# Bernoulli’s Equation

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

## Abstarct

The Bernoulli equation can be viewed as an energy law. It relates blood pressure (P) to flow velocity (v). Bernoulli’s law says that if we follow a blood particle along its path (dashed line in left Figure in the box) the following sum remains constant:
$$P+{\scriptscriptstyle \raisebox{1ex}{1}\!\left/ \!\raisebox{-1ex}{2}\right.}\cdot r\cdot {u}^{2}+r\cdot g\cdot z=\text {constant}$$
where ρ is blood density, g acceleration of gravity, and z elevation with respect to a horizontal reference surface (i.e., ground level or heart level). The equation of Bernoulli says that as a fluid particle flows, the sum of the hydrostatic pressure, P, potential energy, ρ · g · z, and the dynamic pressure or kinetic energy, ½ · ρ · v 2, remains constant. One can easily derive Bernoulli’s equation from Newton’s law: Pressure forces + gravitational forces = mass × acceleration.

Velocimetry

### References

1. 1.
Gorlin R, Gorlin SG. Hydraulic formula for calculations of the area of the stenotic mitral valve value, orthocardiac values and central circulating shunts. Am Heart J 1951;41:1–29.
2. 2.
Wilkinson JL. Haemodynamic calculations in the catheter laboratory. Heart 2001;85:113–120.
3. 3.
Burton AC. Physiology and Biophysics of the Circulation. 1972, Chicago, Year Book Medical Publ., 2nd edn.Google Scholar

## 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