Getting Started With A Blood Vessel Problem

[Tom McCurdy]

i was hoping for someone to help me get started with this problem
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A blood vessel is cylindrical with radius R. and length l. the blood near the boundary moves slowly; blood at the cetner moves the fastest. The velocity, v, of the blood at a distance r from the center of the artery is given by

$$v = \frac{P}{4nl}(R^2-r^2)$$

where P is the pressure difference between the ends of the blood vessel and n is the viscosity of the blood.

a.) Find the reate at which the blood is flowign down the blood vessel. (Give your answer as a volume per unit time.)

b.) Show that your result agrees with Poiseuille's Law which says that the rate at which blood is flowing down the blood vessel is proportional to the radius of the blood vessel to the fourth power.

I am not really sure how to begin this problem.. seeing that the power needs to be rasied from 2 to 4 suggested to me that perhaps two integrals were needed, although how to use two integrals on this problem didn't make any sense to me.

 

[vaishakh]

B has some mistakes I would say.
Rate = v*pi^2*R^2 = k*(R^4 - r^2*R^2)
Rate is directly proportional to R^2(R^2 - r^2).

 

[Tom McCurdy]

I'm pretty sure ... other than perhaps spelling (lol) that both a and b don't have mistakes in them.

 

[quantumdude]

Sketch the velocity profile (graph of $r$ vs. $v(r)$). Remembering that the tube has a circular cross section, find an expression for the volume of a cylindrical shell of radius $r$, length $l$ and thickness $dr$. Then figure out how long it would take for the shell to leave the artery. That should allow you to write down a differential volumetric flow rate, which you can integrate.

 

[Tom McCurdy]

still confused

Alright so volume would be equal to

$$volume=2 \pi l\int_0^R{r*dr}$$

then would you take the deravative of the velocity to get position to solve for time?