Poiseuille's Equation

For laminar, non-pulsatile fluid flow through a uniform straight pipe, the flow rate (volume per unit time) is given by Poiseuille's Equation:

F = DP p r 4 / 8 h l.

Obviously, the use of Poiseuille's Equation on the human circulatory system is highly suspect. While the flow is essentially laminar outside of the capillaries, it is definitely pulsatile throughout the arterial subsystem. In addition, the equation is based on the parabolic velocity gradient discussed in the last section. But since pressure waves in arterial walls propagate more quickly than those in blood, the velocity profile is more uniform than parabolic. Beyond that, Poiseuille's Equation assumes a constant viscosity, whereas the viscosity of blood actually changes with velocity, since blood is not a uniform fluid. In fact, the viscosity is much lower in the capillaries than in the rest of the system, since the red blood cells line up in single file to pass through them. On top of everything else, the blood vessels are not straight, uniform pipes!

All of these reservations notwithstanding, we will apply Poiseuille's Equation to the cardiovascular system to compute order of magnitude quantities which we otherwise could not compute at all. For instance, the problem set below asks that you compute the number of capillaries in your body. The number will come out to be in the billions. The alternative to this method is dissection and counting, which is not only unhealthy for the patient, but in fact would take on the order of 500 years, assuming that you could count one capillary a second for 40 hours a week!

As an example of its utility, consider an arterial branch. Assume that a major artery of radius 0.75 cm branches off into two smaller arteries of radius 0.5 cm. Since all of the blood which flows from the larger artery into the junction of the arteries must come out into the two smaller ones, the flow rate of the larger must be twice that of the smaller. Hence

DPlarger p rlarger 4 / 8 h l larger= 2 DPsmaller p rsmaller 4 / 8 h l smaller,

or

DPsmaller / l smaller = (1/2) (rlarger / rsmaller) 4 (DPlarger / l larger),

or

DPsmaller / l smaller = 2.53 DPlarger / l larger.

That is, the pressure drop per unit length due to viscous dissipation in the smaller arteries is over two and a half times that of the larger artery, due to the change in radius. This indicates that we should expect the average length of vessels to decrease rapidly as their radii decrease: otherwise the total pressure drop would leave insufficient pressure for the return trip to the heart!

Since flow is equal to velocity times cross sectional area,

vsmaller p rsmaller 2 = (1/2) vlargerp rlarger 2,

or

vsmaller / vlarger = (1/2) (rlarger / rsmaller) 2 = 1.13,

and we see that the velocity of blood in a smaller vessel is larger than that in a larger vessel with the same flow. Of course, these results apply to any fluid flowing through any "pipes", including for instance, air through bronchi and bronchioles in the lungs, providing that we ignore variations in flow rate over time and variations in size and shape of the pipes. So Poiseuille's Equation allows us to learn both qualitative and quantitative order of magnitude information about the effect of vessel size on flow.

Problems

The next section is about pulsatile flow.



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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