Bernoulli's Principle and Poiseuille's Law

[Mission Medical]

According to Hyperphysics, Poiseuille's Law says:
Volume Flowrate = [π(pressure difference)(radius)^4] / [8(viscosity)(length)]

...and Bernoulli's Principle says:
"In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy."

What's confusing me is that from Poiseuille's Law the formula tells us that an increase in radius would cause higher volume flowrate, but Bernoulli's principle is saying that in regions of constriction (decreased radius) there's higher velocity. Don't these ideas contradict?

---

Members don't see this ad.

 

[Gauss44]

According to Hyperphysics, Poiseuille's Law says:
Volume Flowrate = [π(pressure difference)(radius)^4] / [8(viscosity)(length)]

...and Bernoulli's Principle says:
"In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy."

What's confusing me is that from Poiseuille's Law the formula tells us that an increase in radius would cause higher volume flowrate, but Bernoulli's principle is saying that in regions of constriction (decreased radius) there's higher velocity. Don't these ideas contradict?

I don't think that they contradict.

If the same pipe (same system) has wider areas and narrower areas, then the velocity would be the greatest in the narrowest area.

I think: In different pipes, a larger radius means a greater rate of flow (not necessarily a greater velocity): If pipe A is wider than pipe B, the pipe A will allow for a greater rate of flow than pipe B, assuming all other factors in the system are the same.

 

[Mission Medical]

I don't think that they contradict.

If the same pipe (same system) has wider areas and narrower areas, then the velocity would be the greatest in the narrowest area.

I think: In different pipes, a larger radius means a greater rate of flow (not necessarily a greater velocity): If pipe A is wider than pipe B, the pipe A will allow for a greater rate of flow than pipe B, assuming all other factors in the system are the same.

How does rate of flow differ from velocity? I was thinking they're the same since both are rates and the velocity is (I think) referring to the rate at which the fluid is flowing.

 

[Gauss44]

How does rate of flow differ from velocity? I was thinking they're the same since both are rates and the velocity is (I think) referring to the rate at which the fluid is flowing.

Q=vA

rate of flow = velocity times area

 

[Mission Medical]

Q=vA

rate of flow = velocity times area

That makes sense. So does this mean that in a region of constriction (decreased radius), the velocity will increase, but the volume flow rate will decrease? And the volume flow rate would be decreasing due to the decrease in radius causing a significant decrease in area which offsets the increase in velocity?

 

[HesitantManatee]

I don't think that they contradict.

If the same pipe (same system) has wider areas and narrower areas, then the velocity would be the greatest in the narrowest area.

I think: In different pipes, a larger radius means a greater rate of flow (not necessarily a greater velocity): If pipe A is wider than pipe B, the pipe A will allow for a greater rate of flow than pipe B, assuming all other factors in the system are the same.

+1

OP you should also know that Bernoulli's Principle applied only to ideal fluids whiles Poiseuille's equations only applies to non-ideal fluids so I don't think that you will ever be asked to make a comparison like this.

 

[JohnEC]

That makes sense. So does this mean that in a region of constriction (decreased radius), the velocity will increase, but the volume flow rate will decrease? And the volume flow rate would be decreasing due to the decrease in radius causing a significant decrease in area which offsets the increase in velocity?

If you are talking about a SINGLE pipe, the volume flow rate must always stay the same, it will not increase or decrease. This is because if it was not the same everywhere, you would essentially have empty spaces in the pipe, where there is no fluid. This is what is meant by the continuity equation, A1V1=A2V2, and that is the reason that when you have a constriction (A2 is smaller than A1) the velocity (not volume flow rate) must increase (V2 is larger than V1).

Poiseuille's law is dealing with different pipes, where the volume flow rate need not be the same. Its saying that if you have two completely separate pipes with different radii, then the one with the larger radius will have the increased volume flow rate, if all else is the same.

Does that help at all?

 

[Mission Medical]

+1

OP you should also know that Bernoulli's Principle applied only to ideal fluids whiles Poiseuille's equations only applies to non-ideal fluids so I don't think that you will ever be asked to make a comparison like this.

What is the difference between ideal and non-ideal fluids, and how do you tell which you're dealing with?

If you are talking about a SINGLE pipe, the volume flow rate must always stay the same, it will not increase or decrease. This is because if it was not the same everywhere, you would essentially have empty spaces in the pipe, where there is no fluid. This is what is meant by the continuity equation, A1V1=A2V2, and that is the reason that when you have a constriction (A2 is smaller than A1) the velocity (not volume flow rate) must increase (V2 is larger than V1).

Poiseuille's law is dealing with different pipes, where the volume flow rate need not be the same. Its saying that if you have two completely separate pipes with different radii, then the one with the larger radius will have the increased volume flow rate, if all else is the same.

Does that help at all?

Yes, this makes sense, but when you say different pipes, do you literally mean different pipes that are connected? Or could it be just one pipe that has a different cross-sectional area at certain regions? Also, why would empty spaces result if the volume flowrate stays the same?

Sorry. My Physics I class at college never covered fluid mechanics at all for some reason.

 

[JohnEC]

Yes, this makes sense, but when you say different pipes, do you literally mean different pipes that are connected? Or could it be just one pipe that has a different cross-sectional area at certain regions?

Poiseuille’s Law: You have two different pipes that aren't connected, with two different radii. If the same type of fluid (same viscosity) flows through both pipes, and the same pressure difference exists across the ends of the pipe, then the pipe with the larger radius will carry a greater volume of fluid in a given time. The larger the area, the greater the VOLUME FLOW RATE.

Bernoulli’s Principle: You have a single pipe, that may have different cross-sectional areas at different points. The same volume flow rate exists in all portions of this pipe. In order to deliver this same volume, fluid in a narrower portion of the pipe must travel faster than fluid in a wider portion. The larger the area, the smaller the FLUID VELOCITY.

Also, why would empty spaces result if the volume flowrate stays the same?

You would have empty space if the volume flow rate wasn’t the same throughout the pipe. Image that you have a section of pipe. No matter what the pipe looks like (it can be changing raddi throughout the section), if you put say 1 liter of fluid into the pipe over the course of 1 second, then 1 liter of the fluid MUST also leave the pipe in that same 1 second. It’s impossible for less than 1 liter to leave the pipe because that would mean you are adding more total fluid to the pipe, which isn’t possible because the pipe is a set volume. You also can’t have more than 1 liter leave the pipe because then less fluid is entering the pipe than is leaving the pipe, which means the pipe is no longer full (that’s what I meant by empty space).

Sorry if this is confusing! let me know if something doesn't make sense!

 

[Mission Medical]

Poiseuille’s Law: You have two different pipes that aren't connected, with two different radii. If the same type of fluid (same viscosity) flows through both pipes, and the same pressure difference exists across the ends of the pipe, then the pipe with the larger radius will carry a greater volume of fluid in a given time. The larger the area, the greater the VOLUME FLOW RATE.

Bernoulli’s Principle: You have a single pipe, that may have different cross-sectional areas at different points. The same volume flow rate exists in all portions of this pipe. In order to deliver this same volume, fluid in a narrower portion of the pipe must travel faster than fluid in a wider portion. The larger the area, the smaller the FLUID VELOCITY.




You would have empty space if the volume flow rate wasn’t the same throughout the pipe. Image that you have a section of pipe. No matter what the pipe looks like (it can be changing raddi throughout the section), if you put say 1 liter of fluid into the pipe over the course of 1 second, then 1 liter of the fluid MUST also leave the pipe in that same 1 second. It’s impossible for less than 1 liter to leave the pipe because that would mean you are adding more total fluid to the pipe, which isn’t possible because the pipe is a set volume. You also can’t have more than 1 liter leave the pipe because then less fluid is entering the pipe than is leaving the pipe, which means the pipe is no longer full (that’s what I meant by empty space).

Sorry if this is confusing! let me know if something doesn't make sense!

Thanks. This has all been very helpful. I'm just a little bit confused about that last part of your explanation for my second question. Are you basically saying that since more fluid would be leaving the pipe than entering in that scenario, the pipe is no longer full because there's a net decrease in the amount of fluid in the pipe? As opposed to equal amounts of fluid flowing in an out, which would lead to a situation in which there's always fluid in the pipe.

 

[JohnEC]

Thanks. This has all been very helpful. I'm just a little bit confused about that last part of your explanation for my second question. Are you basically saying that since more fluid would be leaving the pipe than entering in that scenario, the pipe is no longer full because there's a net decrease in the amount of fluid in the pipe? As opposed to equal amounts of fluid flowing in an out, which would lead to a situation in which there's always fluid in the pipe.

Yes thats right. But with the types of problems done in intro physics you are assuming steady flow, so the pipe isn't going to be partially empty. You are assuming that the pipe remains full at all points.

 

[Mission Medical]

Yes thats right. But with the types of problems done in intro physics you are assuming steady flow, so the pipe isn't going to be partially empty. You are assuming that the pipe remains full at all points.

Thanks!