Laplace's Law and Aneurysms

What it shows:

Blow up a long cylindrical balloon and it inflates according to Laplace's law. Arteries, which also need to be flexible, are designed to fight against the kind of aneurysms seen in inflating rubber balloons.

balloonballoon

How it works:

Laplace's law for the gauge pressure inside a cylindrical membrane is given by ΔP = γ/r, where γ is the surface tension and r the radius of the cylinder. Note the inverse relation between pressure and radius.

When you blow up a balloon, only one part initially expands into an aneurysm. Continue inflating it and the aneurysm grows towards the balloon's end. (A bicycle inner tube behaves similarly.) The pressure inside the balloon is the same throughout, but when you feel the surface of a partially inflated balloon, the tension of the inflated part is considerably greater than the rest of the balloon. Granted, it's not surface tension, but the analogy to Laplace's relation between tension and radius is clear—for a fixed pressure, a larger radius cylindrical membrane will have a larger tension in the membrane wall and will consequently need to be "stronger" if it is not to burst. A bicycle tire withstands 100 psi with a relatively thin wall compared to automobile tires, which are inflated to only 35 psi. True, the automobile tire must be strong as it suffers all the insults of the road, but the walls of the tire also need to be strong to simply hold the pressure, given the large radius of the interior circle of the torus. Inflating an automobile tire to 100 psi would be disastrous, notwithstanding its thick walls.

Setting it up:

This is simple—you just need one of those long cylindrical balloons to blow up. Make sure one is available beforehand!

Comments:

In physiology, the gauge pressure is called the transmural pressure (the pressure difference across the walls of a blood vessel). A diseased weak spot on an artery wall will expand due to the transmural pressure and the radius of the wall increases at that spot. We now have a runaway situation: an increase in radius requires a stronger wall to contain the pressure, but it was a weaker wall that caused the problem in the first place. See I.P. Herman for a detailed discussion.1

1 Irving Herman, Physics of the Human Body, (Springer, 2007), chapter 8: Cardiovascular System.