Question

When blood flows along a blood vessel, the flux F (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel:

F = kR⁴

(This is known as Poiseuille’s Law; we will show why it is true in Section 8.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is inflated inside the artery in order to widen it and restore normal blood flow. Show that the relative change in Fis about four times the relative change in R. How will a 5% increase in the radius affect the flow of blood?

Short answer

A 5% increase in radius will reflect to 20% increase in blood flow.

Step-by-step solution

Describe the given information

It is given as \(F = k{R^4}\). The differential flux is given as \(dF = 4k{R^2}dR\).

Find the relative error influx

The relative error in flux can be obtained as \(\frac{{\Delta F}}{F} \approx \frac{{dF}}{F}\). On Plugging the expression of \(dF\), \(F\)and simplifying, we get,

\(\begin{aligned}{c}\frac{{dF}}{F} &= \frac{{4k{R^2}dR}}{{k{R^4}}}\\ &= 4\left( {\frac{{dR}}{R}} \right)\end{aligned}\)

Thus, the relative error in current F is 4 times relative error in radius R.Therefore, 5% increase in radius will reflect to 20% increase in blood flow.