Question

The velocity v of blood that flows in a blood vessel withradius R and length l at a distance rfrom the central axis is v(r) = (P/4ηl)(R2 - r2) where P is pressure difference between the ends of the vessel andη is the viscosity of the blood. Find the average velocity(with respect to r) over the interval 0 ≤ r ≤ R.Compare the average velocity with the maximum velocity.

Answer 1

To find the average velocity over the interval 0 ≤ r ≤ R, integrate the velocity function and divide by the length of the interval. The maximum velocity can be found by substituting r = 0 into the velocity function.

Step-by-step explanation:

The average velocity over the interval 0 ≤ r ≤ R can be found by taking the average of the velocity function v(r) = (P/4ηl)(R2 - r2) with respect to r. To find the average velocity, we need to integrate v(r) over the interval 0 ≤ r ≤ R and divide by the length of the interval (R - 0). The formula for average velocity is:

Average velocity = (1/(R - 0)) * integral[(P/4ηl)(R2 - r2) dr] from r = 0 to r = R

To compare the average velocity with the maximum velocity, we can find the maximum velocity by substituting r = 0 into the velocity function v(r) = (P/4ηl)(R2 - r2).