Paradox of Bernoulli's Theorem and Moving Frames

[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.