How does a difference in air pressure induce a force?

[eddiezhang]

Homework Statement

How does a difference in air pressure induce a force? (See picture)

Relevant Equations

P = F / A, I suppose

Another likely very basic question from me

Suppose I pinch a piece of A4 paper and pull it up, at a velocity of let’s say 1 or 2m/s (i.e. order of magnitude 10^0). The paper, of course, encounters air resistance / drag as it moves through the air, but there’s also a (larger) force that is produced by the pressure difference as, momentarily, the air pressure beneath the paper is lower than above.

1) Are these different forces, or am I just describing the same thing… if not, does the second force have a name? I’ve seen it labelled as ‘drag’, so I’m confused on what actually is what here. Intuitively, they’re different forces, as the mechanism for producing air resistance is not really related to pressure.

2) Am I correct in guessing that the magnitude of air resistance here is much smaller than the force created by the difference in air pressure?

3) The million-euro question - can someone walk me through the mathematical justification for how the second force is produced? More precisely, how does a difference in air pressure across two sides of a body produce a net force? How can it be quantified?

I’m aware that anything to do with the physics of flowing air is a nightmare to model out, but I’m willing to try follow along… (am acquainted with Bernoulli's principal)

Thanks for your time

 

[haruspex]

how does a difference in air pressure across two sides of a body produce a net force

That’s the easy bit. Pressure exerts a force normal to an adjacent surface. On any small area element of the surface, considered as a vector $\vec{dA}$ normal (into) the surface with a magnitude in proportion to the area, pressure p exerts force $p\vec {dA}$.

Wrt the pinched paper, I assume you mean it has a crease in the middle making it stand up a little. You pinch at the crease and pull the paper up.
The air pressure will both tend to hold the two pinched sides together and hold the portions still horizontal flat on the table. If the paper were completely limp, there would be no resistance since these forces are normal to the directions of travel.
In reality, some stiffness in the paper will lift it off the table as it approaches the crease. This will lead to a pressure difference on the curved section, creating tension in the paper. That tension will oppose your attempt to lift it.

 

[eddiezhang]

That’s the easy bit. Pressure exerts a force normal to an adjacent surface. On any small area element of the surface, considered as a vector normal (into) the surface with a magnitude in proportion to the area, pressure p exerts force $p\vec A$.

Right... I should have phrased that question better. I'm more interested in specifically the air pressure underneath the paper. Knowing, for example, the speed at which I bring it upwards and the air pressure of the air round me, is there some mathematical method of predicting roughly what the (resultant) air pressure underneath the paper is (or even better, an approximation)?

I'm... not very optimistic that there is a nice analytical answer, so a nudge in the right direction as to how I might approximate this would be wonderful

Edit: just saw your additional point about how the tension of the paper opposing my attempt to lift it. How might I model / approximate the pressure difference at the edges of the paper, and is the pressure difference between the curved section more significant than the pressure difference above and below the paper?

tldr: suggestions on how to approximate the relevant forces?

 

[haruspex]

Right... I should have phrased that question better. I'm more interested in specifically the air pressure underneath the paper. Knowing, for example, the speed at which I bring it upwards and the air pressure of the air round me, is there some mathematical method of predicting roughly what the (resultant) air pressure underneath the paper is (or even better, an approximation)?

I'm... not very optimistic that there is a nice analytical answer, so a nudge in the right direction as to how I might approximate this would be wonderful

Edit: just saw your additional point about how the tension of the paper opposing my attempt to lift it. How might I model / approximate the pressure difference at the edges of the paper, and is the pressure difference between the curved section more significant than the pressure difference above and below the paper?

tldr: suggestions on how to approximate the relevant forces?

I should also have mentioned that air pressure holding the horizontal sections down means there will be some friction there. This helps keep open the space below the crease.
One messy part is that it depends on speed. If you pick it up very slowly air will get in quickly enough to equalise the pressure. Another is the dependency on stiffness of the paper, but with friction holding the cavity open, maybe we can ignore that.

 

[Lnewqban]

That is a very rigid piece of paper.
The net force depends on the pressure differential (P above - P below).

Please, see:
https://www.grc.nasa.gov/www/k-12/airplane/wrights/dragold.html

 

[haruspex]

That is a very rigid piece of paper.

On what basis do you deduce that?

The net force depends on the pressure differential (P above - P below).

Yes, but not directly, unless the paper is very rigid. As I wrote, it is more likely that the pressure difference results in an increased normal force from the table, in turn resulting in friction.

 

[Lnewqban]

On what basis do you deduce that?

It was a sarcastic remark based on the shown diagram.
What is represented, seems to behave as a rigid panel.
The link that I then provided follows that idea.

Yes, but not directly, unless the paper is very rigid. As I wrote, it is more likely that the pressure difference results in an increased normal force from the table, in turn resulting in friction.

Very accute and valid observation, like usually.
Again, the diagram shows instead a force that is far from normal to the table.

I hope the OP grasps the two very different scenarios and decide on which one to discuss further.

 

[Mister T]

Intuitively, they’re different forces, as the mechanism for producing air resistance is not really related to pressure.

Both are due to collisions with air molecules.

 

[haruspex]

based on the shown diagram.

Ah, ok. I hadn't taken that as remotely accurate. In post #2 I suggested a rather different picture and @eddiezhang didn't comment on it either way.

 

[256bits]

Homework Statement: How does a difference in air pressure induce a force? (See picture)
Relevant Equations: P = F / A, I suppose

1) Are these different forces, or am I just describing the same thing… if not, does the second force have a name? I’ve seen it labelled as ‘drag’, so I’m confused on what actually is what here. Intuitively, they’re different forces, as the mechanism for producing air resistance is not really related to pressure.

What are you describing? The ruler under the newspaper on a table experiment?
In that case, if you hit the ruler to try to move the ruler quickly downward, a suction is created underneath the newspaper leading to an air pressure difference of approx up to 1 atmosphere. The ruler attempts to left the newspaper creating more volume. Since air cannot rush in quickly enough through the sides of the newspaper to fill that volume in the Ft hit contact time, the pressure drops under the newspaper creating a pressure differential. The ruler is held in place.

The effect is called suction.

At some impulse where F is large and t is small, the ruler breaks or the paper tears.

At some impulse where F is small and t is large, the ruler lifts off the table.

Instead of pinching your paper, try lifting it up with sticky tape.

 

[eddiezhang]

Ah, ok. I hadn't taken that as remotely accurate. In post #2 I suggested a rather different picture and @eddiezhang didn't comment on it either way.

Fair enough - the picture is pretty misleading

 

[eddiezhang]

What are you describing? The ruler under the newspaper on a table experiment?

Pretty much... yeah.
I asked this question in this rather... different way because I thought an explanation for this scenario would provide a more 'generalised' explanation, but as I've learned, they're quite different scenarios...

I hope I don't sound like a broken record, but I'm curious about the mathematics of this 'suction' force, assuming certain parameters like the speed at which the newspaper is lifted up (i.e. the resultant velocity of the ruler), the surrounding air pressure, the angle swept out by the ruler, the rigidity of the paper etc.

 

[eddiezhang]

I don't think I can edit my original post, so here's a more accurate version of what I'm describing

 

[eddiezhang]

Is this an aerodynamic drag problem?

This is sort of why I'm asking this question, but are there two distinct forces acting here, i.e. drag / air resistance and then the force induced by the pressure difference (which I will call the 'pressure differential force' from now on)? Or, as I suspect, am I just describing how air resistance / drag is created (i.e. via the pressure difference)? If these two forces are distinct, however, would I be correct in my estimation that the drag force is much less significant than the 'pressure differential force'?

(I'm vaguely aware of there being a difference being 'form drag' and 'pressure drag')

 

[haruspex]

the mathematics of this 'suction' force, assuming certain parameters like the speed at which the newspaper is lifted up (i.e. the resultant velocity of the ruler), the surrounding air pressure, the angle swept out by the ruler, the rigidity of the paper etc.

The ruler case is a bit different from the pinched fingers case.

In the latter, the air pressure difference acts on the paper that's flat on the table, increasing the frictional force. If the surfaces are very smooth, and the paper quite flimsy, you would not feel much resistance. You could try this with cling film, comparing a dry smooth surface with an oiled one.

In the former, the above may still be significant, but now you have a substantial cavity under the paper. Much of the torque needed is in fighting the air pressure above the ruler. Air has to flow in under the paper to fill that cavity, making this case partly an aerodynamic drag problem. Drag slows the inflow, so the air pressure inside the cavity drops.
You have to assume some initial air gap for the flow to start. The size of that will significantly affect the result. As the cavity grows, it will also widen at the base, reducing the drag. Also, each equal rise of $\Delta\theta$ of the ruler does not increase the volume of the cavity by as great a fraction as at first, so the pressure drop is less.
Compare a smooth surface with a rough one (a larger initial channel for the air).

 

[Lnewqban]

As the air rushes in from the edges of the plate/board/paper toward the central area, its velocity depresses the value of the static pressure under the paper.
That is a brief condition: as the inter-surfaces gap increases, the air velocity decreases and the static pressure increases (not in the case of the ruler under paper experiment, in which the gap remains small).

The following diagrams show an experiment in which the airflow goes in the opposite direction (from center to edges), also creating a suction effect based on Bernoulli’s.