Different Versions of Poiseuille's Law?

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Dec 9, 2017
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In my studying I came across a different version of Poiseuille's Law than I'm used to seeing.

While I am used to seeing it in this form:

I just saw a problem easily solved by using the form below, which is showing a "resistance to flow" value 'R' rather than a flow rate 'Q':

How can one of them include a flow rate yet the other includes resistance, while appearing to be the same equation? Would anybody be able to break this down and explain it to me?

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The two forms are the same because the flow rate is equal to the pressure difference divided by the resistance to flow.

Q = flow rate
ΔP = pressure difference
R = resistance to flow

Q = ΔP / R (basic flow equation)

Poiseuille's Law expresses the resistance in terms of the vessel/pipe and flow parameters.

R = (8*η*L) / (π*r^4), where:

η = fluid viscosity (viscosity is a measure expressing resistance to stress)
L = vessel/pipe length
r = vessel/pipe radius

When you combine the two equations together:

R = (8*η*L) / (π*r^4) -->
1/R = (π*r^4) / (8*η*L) -->

ΔP/R = ΔP * (π*r^4) / (8*η*L) = Q, which is the form you're familiar

Here's a useful resource that ties this to medicine: Physiology Tutorial - Blood Flow