interaction due to viscosity and momentum. I speculated that the flow would be turbulent. For a simpler case, consider a flow exiting from a thin walled tube into ambient air, what does the flow in the vicinity of the end of the tube look like?

- - -

I'm hoping no one has missed the fact that the original premise was wrong:

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.

The sheets will not be brought closer together. The static pressure of the blown air will be slightly greater than ambient. As mentioned before, you can test this with a hair dryer in cold mode, holding two sheets of paper on opposite sides of the nozzle. Almost nothing happens when you turn on the blow dryer, at least along the main flow from the nozzle (some air from outside the main flow will be drawn into the flow because of viscosity).

My guess is that the actual experiement involved coanda or venturi effects.

 

[Loro]

Several responders have now stated this (or equivalent) in different ways

Ok, thanks a lot, I think I've unsterstood it correctly then

So, there is a big difference between the finite sheet problem and the infinite sheets

the flow has to sort itself out at some point (and I assume the pressure and velocity will have to join up with ambient eventually)

I don't see how there could be a big difference - we can take longer and longer sheets, and that should make the finite sheet behaviour more and more similar to the infinite sheet one.

Isn't it that if we're still considering the case of inviscid flow - it wouldn't sort itself out? The streamlines from in between the sheets would always have a higher H-quantity that the outside ones, no matter how far away from the sheets we look? (And in that way the infinite sheets would be a limiting case for the finite sheet problem)

Even the Bernoulli equation is valid along a single streamline in a viscous flow so long as it is not turbulent.

Doesn't the flow have to be irrotational? - which I'm guessing it wouldn't be if there was viscosity - because there would be vortices behind the sheets.

The sheets will not be brought closer together.

So are you saying that it's partially viscosity, what's responsible for this effect, and it can also partially be the pressure difference, if the amount of work done on the air to make it move is just right to make the pressure in between smaller?

(I need to check it myself with a hair dryer! But tomorrow - now it's too late at night here...)

 

[boneh3ad]

So to avoid using the term streamline, there was a question as to what happens when the flow goes past the end of the sheets into the ambient air, and the effect on the surrounding air outside but near the end of the sheets. Since the flows are at different pressures and speeds (similar to what happens at the trailing edge of a wing), there's an interaction due to viscosity and momentum. I speculated that the flow would be turbulent. For a simpler case, consider a flow exiting from a thin walled tube into ambient air, what does the flow in the vicinity of the end of the tube look like?

It would simply be a stratified flow and would be subject to the Kelvin-Helmholtz instability. It still may or may not become turbulent depending on a number of factors, such as the relative velocities of the strata, the densities of the strata and the wavenumbers of whatever perturbations are present in the flow. There simply isn't enough information to determine if the flow will or will not become turbulent.

Doesn't the flow have to be irrotational? - which I'm guessing it wouldn't be if there was viscosity - because there would be vortices behind the sheets.

It has to be irrotational to use Bernoulli in the general sense it is often applied, but even in a rotational flow, Bernoulli can be applied along a streamline because the flow along a streamline is irrotational assuming it is steady and isn't turbulent. Turbulence throws it all out the window.

 

[olivermsun]

Isn't it that if we're still considering the case of inviscid flow - it wouldn't sort itself out? The streamlines from in between the sheets would always have a higher H-quantity that the outside ones, no matter how far away from the sheets we look? (And in that way the infinite sheets would be a limiting case for the finite sheet problem)

Maybe there's another way to think about this that will make it clearer.

Let's say that your inviscid flow is a "jet" flowing in the positive x direction which remains steady (and parallel) after it exits the channel formed by the sheets of paper. What this means (via Bernoulli or any other way you look at it) is that the velocity and pressure in the jet are constant for all x. It also means that jet and the "ambient" fluid have equal pressures along their boundary (otherwise, the flow would change shape). It doesn't really matter what you call static pressure and what you call dynamic pressure, since the pressure is constant throughout the flow (in fact, throughout the entire domain in your example).

If this is the scenario you're envisioning, then the sheets of paper are completely redundant. The pressure is equal on all sides of the sheets, and there will be no net force on the sheets.

However, if you have a clearly defined idea of an "ambient" pressure and some other thing going in your jet (so that, for example, there is a net inward force on the sheets) then you're going to have to explain how the jet joins with the ambient flow at the ends of the channel. There's will be a pressure differential somewhere, causing the flow to accelerate (or decelerate).

 

[olivermsun]

It would simply be a stratified flow and would be subject to the Kelvin-Helmholtz instability.

I don't recall the problem specifying a density difference, but either way, if you are thinking something like a turbulent jet then the details of the flow get exciting. I don't think it's really necessary for addressing the OP's confusion, however.

 

[rcgldr]

The sheets will not be brought closer together. ... My guess is that the actual experiement involved coanda or venturi effects.

So are you saying that it's partially viscosity, what's responsible for this effect ...

No I'm stating the effect doesn't occur. I tried this with a blow dryer as described, and the sheets do not converge.

My guess it that curved shapes were involved, like the balloons in this youtube video:

http://www.youtube.com/watch?v=jZBp59KnwvM&hd=1

The video claims the balloons converge because of Bernoulli effect, without much explanation. To be clarify this, note the flow out of the blower has slightly higher than ambient pressure (otherwise there woudl be no flow out of the nozzle). Coanda effect causes the air from the flow to be diverted outwards at it curves around and separates on the down stream surfaces of the balloons. This results in the balloons exerting an outwards force on the air, and the air exerting and equal and opposing inwards force on the balloons, causing them to converge. There's some venturi effect, but since the flow is not constrained vertically, I'm not sure how much it contributes to the overall effect.

update- removed link to questionable website regarding spool and card effect. Although the particular arcitle seems valid (that the reduced pressure of the flow is related to radial expansion of that flow from the center of the spool), the website seems to be promoting an alternate theory for lift, and I don't have the knowledge to know if all the articles at that website are valid.

So I've learned to stick with venturi based devices to demonstrate Bernoulli effect. (I added links to an example of such a device in my next post).

considering the case of inviscid flow

Inviscid flow is unpredictable. Since there's no viscosity, adjacent flows at different speeds do not interact which allows them to coexist. If there was inviscid flow between the sheets what happens beyond the end of the sheets depends on the existing flow downstream of the sheets. One possibility is that an existing flow continues endlessly never interacting with the surrounding inviscid fluid or gas. Another possibility is that the flow collides with a non moving volume of fluid or gas, and I don't think that the outcome is predictable for an inviscid flow.

 

[boneh3ad]

I don't recall the problem specifying a density difference, but either way, if you are thinking something like a turbulent jet then the details of the flow get exciting. I don't think it's really necessary for addressing the OP's confusion, however.

That response was to rcgldr, not the OP. My bad if that caused any confusion. Either way, stratified flows don't have to have a density difference. They must simply have discontinuous layers. The discontinuity can be in density, velocity, viscosity, etc.

 

[harrylin]

[..]
A much better demonstration of Bernoulli is a "Bernoulli levitator", which can be made using a spool and card (the website suggests a sewing pin used to keep the card from sliding sideways, but a small tack might be safer (since it couldn't be accidentally sucked through the card)):

http://www.seykota.com/rm/spool_card/spool_card.htm
[..]

Sorry but according to the website you refer to, that is a poor example of the Bernouilli effect; and it appears to me that the author is right because the air flow is expanding.

 

[harrylin]

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:

[...]

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:

[...]

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).

I think that your last remark is the essential one: Bernouilli is about a change of pressure when a fluid changes speed, due to conservation of its total energy.
The picture that you refer to is about a pressure difference between the fluid and the environment, and that is not directly applicable. However Bernouilli can be used for that kind of situations too, for example by introducing a Venturi. See http://en.wikipedia.org/wiki/Bernoulli's_principle

Harald

 

[rcgldr]

A much better demonstration of Bernoulli is a "Bernoulli levitator", which can be made using a spool and card (the website suggests a sewing pin used to keep the card from sliding sideways, but a small tack might be safer (since it couldn't be accidentally sucked through the card)):

http://www.seykota.com/rm/spool_card/spool_card.htm

Sorry but according to the website you refer to, that is a poor example of the Bernouilli effect; and it appears to me that the author is right because the air flow is expanding.

You're correct, the radial flow explanation makes sense. I updated my previous post. I had been misled by similar sites and hadn't paid attention to the explanation given at that site. It was just the first one I found with a photo, and I hadn't read all of the article. Thanks for pointing this out.

However that website seems to be promoting radial momentum as an explanation in other scenarios, and an alternate theory for lift. I don't have the knowledge to know if the rest of the articles at that website are valid, so reader beware. I removed the link from my previous post.

Venturi effect seems to be a valid example of Bernoulli principle, so here's a link to a device that uses venturi effect to reduce pressure, normally used to drain water water from an aquarium. There's also a diffuser effect at the exit path that allows the pressure inside to remain below ambient while providing an exit path where the flow slows and it's pressure returns to ambient.

http://andysworld.org.uk/aquablog/?postid=247

Link to image of the internals, figure 4 shows the device in venturi + diffuser (drain) mode:

psdrawing.gif

 

[harrylin]

Generally on the way to having a stream online, that is not what can come. Provide the conclude of every head tumble dryer (through chilly temperature approach) between double pieces of paper and in addition against downhill. When you finally go the very blow dryer with regards to (present in frigid mode), most of the documents are not fascinated along.

Often the Bernoulli standards which applies sooner proceeding o2 to lessen worry basically keeps any time you are you won't notice any external pushes connected and so neo improvements on you've got a power. In case there are the blow dryer, you've got a stamina from your area is really raised, together weight as well as the velocity are really bigger in your mist nozzle of a hair dryer.

If you happen to customise the experiement in order to really blow the air amid multiple revoked balloons or even cleaned out those things can actually bottles, lots of people of coanda plus venturi outcome can the balloons and / or bins in meet. The very coanda end result is just ultimately concerned with Bernoulli.

Hi it looks like you use a translation program that doesn't work well. Here's a way to improve (I tested it myself): copy-paste the result back into the translation program and translate backward. Modify everything that has been messed up, and then try again, until the result is satisfying.

 

[RandomGuy88]

It has to be irrotational to use Bernoulli in the general sense it is often applied, but even in a rotational flow, Bernoulli can be applied along a streamline because the flow along a streamline is irrotational assuming it is steady and isn't turbulent. Turbulence throws it all out the window.

Be careful saying you can apply Bernoulli along a streamline in a viscous flow. You can use a modified version of the equation if you add a term that accounts for frictional losses. In a pipe this term is a function of the length, diameter and a constant known as the friction factor which is usually determined from a chart.

 

[boneh3ad]

Be careful saying you can apply Bernoulli along a streamline in a viscous flow. You can use a modified version of the equation if you add a term that accounts for frictional losses. In a pipe this term is a function of the length, diameter and a constant known as the friction factor which is usually determined from a chart.

Fair enough. I should add the caveat that what I said only holds in portions where viscous dissipation is unimportant, i.e. not in the boundary layer and not in pipe or duct flow except in certain cases.

 

[rcgldr]

So before we go much further off on these tangents, is Loro happy with the answers he's received about his original questions? I'm wondering if Loro has tried to reproduce the experiement using two sheets of paper and a blow dryer yet.

 

[Loro]

Thanks,

To be honest I'm happy with the explanation of Studiot (although it explains the error of my original line of thought - but doesn't explain why the effect occurs)

I'm not so sure about what rcgldr says. I've just tried it - I hung two sheets of paper using clips, next to each other (so that they weren't curved and nothing held them rigidly) - I tried blowing myself and also with a hair-dryer (unfortunately it can only blow hot air) - in all cases the sheets do come closer to each other.

 

[Studiot]

Hi, Loro, thanks for the comment.

During the course of this thread one or two misconceptions have crept in and it would be useful at this stage to round them all up into one post.
Before I do this it would also be useful to know at what level to pitch this, mathematically and physically ie where are you at and are you happy with calculus and vectors (del, cross product etc)?
I also think it is a good idea to get a good grasp of what is happening physically before embarking on long mathematical excursions.

 

[Loro]

I'm a third year physics student. Yeah I'm ok with calculus, vector calculus, a bit of complex analysis. Our fluids course was like a lot of stuff compressed into not too many lectures - but I am (or should be) familiar with the most elementary things like solving Navier-Stokes equations for simple geometries, Kelvin's circulation theorem, streamlines, and qualitatively: starting vortices, turbulence, ...

I also think it is a good idea to get a good grasp of what is happening physically before embarking on long mathematical excursions.

Yeah I totally agree - however here after reading all these responses it seems to me that one can think of at least several mechanisms that could give rise to this effect.

 

[rcgldr]

I'm not so sure about what rcgldr says. I've just tried it - I hung two sheets of paper using clips, next to each other (so that they weren't curved and nothing held them rigidly) - I tried blowing myself and also with a hair-dryer (unfortunately it can only blow hot air) - in all cases the sheets do come closer to each other.

In my case, I sandwiched the nozzle between two sheets of paper (about 2 inches of overlap between end of sheet and end of nozzle, with the remaining 9 inches hanging), held it all together, aimed it downwards and turn on the hair dryer, not much happened.

Since my results were different than yours, I tried with the sheets of paper taped to the sides of shower curtain hooks (each hook is larger than the curtain rod, so each only touches at one point and can freely rotate +/- 10 to 15 degrees), spaced so that they were separated from each other by the diameter of the nozzle. Each sheet of paper was oriented so was 21.6 cm (8.5 inches) horizontal and 27.9 cm (11 inches) vertical.

With human powered flow (I hadn't tried this before), if the sheets were oriented to narrow in the direction of flow, the far ends of the sheets would tend to converge to a fixed distance apart at the point where the flow passed the end of the sheets, regardless if the ends of the sheets were initially closer or farther apart than the distance they tended to converge to when a flow was present. If the sheets were parallel or too far apart, there was no noticable reaction.

I tested with the hair dryer again and got mixed results. Again I sandwiched the nozzle between the sheets of paper, then turned it on. The results with the hair dryer seemed to be sensitive to the direction of the flow relative to the sheets. Blowing across the width of the sheets (21.6 cm, 8.5 inches), there was no noticable or consistent effect. Blowing down along the length of the sheets (27.9 cm, 11 inches) the sheets usually converged to form a sort of elliptical cone around the flow from the blower.

Note if the flow from the hair dryer is sufficiently higher than ambient, then along the path of the flow, the flow decreases in pressure as it accelerates and the flow cross sectional area will decrease since mass flow is constant except for the surrounding air being drawn into the flow due to viscosity. I don't know if the pressure flow from a hair dryer is suffciently above ambient to produce a noticable effect.

The hair dryer I used is tapered, with an intake in the back. At the fan, the diameter of the nozzle is 7.6 cm (3 inches) and at the nozzle, 5.7 cm (2.25 inches). I don't know how this would effect the results.

Anyway, based on my experiments, when there is a reaction between the sheets, the result is that the sheets tend to form a fixed shape around the flow, regradless if they were initially narrower or wider than the flow.