Is there a contradiction between the continuity equation and Poiseuilles Law? The continuity equati

Blaine Stein 2022-05-15 Answered
Is there a contradiction between the continuity equation and Poiseuilles Law?
The continuity equation states that flow rate should be conserved in different areas of a pipe:
Q = v 1 A 1 = v 2 A 2 = v π r 2
We can see from this equation that velocity and pipe radius are inversely proportional. If radius is doubled, velocity of flow is quartered.
Another way I was taught to describe flow rate is through Poiseuilles Law:
Q = π r 4 Δ P 8 η L
So if I were to plug in the continuity equations definition of flow rate into Poiseuilles Law:
v A = v π r 2 = π r 4 Δ P 8 η L
Therefore:
v = r 2 Δ P 8 η L
Now in this case, the velocity is proportional to the radius of the pipe. If the radius is doubled, then velocity is qaudrupled.
What am I misunderstanding here? I would prefer a conceptual explanation because I feel that these equations are probably used with different assumptions/in different contexts.
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Answers (2)

hodowlanyb1rq2
Answered 2022-05-16 Author has 12 answers
When you write down Q = v A, it's implicit that the velocity profile is uniform over the cross-section A (and purely perpendicular to it).
In general,
Q = v d A
This no longer implies that v 1 R 2
If we assume v has purely radial dependence and is aligned with d A as in Poiseuille flow, then we have:
Q = 2 π 0 R v ( r ) r d r
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Daphne Haney
Answered 2022-05-17 Author has 4 answers
For a fixed volumetric throughput rate Q, Δ P decreases as r 4 , so v decreases as r 2 , exactly what you would expect from the continuity equation.
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