Does pressure have anything to do with flow rate?

[needingtoknow]

Homework Statement

For example, let's say we have a container and then we have a hole in the side of the container. Water that is in the container will have a greater flow speed at the hole of the container than let's say at the top of the container, Point A. (point A is subject to atmospheric pressure)

But isn't pressure consistent throughout the container, even if Point A and the hole were not both subject to atmospheric pressure only (which they are), gauge pressure is consistent throughout the container anyways right.

So if the pressure is the same at both Point A and the hole, how does it make sense that one has a greater flow speed than the other with respect to pressure? (keeping in mind I understand how it makes sense with respect to the flow rate formula f = Av)

Thank you for your help!

 

[rude man]

Pressure is not the same at the top and at the hole.

At the top it's 1 at. At the hole on the INSIDE it's 1 at + the water pressure at that depth (rho g h). At the OUTSIDE of the hole the pressure is again 1 at.

So use Bernoulli's equation to tell you how pressure, flow (kinetic energy) and height interrrelate along a streamline from the top to the outside of the hole. Hint: at the hole inside the container, the pressure goes up by rho g h but the potential energy drops by the same amount.

 

[needingtoknow]

I can tell you what my solutions book says because there could be a chance that it is wrong. "Let the bottom of the tank be our horizontal reference level, and choose Point 1 to be at the surface of the liquid and Point 2 to be at the hole where the water shoots out. First, the pressure at Point 1 is atmospheric pressure; and the emerging stream at Point 2 is open to the air, so it's at atmospheric pressure, too."

Also you say the potential energy drops as well, is that because the water is flowing from a point that is closer to the ground?

 

[needingtoknow]

Also according to the Bernoulli effect, where pressure is greater flow speed is lower so then even if like you said there is greater pressure at the hole, why is the flow speed greater?

 

[Nathanael]

Also you say the potential energy drops as well, is that because the water is flowing from a point that is closer to the ground?

He's talking about the gravitationa potentiall energy within the container, ignoring the hole.
The lower you are in the container, the lower the gravitational potential energy is.
But the lower you are in the container, the higher the pressure energy is.

Also according to the Bernoulli effect, where pressure is greater flow speed is lower so then even if like you said there is greater pressure at the hole, why is the flow speed greater?

Yes, but he did not say there is greater pressure at the hole, he said there is greater pressure within the container at the depth of the hole.

At the hole, however, he said that the pressure is only one atmosphere.
(Because, at the hole, the only thing "stopping the water" (causing pressure) is the atmosphere.)So, at the hole, there is the same pressure energy, but less gravitational potential energy.
So Bernouilli's equation says that at the difference in GPE (it has less) must equal the difference in kinetic energy (it has more).
(all of this is per volume)

 

[needingtoknow]

Actually wait I think I am mistaken then. The pressure is the same for all points that are in the same line, same depth. But these are clearly at two different depths. At the depth of the hole there is greater pressure. Ohh and what you are saying is that because the water at the hole is exposed to air and not so much to water, it only experiences atmospheric pressure so that is why it still has a high flow rate because the atmospheric pressure isn't that high so it still abides by Bernoulli's Principle.

My question now I guess is that why does the water near the top of the container that has the same atmospheric pressure which is quite low, why is that point not experiencing a high flow speed. After all there is low pressure there right and according to Bernoulli's Principle it should right? (I know why it flows slowly according to F = Av because the area to flow into is much larger at the top than it is at the hole)

 

[Nathanael]

Well Bernouilli's equation is not only about pressure and velocity. A low pressure does not necessarily mean a high velocity.

Bernouilli's equation just says something along the lines of "the density of energy is conserved for any given volume of a fluid"

So that means if you have a decreased pressure, you must have an equivalent increase in some other form of energy.
In a lot of cases this other form of energy that is increased is Kinetic Energy, but it can also be gravitational potential energy.So even though the top of the container has a "low pressure"*, it has a (relatively) high GPE and thus has a (relatively) low Kinetic Energy.
(Relative to the other point we're comparing it to, which is the hole that is lower down.)P.S.
*It does not have a relatively low pressure, and it is really only relative amounts of energy that are important.
(Low pressure, or low kinetic energy, or low GPE, only makes "useful sense" when compared to somewhere else.)

 

[needingtoknow]

What you said makes sense but for it to make 100 percent sense I need to clear one last thing up. Why do you say pressure is energy? Isn't it just force divided by area?

 

[Nathanael]

What you said makes sense but for it to make 100 percent sense I need to clear one last thing up. Why do you say pressure is energy? Isn't it just force divided by area?

Yes that is one way of looking at pressure.

That perspective of pressure is useful for things like the pressure between my feet and the ground.
(Where there is only an "area of interaction")

But that (F/A) doesn't make complete sense when talking about pressure in a volume, does it?

There is another way of looking at pressure that is more useful for volumes. Instead of Force per Area, you can say pressure is Energy per Volume.

(We can see from the units that this is the same thing:)
[itex]Pressure=\frac{Energy}{Volume}=\frac{N\cdot m}{m^3}=\frac{N}{m^2}=\frac{Force}{area}[/itex]A side note: I personally imagine pressure to be a sort of "internal kinetic energy"
What I mean by that is, even though the center of mass of a volume of liquid is still (and so you would say it's kinetic energy is zero, because there's no net momentum) there are still random internal motions of molecules that have energy.
This "net-zero-momentum kinetic energy" is what I imagine when people say "pressure"

I don't know if that made sense, I don't know if that's a typical scientific view of pressure, but I find it to be very useful for imagining (and solving) fluid problems.Hopefully this helps and hopefully I didn't make that last part sound confusing (it's really a simple and useful idea)

 

[needingtoknow]

No that made more sense than it before ! Thank you very much for your help Nathaneal!

 

[Nathanael]

You're very welcome. I'm always happy to help with someone's understanding