# Law of Urination

Last week I lectured on how bone size scales with the size of animals, and how we can use simple physics to understand this (notes are here if you’re interested). So, this New Scientist article is extremely well timed. At a conference on fluid dynamics next month, Patricia Yang and coworkers from Georgia Tech will present a (kind of) Universal Law of Urination. They were intrigued by the observation that although bladders vary enormously in volume among mammals, the time the mammals take to relieve themselves varies very little. I confess that I had never thought about this, but now that the subject is brought up, I am curious.

I don’t know what their theory is, but with a single handy equation from the physics of flowing liquids, one can at least get a rough idea of what is going on. This equation is the very handy equation for what is called Poiseuille or Hagen-Poiseulle flow. This equation concerns liquid flowing down a pipe of radius R and length L. Roughly speaking, the equation is:

Q ~ (ΔP/ηL)R4

where:

Q = volume of liquid flowing down a pipe per second

ΔP = pressure drop along pipe, i.e., pressure at one end minus pressure at the other end

L = length of pipe

η = viscosity, i.e., how thick the liquid is

Now, if we assume that this is OK for urethra then the time for an animal to relieve itself is V/Q, for V the volume of the bladder.

Now for the scaling bit. We assume that as the size of an animal, call it s, increases then the volume of the bladder scales with the volume of animal, i.e., V ~ s3. We can also guess that the urethra length and radius also scale with s, i.e., L ~ s and R ~ s. All three assumptions are just what you get if as animals get bigger then their bladder, urethra just scale up in proportion.

Putting all this together

time to empty bladder ~ s3 / [ (ΔP/s)s3] ~ 1/ (ΔP)

All the powers of s have cancelled! The remaining dependence is on the pressure difference. Now the forces in an elephant are of course much larger in a elephant than in a mouse, but pressure is a force per unit area which should be more or less a function of the properties of muscles – which is similar in all animals. So ΔP is probably not very different between mice and elephants.

This is a very simple minded analysis, but it does show that you should expect the time to empty a bladder to vary much more slowly than you would expect. Mice’s bladders are around 0.00001 litres, elephants a million times larger at around 10 litres, but it certainly doesn’t take a elephant a million times longer to go.