Darcy's Law in Pipes

It’s common knowledge that most pipes, if they’re in use, have some kind of fluid flowing in it; and if you’re wondering how flow would be different if your liquid’s a bit dirty or dense with other stuff, don’t worry,

Darcy’s got you covered.

In filtration pipes, Darcy's Law is used to calculate the permeability of filtration layers, or a permeable membrane[1]. Both the filtration layer and the permeable membrane are porous layers. Filtration layers often consist of grains of varying shapes and sizes, whereas a permeable membrane is a thin porous sheet.

At this point you may also be wondering,

“How in dam-nation does Darcy affect me, personally?”

"apart from his good looks and physique, of course."

Well, all over your body, you have pipes (over 100,000 km!).

People like to call them blood vessels. They supply the oxygen your body needs and get rid of all the nasty stuff it produces. So they’re quite important, and if they stop transporting blood, you wouldn’t be able to finish reading this article ☹.

Thankfully, these vessels work best under pressure!

These "pipes" can be blood vessels which carry blood [2].

It can also be extracellular space (space between the cells in the body) where interstitial fluids flow [3]

If you have read our secret article (or if you happened to be a Darcian logic genius...), you will know that the volumetric flowrate of "pipes" in a human body can be calculated using Darcy's Law:

Where η is the dynamic viscosity, L is the Vessel length and r is the radius of the vessel.

So the radius seems to have the biggest impact on the resistance, which is a good thing because our body can simply make a small change to the radius (vasodilation) to make a big change the flow rate, or reduce the pressure, so our vessels don't burst open.

It is also influenced by the viscosity, which is quite complicated, since the composition of blood changes all the time and has lots of things inside it, like red blood cells that are suspended in the fluid. Viscosity depends most however on the amount of water in the blood, so to keep the resistance low, we can drink some more water which will decrease the viscosity, decreasing the resistance.

To make the matter even more complex, the blood system isn’t just one pipe, but rather many combined in series and parallel. So, we need to find the total resistance of the vessels.

However, we can model these vessels just like resistors in electrical circuits. The blood system starts off as one vessel (leaving the heart) and branches into multiple vessels that go to each organ, which then branch off again to supply blood to each area of the organ. We can think of these being resistors in parallel, with a total resistance of one collection of branches being found by,