Poiseuille's Law Questions?

Question

This is one of my physics homework questions. I know that I need to use Poiseuille's Law, but I have no idea how to go about it.
Any suggestions?

1) Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what factor must the radius of an artery be increased in order to increase blood flow by a factor of 19?


2)When physicians diagnose arterial blockages, they quote the reduction in flow rate. If the flow rate in an artery has been reduced to 70% of its normal value due to plaque formation, and the average pressure difference has increased by 10%, by what factor has the plaque reduced the radius of the artery?

Thank you very much!

Answer 1

Poiseulle's law states that the flow rate Q is dependant upon fluid viscosity n, pipe length L and the pressure difference between the ends P as in

Q=pi R^4 P/(8 n L)

Q- Flow rate
R – Internal radius of the pipe
n – Viscosity of the fluid
L – Length of the pipe
P – Pressure difference

Based on problem statement
Q2/Q1=19= [pi R2^4 P/(8 n L)]/[ pi R1^4 P/(8 n L)]
19=(R2/R1)^4

Then R2/R1=(19)^(1/4)= 2.09 or R2=2.09R1 we have to increase the size of the artery by more than a factor of 2.

Use the same ratio with some modifications
P2=.9P1
Q2=.7Q1

Then

Q2/Q1=.7= [pi R2^4 P2/(8 n L)]/[ pi R1^4 P2/(8 n L)]
.7 = (R2/R1)^4 (P2/P1)
R2/R1= (.7 x (P2/P1))^(1/4)
R2/R1=(.7 x .9)^(1/4
R2/R1= 0.891
Or
R2=0.891 R1