Fluid Dynamics - Continuity vs Poiseuille

[adsf123]

Am I right in assuming Poiseuille's law is for real fluids whereas continuity only applies to ideal fluids? These two equations seem to contradict each other and previous forum posts don't really answer the question of why they contradict.

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[loveanswers]

Am I right in assuming Poiseuille's law is for real fluids whereas continuity only applies to ideal fluids? These two equations seem to contradict each other and previous forum posts don't really answer the question of why they contradict.

Wondering the same exact thing. One law says that Q=AV and the other says Q varies with respect to V^4

 

[Platonic]

Wondering the same exact thing. One law says that Q=AV and the other says Q varies with respect to V^4

they don't contradict. Continuity says Q=AV, and the other says Q is proportional to R^4. Both are correct, and they both apply to ideal fluids. Not that it matters much whether it's ideal or real. Real fluids behave very similarly to ideal fluids, so you wouldn't get radically different equations to describe the two. For example, the van der walls equation describing real gases is very similar to the ideal gas law. And if you set certain constants in Van der Walls equal to zero, the two equations are identical.

 

[adsf123]

How can they not contradict. If a pipe gets smaller, according to continuity the velocity increases to offset the smaller area. However, according to Poiseuille, the flow rate will decrease by r^4 so continuity cannot be correct?

 

[SpaceBuff12]

Poiseuille's equation only holds for fully developed flow. There is a point far from the entrance of the tube at which the radial velocity distribution is identical for all points farther downstream; this is Poiseuille's flow and the mean velocity is given by:

mean velocity: -[ (r^2) / (8*mu) ] *(dp/dz)

Note: p is pressure, z is the length along the tube.

You cannot apply Poiseuille's equation at the entrance and exits because there is a net force on the fluid until the flow becomes fully developed.

 

[Platonic]

How can they not contradict. If a pipe gets smaller, according to continuity the velocity increases to offset the smaller area. However, according to Poiseuille, the flow rate will decrease by r^4 so continuity cannot be correct?

To conclude that Q decreases by a factor of R^4, you must assume that everything else in the equation (other than Q and R) is constant. This isn't true, though. In fact, pressure drop delta P changes to compensate for the change in R.

The continuity equation is based on the assumption that flow rate Q is a constant. This assumption is based on the incompressibility of fluids. Since fluid can't compress and collect within a segment of the pipe, the rate of flow into the segment must equal the rate of flow out of the segment. This assumption is ALWAYS true, including in the cases where Poiseuille applies.

When we consider a case where the pipe gets smaller, we must start with the assumption that Q doesn't change(otherwise, we've got a compressibility issue). Looking at Poiseuille equation, that means delta P should increase to offset the decrease in thickness. So the pressure gradient over a skinny section of pipe will be greater than the pressure gradient over a section of thicker pipe. But Q will not change.

You cannot apply continuity at the entrance and exits because there is a net force on the fluid until the flow becomes fully developed.

I'm assuming this is a typo and you meant to say Poiseuille? Continuity has nothing to do with force, it's based on incompressibility of the fluid. The equation Q=AV doesn't apply to real flow though, since the speed of flow is different based on proximity to the surface of the pipe. But I believe the equation is accurate if you use average velocity.

 

[SpaceBuff12]

I'm assuming this is a typo and you meant to say Poiseuille?

Good catch. I corrected the statement.

 

[adsf123]

" To conclude that Q decreases by a factor of R^4, you must assume that everything else in the equation (other than Q and R) is constant. This isn't true, though. In fact, pressure drop delta P changes to compensate for the change in R."

This is what I thought happened until I read the heart & lungs chapter in BR. They claim that "notice that the flow rate is proportional to R^4. This tells us that the rate of blood flow is extremely dependent on the radius of the vessel. If the radius of the vessel were reduced by a factor of 2, then the rate of blood flow will be reduced by a factor of 16."

 

[wjs010]

" To conclude that Q decreases by a factor of R^4, you must assume that everything else in the equation (other than Q and R) is constant. This isn't true, though. In fact, pressure drop delta P changes to compensate for the change in R."

This is what I thought happened until I read the heart & lungs chapter in BR. They claim that "notice that the flow rate is proportional to R^4. This tells us that the rate of blood flow is extremely dependent on the radius of the vessel. If the radius of the vessel were reduced by a factor of 2, then the rate of blood flow will be reduced by a factor of 16."

In blood, it is not ideal fluid. So, q is constand. AV1=AV2

 

[Platonic]

" To conclude that Q decreases by a factor of R^4, you must assume that everything else in the equation (other than Q and R) is constant. This isn't true, though. In fact, pressure drop delta P changes to compensate for the change in R."

This is what I thought happened until I read the heart & lungs chapter in BR. They claim that "notice that the flow rate is proportional to R^4. This tells us that the rate of blood flow is extremely dependent on the radius of the vessel. If the radius of the vessel were reduced by a factor of 2, then the rate of blood flow will be reduced by a factor of 16."

I think I see your issue. I thought you were saying that for a particular tube, Poiseuille predicts that flow rate Q was different for different segments depending on their thickness R. But you're saying that Poiseuille predicts and increased flow rate Q with the increasing R, and that the continuity equation predicts a decreasing velocity with increasing R, and that this appears to be a contradiction.

But the continuity equation does not say that flow rate will decrease if you increase thickness, it simply says that flow rate is constant throughout the tube, and that the velocity of flow is slower in thicker segments of the tube. This is not inconsistent with Poiseuille. The Poiseuille equation can be used to calculate Q, and the continuity equation can be used to calculate the velocity of flow at different segments of the tube based on the value of Q.


"How can they not contradict. If a pipe gets smaller, according to continuity the velocity increases to offset the smaller area. However, according to Poiseuille, the flow rate will decrease by r^4 so continuity cannot be correct? "

Velocity and flow rate are different, so this isn't really a contradiction, but that's besides the point. The key point is that the continuity equation isn't used to calculate the flow rate for different tubes of different sizes. It's used to calculate the speed of flow in one tube based on the flow rate of the tube and the thickness of a segment.

 

[adsf123]

I think I see your issue. I thought you were saying that for a particular tube, Poiseuille predicts that flow rate Q was different for different segments depending on their thickness R. But you're saying that Poiseuille predicts and increased flow rate Q with the increasing R, and that the continuity equation predicts a decreasing velocity with increasing R, and that this appears to be a contradiction.

But the continuity equation does not say that flow rate will decrease if you increase thickness, it simply says that flow rate is constant throughout the tube, and that the velocity of flow is slower in thicker segments of the tube. This is not inconsistent with Poiseuille. The Poiseuille equation can be used to calculate Q, and the continuity equation can be used to calculate the velocity of flow at different segments of the tube based on the value of Q.


"How can they not contradict. If a pipe gets smaller, according to continuity the velocity increases to offset the smaller area. However, according to Poiseuille, the flow rate will decrease by r^4 so continuity cannot be correct? "

Velocity and flow rate are different, so this isn't really a contradiction, but that's besides the point. The key point is that the continuity equation isn't used to calculate the flow rate for different tubes of different sizes. It's used to calculate the speed of flow in one tube based on the flow rate of the tube and the thickness of a segment.

Aw, this makes sense now. Thanks.