Viscous fingering is a fractal pattern that occurs when a less viscous (or thick) fluid spreads through a more viscous (or thick) fluid. Such systems are present in oil extraction, when we pump one fluid underground to push another one out. Fractals are common in nature even though they’re new to our mathematics, and they are beautiful.

The pictures in this post were created with basic watercolor paints using one simple principle: water containing paint is more viscous than regular water. It’s easy to try at home!

For the top picture, I laid down red paint. Before the paint dried, I added salt, then let the square dry. Water from the still-damp paper rushed to the salt (because of entropy, systems tend towards uniform distributions of things if they can help it– in this case, the lowest energy state is to have a uniform distribution of salt). But because paint molecules are larger than water molecules, they don’t move as well. The water that accumulates around the salt has less paint than the water in the rest of the paper, and thus we have a less viscous fluid spreading into a more viscous one. Try it at home! If the paint is too wet or too dry when you add the salt, the results won’t be as dramatic, so play around a bit. Larger salt crystals can be especially fun.

For the three pictures below, I simply placed a drop of water into a damp square of paint. The patterns vary depending upon the size of my drop, the wetness of the paint, and the paint color (the chemistry of which influences the viscosity of the paint).

Below are a couple of examples from the University of Alberta of viscous fingering with pentane into oil and water into oil. This particular research aims to improve the flow rate of oil during extraction. And it looks pretty similar to some humble watercolors.

Left: pentane displacing mineral oil. Right: Water displacing mineral oil (University of Alberta).