Mathematics and Biology

Pressure Dependence Velocity Profile

Math Awareness Month 1999

Poiseuille's Law: A Derivation using the Velocity Profile

A full understanding of the velocity profile requires an understanding of calculus. The law for the velocity can be derived as a solution to a differential equation. One way to do this is to use an equation known as the Navier-Stokes equation, simplified to handle our case. An alternative method is to derive a differential equation using Newton's second law. If you have some background in calculus, you may want to look at these outside sources:

A consequence of the velocity profile law is that the average velocity of the blood in the blood vessel is exactly half of the maximum (or central) velocity:

This means that the we get the same amount of blood flowing through a blood vessel using the actual velocity profile as though we had blood all flowing at the same average velocity. But for this imaginary blood vessel with everything moving at the same speed, it is easy to calculate the blood flow. The rate of flow is the cross-sectional area times the average velocity:

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D. Brian Walton is supported by a National Science Foundation Graduate Research Fellowship.