Continuity & Bernoulli

[MedPR]

Confused, again.
So Bernoulli says that Pressure decreases as Area increases. Cool, that makes sense. The continuity equation says that velocity decreases as area increases, since the volume of flow must be equal. That makes sense too. However, Bernoulli also says that as velocity increases, pressure decreases.

Am I missing something here?

Edit Ignore the above! :

http://hyperphysics.phy-astr.gsu.edu/hbase/pber.html#beq

So continuity says Area up = velocity down.
Bernoulli says pressure up = velocity down

Based on that you can reason that pressure up = area up?

But
There's a problem in TBR that says "The relationship between velocity and pressure difference is linear and both should be equal at the same time." So as pressure increases, velocity increases. This contradicts Bernoulli's equation, doesn't it?

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[milski]

Confused, again.

So Bernoulli says that Pressure decreases as Area increases. Cool, that makes sense.

The continuity equation says that velocity decreases as area increases, since the volume of flow must be equal. That makes sense too.

However, Bernoulli also says that as velocity increases, pressure decreases.

Am I missing something here?

There is no area in the Bernoulli's equation:

P+1/2*ρ*v^2+ρ*g*y=const (disregard ρ*g*y for horizontal pipe).

Where does your first statement about pressure decreasing when the area increases come from?

 

[MedPR]

ok, fixed OP.

 

[milski]

Yes, the pressure will be higher in the sections with larger area. (For constant laminar flow in a horizontal tube).

"The relationship between velocity and pressure difference is linear and both should be equal at the same time."

Can you post a bit more context? They are talking about pressure difference, not the pressure itself. That still does not make it right, since the pressure difference is proportional to v^2, not v. So it has to be about something else?

 

[MedPR]

Yes, the pressure will be higher in the sections with larger area. (For constant laminar flow in a horizontal tube).



Can you post a bit more context? They are talking about pressure difference, not the pressure itself. That still does not make it right, since the pressure difference is proportional to v^2, not v. So it has to be about something else?

http://imgur.com/DOqhN

"Fluids flow from regions of higher pressure to regions of lower pressure, so in order to get a fluid to flow, there must be a pressure difference across a length of the pipe. It should seem intuitive that a greater pressure difference would push fluid faster, so D can be eliminated. Also, choice C implies that the speed can increase despite no increase in pressure difference past a threshold, which is not true. Choice C is also eliminated. A change in pressure from a negative value to a positive value implies that pressure gradient reversed directions. This would reverse the direction of the fluid flow, and is not what choice A shows. The relationship between velocity and pressure difference is linear and both should be equal to zero at the same time. Only choice B shows this.

Thus, B is correct.

 

[milski]

They are talking about total pressure drop across the whole pipe, not about the pressure at a specific place. That's Poiseuille's law, not Bernoulli's.

Poiseuille's tells you how much pressure you are going to lose over a pipe with constant radius for a given flow rate, length and radius of the pipe and specific fluid.

Bernoulli is about how the pressure in the pipe changes as it changes radius.

In reality you will experience both of them over a single pipe - the further away you go, the lower the pressure will be. And in the more narrow sections you'll get lower pressure as well.

 

[MedPR]

I guess I'm confused about being confused..? It makes sense intuitively that higher pressure will increase velocity. But bernoulli's equation pretty blatantly states the opposite. So I'm not sure when to apply which rule?

 

[MedPR]

So Poiseuille's is talking about a single pipe with constant area and continuity + bernoulli is talking about a pipe where area isn't constant?

 

[milski]

That's a bit of overgeneralization but you can think that Poiseuille is about what happens when you vary how much you push things from one end of the pipe while Bernoulli deals with what happens in the pipe for a specific, constant push from the end.

Another analogy that you can think about is that in a way Bernoulli is like the conservation of momentum - a smaller mass would start moving faster and vice versa. Poiseuille is like introducing friction - how much you'll slow down after moving a certain distance.

So Poiseuille determines how fast you start moving at the entrance of the pipe, Bernoulli tells you how you speed up and down as the cross-section changes. Again, this paragraph simplifies it a bit since Poiseuille is only for a fixed radius pipe but it should give you idea about the difference between the two.

 

[milski]

So Poiseuille's is talking about a single pipe with constant area and continuity + bernoulli is talking about a pipe where area isn't constant?

That is correct. The more interesting part is that Poiseuille is talking about what happens at both ends of the pipe while Bernoulli talks about what's going on in the pipe. If you have to apply Poiseuille for a pipe with non-constant area, you just break it into sections with constant radius and add the pressure differences for each of them.

PS: Poiseuille is not only for constant area but for constant circular area.

 

[MedPR]

That is correct. The more interesting part is that Poiseuille is talking about what happens at both ends of the pipe while Bernoulli talks about what's going on in the pipe. If you have to apply Poiseuille for a pipe with non-constant area, you just break it into sections with constant radius and add the pressure differences for each of them.

PS: Poiseuille is not only for constant area but for constant circular area.

Ok thank you for clarifying. So in a pipe with two different cross-sectional areas (same as circular area, right?) the velocity will be greater where the area of the pipe is smallest (by Q=Av=v*pi*r^2), and by bernoulli's equation, we know that when the velocity increases, the pressure decreases.

Is that the only way to explain why pressure decreases as area decreases? I mean, intuitively I would think that pressure would be greater, since there is more fluid in a smaller cross-sectional area... Am I confusing Poiseuille and Bernoulli again or something?

 

[davcro]

There's a problem in TBR that says "The relationship between velocity and pressure difference is linear and both should be equal at the same time." So as pressure increases, velocity increases. This contradicts Bernoulli's equation, doesn't it?

No. If you increase P1 on the left side of the equation (http://hyperphysics.phy-astr.gsu.edu/hbase/pber.html#beq), the right side of the equation must compensate by increasing the velocity. The right side could compensate by increasing the pressure, but then the pressure difference wouldn't change. It could also compensate by increasing the height.

 

[davcro]

Is that the only way to explain why pressure decreases as area decreases? I mean, intuitively I would think that pressure would be greater, since there is more fluid in a smaller cross-sectional area... Am I confusing Poiseuille and Bernoulli again or something?

Think of it in terms of energy. When the radius decreases, the fluid must move quicker, the kinetic energy increases. Where did this energy come from? It came from pressure energy (or energy density). The pressure is converted into kinetic energy. It must go down.

 

[milski]

Ok thank you for clarifying. So in a pipe with two different cross-sectional areas (same as circular area, right?) the velocity will be greater where the area of the pipe is smallest (by Q=Av=v*pi*r^2), and by bernoulli's equation, we know that when the velocity increases, the pressure decreases.

Yes, this is correct.

I mentioned circular because for Poiseuille's law the shape of the pipe matters. A rectangular pipe with the same area will have a different pressure drop compared to a circular one. That should make sense, since the pressure change is mostly due to the friction with the walls of the pipe.

Is that the only way to explain why pressure decreases as area decreases? I mean, intuitively I would think that pressure would be greater, since there is more fluid in a smaller cross-sectional area... Am I confusing Poiseuille and Bernoulli again or something?

Yes, you are. Once you are inside the pipe, it's Bernoulli only. Try to think about it as flow rate, not as velocity. Inside the pipe the flow rate is constant, the only pressure changes happen due to area changes. If you're changing the pressure at the end, you're really changing the flow rate, not simply the velocity.

Actually, there is not more fluid in the narrower part. The fluid does not compress, so the same volume of fluid takes a longer part of the pipe.

 

[MedPR]

Yes, this is correct.

I mentioned circular because for Poiseuille's law the shape of the pipe matters. A rectangular pipe with the same area will have a different pressure drop compared to a circular one. That should make sense, since the pressure change is mostly due to the friction with the walls of the pipe.



Yes, you are. Once you are inside the pipe, it's Bernoulli only. Try to think about it as flow rate, not as velocity. Inside the pipe the flow rate is constant, the only pressure changes happen due to area changes. If you're changing the pressure at the end, you're really changing the flow rate, not simply the velocity.

Actually, there is not more fluid in the narrower part. The fluid does not compress, so the same volume of fluid takes a longer part of the pipe.

Right, I forgot about that. This is why the pressure in a capillary is not greater than the pressure in an artery or vein..right?

 

[milski]

Right, I forgot about that. This is why the pressure in a capillary is not greater than the pressure in an artery or vein..right?

That is correct. But let's go with the pressure in the smaller veins and arteries close to the capillaries. The capillaries are a non-ideal case due to being really narrow, probably better not to go there.

 

[Ltk2015]

"Fluids flow from regions of higher pressure to regions of lower pressure, so in order to get a fluid to flow, there must be a pressure difference across a length of the pipe. It should seem intuitive that a greater pressure difference would push fluid faster, so D can be eliminated. Also, choice C implies that the speed can increase despite no increase in pressure difference past a threshold, which is not true. Choice C is also eliminated. A change in pressure from a negative value to a positive value implies that pressure gradient reversed directions. This would reverse the direction of the fluid flow, and is not what choice A shows. The relationship between velocity and pressure difference is linear and both should be equal to zero at the same time. Only choice B shows this.

Thus, B is correct.

WRONG.

You cannot generalize fluid flow by saying: "fluids flow from regions of lower pressure to regions of higher pressure" EVER.

 

[David513]

Hoping to clarify something about this...

Is it fair to say that you should consider the relationship between pressure and area as a function of whether a fluid is moving or standing still?

In other words, when a fluid is moving and you decrease the area, the pressure goes down because there is more uniform translational motion in a moving fluid, therefore fewer molecules of the fluid collide with each other.

Meanwhile, when a fluid is standing still and you decrease the area, the pressure goes up because there is more random translation motion in a resting fluid, therefore more molecules of the fluid collide with each other.

This seems to defy P=F/A and Bernoulli's equation but makes sense intuitively... help!