Law of Poiseuille

Abstarct

With laminar and steady flow through a uniform tube of radius r _i , the velocity profile over the cross-section is a parabola. The formula that describes the velocity (v) as a function of the radius, r is:

$$ {v}_{r}=\frac{\Delta P \cdot \text ({r}_{i}-{r}^{2})}{4 \cdot \eta \cdot \text l}={v}_{\mathrm{max}}(1-{r}^{2}/{r}_{i})$$

ΔP is the pressure drop over the tube of length (l), and η is blood viscosity. At the axis (r = 0), velocity is maximal, v _max , with v _max = ΔPr _i ²/4ηl, while at the wall (r = r _i ) the velocity is assumed to be zero. Mean velocity is:

$$ {v}_{mean}=\Delta P \cdot \text {r}_{i}^{2}/8\text h \cdot \text l=v_{\mathrm{max}}\text /2=Q/\pi {r}_{i}^{2}$$

and is found at r ≈ 0.7 r _i .