A surgeon is faced with the problem of grafting an artery. She wishes to minimize the resistance to the resulting flow.
The resistance R to the laminar flow in a pipe L is given by Poiseuille's law: R = L/ r^4
This is where L and r are the length and radius respectively of the pipe. The graft must run from a main artery of radius r_1=0.5 cm to a point 5 cm from the main artery, using a connecting artery of radius r_2=0.46.
Coordinate the problem by assuming that the main artery runs along the x-axis, so the connecting artery needs to run to the point (10,5) and the graft occurs at (x,0).
Answer the following questions in terms of x.
- What is the length of an artery running from (0,0) to (x,0)?
- What is the resistance of the artery running from (0,0) to (x,0)?
- What is the length of an artery running from (x,0) to (10,5)?
- What is the resistance of the artery running from (x,0) to (10,5)?
- Write the total resistance from the origin to the point (10,5) as a function of x, assuming that the graft occurs at (x,0) and assuming that the resistance at the graft itself is negligible.
- What value of x minimizes this total resistance?
x_{opt} =