Poiseuille's Law

What it Shows

J. L. M. Poiseulle and G. H. L. Hagen determined that the laminar flow rate of an incompressible fluid along a pipe is proportional to the fourth power of the pipe's radius. To test this idea, we'll show that you need sixteen tubes to pass as much water as one tube twice their diameter.

A YouTube video of our Poiseuille's Law apparatus in action (https://youtu.be/wn6eRMIOJ1k)

How it Works

Poiseulle's law says that the flow rate Q depends on fluid viscosity η, pipe length L, and the pressure difference between the ends P by

\(Q = {\pi r^4 P\over 8 \eta L }\)

but all these factors are kept constant for this demo so that the effect of radius r is clear.

The apparatus consists of two 12 liter Plexiglas tanks, one to be emptied through a single 6mm bore capillary tube and the other through sixteen 3mm bore tubes. All tubes are 60cm in length. For direct comparison, all tubes need to be opened to the tanks simultaneously and this is achieved using a valve consisting of a long steel rod with 17 holes drilled through it, corresponding to the 17 tubes (see figure 1b below). The rod runs the length of the tanks and has a handle that rotates it to align the holes in the rod with those in the tank.

top view of tanks and pipes, with details of the valves
Figure 1a (top) Poiseulles's Law apparatus, as seen from above; Figure 1b (bottom) detailed sketch of valve.

Setting it Up

For convenience the two tanks are constructed separately but can be set up side by side with a connector linking the rod valves of each. The apparatus sets up on a bench with a catch tray angled at about 20°, just below the ends of the capillaries. There is a hole in one corner of the tray that allows it to empty to a bucket below. Food coloring can be added to the water to make the levels clear.


Very interesting effect visually, since the 6mm tube doesn't look that much bigger than the 3mm ones—in fact, the total cross-section area of the small tubes is greater than the big one by a factor of 4!