Question

The velocity v of blood that flows in a blood vessel with radius r and length ℓ at a distance r from the central axis is v(r) = p 4ηℓ (r2 − r2) where p is the pressure difference between the ends of the vessel and η is the viscosity of the blood. find the average velocity vave (with respect to r) over the interval 0 ≤ r ≤ r. (use eta for η. use capital p for pressure.)

Answer 1

$v(r)=4P\eta\ell(R^2-r^2)$

(assuming $R$ is the radius of the vessel, and that $0\le r\le R$)

The average velocity of the blood is given by

$\displaystyle\frac1{R-0}\int_{r=0}^{r=R}v(r)\,\mathrm dr$
$=\displaystyle\frac{4P\eta\ell}R\int_{r=0}^{r=R}(R^2-r^2)\,\mathrm dr$
$=\dfrac{4P\eta\ell}R\left(\dfrac23R^3\right)$
$=\dfrac{8P\eta\ell}3R^2$