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Problem 18 Medium Difficulty

The velocity $ v $ of blood that flows in a blood vessel with radius $ R $ and length $ l $ at a distance $ r $ from the central axis is
$$ v(r) = \frac{P}{4\eta l} (R^2 - r^2) $$
where $ P $ is the pressure difference between the ends of the vessel and $ \eta $ is the viscosity of the blood (see Example 3.7.7). Find the average velocity (with respect to $ r $) over the interval $ 0 \le r \le R $. Compare the average velocity with the maximum velocity.

Answer

The average velocity is $\frac{P R^{2}}{6 \eta l}$ which is $\frac{2}{3}$ of the maximum velocity

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Video Transcript

we know that we can use the average value formula and plug in one over R minus zero from zero to our give our d r, which is one over r from zero r. We know now that we end up with p r squared and then we know that we have the degree to therefore the derivative ve promised tea, which is acceleration equals zero indicates when the maximum velocity is reached. They're Farina. The maximum velocity is reached at p r squared for, you know. Therefore, we know the average velocity is this which is 2/3 of the maximum velocity.