Mathematics and Biology

Velocity as Surface in 3-D Physical Conditions

Math Awareness Month 1999

Poiseuille's Law: Velocity Profile

As a first step toward understanding how much blood flows through the arteriole, we will examine how fast the blood (or other fluid) is moving at each point within the vessel. Because the flow is laminar, we can treat the fluid as though made up of thin cylindrical sheets. Using Newton's second law of motion (F=ma) and the precise definition of viscosity, one can use the theory of calculus to find the law that governs the speed of the fluid at each point in the tube. More specifically, we measure the distance of the point from the center of the tube to be at a specific radius (r), at which point the speed is given by the formula

The graph of this formula is easily found to be a parabola. Let us make a few initial observations. First, notice that the blood is not moving when r=a. This means that no slipping is allowed between the blood and the vessel's wall. Secondly, notice that the vertex occurs when r=0. The fastest blood is at the center of the arteriole.

More About Velocity
You can explore the following pages to understand better the velocity and its relationship to the flow:


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