# Law of Poiseuille

• Nicolaas Westerhof
• Nikolaos Stergiopulos
• Mark I. M. Noble
Chapter

## 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 .

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## Authors and Affiliations

• Nicolaas Westerhof
• 1
• Nikolaos Stergiopulos
• 2
• Mark I. M. Noble
• 3
1. 1.Departments of Physiology and Pulmonology ICaR-VUVU University Medical CenterAmsterdamthe Netherlands
2. 2.Laboratory of Hemodynamics and Cardiovascular TechnologySwiss Federal Institute of TechnologyLausanneSwitzerland
3. 3.Cardiovascular MedicineAberdeen UniversityAberdeenScotland