In this video I go over another derivatives application video and show how blood flow can be modeled by considering the rate of change of the blood velocity with distance from the center of the vein or artery as a derivative.

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Derivatives Application: Blood Flow

Blood Flow

When we consider the flow of blood through a blood vessel, such as a vein or artery, we can take the shape of the blood vessel to be a cylindrical tube with radius R and length L as illustrated below.

Because of friction at the walls of the tube, the velocity v of the blood is greatest along the central axis of the tube and decreases as the distance r from the axis increases until v become 0 at the wall.

The relationship between v and r is given by the law of laminar flow discovered by the French physician Jean-Louis-Marie Poiseuille in 1840. This states that:

where η (Greek letter eta) is the viscosity of the blood and P is the pressure difference between the ends of the tube. If P and L are constant, then v is a function of r with domain [0,R] or 0 ≤ r ≤ R.

The average rate of change of the velocity as we move from r = r₁ outward to r = r₂ is given by:

If we let Δr → 0, we obtain the velocity gradient, that is, the instantaneous rate of change of velocity with respect to r:

Example

For one of the smaller human arteries we can take η = 0.027, R = 0.008 cm, L = 2 cm, and P = 4000 dynes/cm². (1 dyne = 1 g·cm/s² = 10-5N)

How fast is the blood flowing at the centerline and at r = 0.002 cm? What is the velocity gradient at r = 0.002 cm?