Fluids - Flow Rate

[Sub Rosa]

I'm having some trouble rationalizing flow rate in physics...

So I have two equations: Pousielle's principle and the continuity equation. The first is telling me that doubling the radius will increase the flow rate by a factor of 16 (since v is proportional to r squared and A is proportional to r squared; multiplying them both will give me an increase of 16). On the other hand, I have the continuity equation which is saying a change in radius will not change the flow rate; it'll just make the fluid flow faster.

???

So which is it - does flow rate change with the radius or not?

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[BerkReviewTeach]

I'm having some trouble rationalizing flow rate in physics...

So I have two equations: Pousielle's principle and the continuity equation. The first is telling me that doubling the radius will increase the flow rate by a factor of 16 (since v is proportional to r squared and A is proportional to r squared; multiplying them both will give me an increase of 16). On the other hand, I have the continuity equation which is saying a change in radius will not change the flow rate; it'll just make the fluid flow faster.

???

So which is it - does flow rate change with the radius or not?

You can answer this question by asking yourself what radius you put into the Pousielle's equation. If the overall pipe gets narrower, then the R is reduced. But if it narrows gradually like the continuity example you are proposing, then we use the radius where the fluid entered the pipe, as that sets the Q for the fluid travelling the length of the pipe. If the pipe narrows as the fluid traverses it, the flow rate remains equal to what it was when it entered the pipe (assuming ideal behavior).

 

[Sub Rosa]

So you're saying the change in radius has to be continuous in order for the continuity equation to apply? But if we abruptly change the radius of the pipe, then we use Poiseuille's equation to find the new flow rate?

Also, does the continuity equation only apply if we're to assume the pressure is held constant throughout the flow?

And thank you for responding so quickly!

 

[milski]

Poiseuille's law tells you what the flow will be for any given pipe and pressure differential, the continuity equation tells you that whatever that flow is, it will stay the same along that pipe.

 

[milski]

So you're saying the change in radius has to be continuous in order for the continuity equation to apply? But if we abruptly change the radius of the pipe, then we use Poiseuille's equation to find the new flow rate?

Also, does the continuity equation only apply if we're to assume the pressure is held constant throughout the flow?

And thank you for responding so quickly!

If you change the radius, you'll change the flow - that comes from Poiseuille's equation. This new flow will be the same along the whole pipe, regardless of the radius - that's what the continuity equation tells you.

In other words, the continuity equation, in the form that you know it, is applicable only for a constant flow.

 

[brood910]

If you change the radius, you'll change the flow - that comes from Poiseuille's equation. This new flow will be the same along the whole pipe, regardless of the radius - that's what the continuity equation tells you.

In other words, the continuity equation, in the form that you know it, is applicable only for a constant flow.

Wrong.

Poiseuille's equation is used when you are talking about TWO different pipes with different radii.

Continuity equation is used when you are talking about ONE pipe that doesnt have a constant radius. Ex: human blood vessels.

 

[milski]

Wrong.

Poiseuille's equation is used when you are talking about TWO different pipes with different radii.

Continuity equation is used when you are talking about ONE pipe that doesnt have a constant radius. Ex: human blood vessels.

You mean a single pipe with different pressures at both ends and same radius.

There is nothing stopping you from treating a single pipe with non-constant radius as a connection of multiple pipes with the same radius, in the worst case you'll end up with some sort of integral.

The point about the continuity equation is that the flow in it is a given, you cannot use the continuity equation to determine how the flow changes when the radius changes.

 

[brood910]

You mean a single pipe with different pressures at both ends and same radius.

There is nothing stopping you from treating a single pipe with non-constant radius as a connection of multiple pipes with the same radius, in the worst case you'll end up with some sort of integral.

The point about the continuity equation is that the flow in it is a given, you cannot use the continuity equation to determine how the flow changes when the radius changes.

Wrong again.

Poiseuille's equation is Q = (P1-P2)x pi x r^2/8nL.
So r DOES affect Q. This works only for laminar flow and ONLY if the pipe has ONE constant radius value.


Continuity equation works only if Q is constant. In other words, you have the same flow rate regardless of changes in radius (area). This means Q = A1V1 = A2V2..

 

[milski]

Wrong again.

Poiseuille's equation is Q = (P1-P2)x pi x r^2/8nL.
So r DOES affect Q. This works only for laminar flow and ONLY if the pipe has ONE constant radius value.


Continuity equation works only if Q is constant. In other words, you have the same flow rate regardless of changes in radius (area). This means Q = A1V1 = A2V2..

That's more or less what I am saying. In the case of the non-constant radius, the flow is determined in a more complicated way than the Poiseuille's above but either way, you need to be talking about a constant Q before you can apply the continuity equation, at least in the laminar case.

Btw, you need it's r^4.