The Rayleigh-Plateau Instability

A simple experiment

 This experiment not only illustrates mathematics that we'll need to understand as we study capillary

surfaces, but also nicely illustrates some of the simple experimental techniques you'll need to develop.

In a recent group meeting we began talking about the Rayleigh or Rayleigh-Plateau instability. Two results

from our discussion will be useful. First, from Rayleigh's analysis, we know that a cylindrical column of

liquid is unstable to spatial perturbations with wavelength satisfying:

Here, is the wavelength and is the radius of the initial un-perturbed cylinder. The second key result

is that one wavelength grows faster than all others. This accounts for the fact that when we perform the

experiment, we always see drops of the same size for initial columns of the same radius. Rayleigh also

showed that the most unstable wavelength is given by:

 We'll derive these results in our group meeting. To test these experimentally, on 6/15/07, I went to the lab

and performed a simple experiment. Here is the setup:

While it is hard to see in the picture, if you look closely you should notice a thin fishing line supported horizontally

in front of the lightbox. The fishing line was marked using a black Sharpie to indicate gaps of one centimeter.

The fishing line is attached to two metal nuts, these are held to the lab stands by magnets.

This makes the system easily adjustable. On the right, is a small vial of motor oil. The experiment is simple. Using

the eye dropper, I took a drop of oil and dragged it across the fishing line giving it a uniform "sheath" of oil.

Then, I took a handheld digital photograph to obtain a photo like this:

 The instability sets in very quickly, you can watch it evolve, but it only takes seconds for the sheath to break up

into drops. Now, the coating is somewhat non-uniform. Notice the larger drops are clearly deformed by gravity.

What I wanted was to data that did not include the influence of gravity, so I extracted the region on the right

where the sheath and drops are very small.

To do this I used IrfanView. I also converted the image to grayscale and negative. The two white spots are

the markings on the line. They are one centimeter apart. Next, I used the image toolbox on Matlab to make measurements.

Again, this was done crudely and quickly. I found that the gap between the white spots was 420 pixels. this meant

that in the figure we had 420 pixels per centimeter. Next, I measured the gap between the peaks of the drops.

This measurement was in pixels, so I converted to centimeters. I took an average of the two data points to find

that the wavelength was about 3.4 mm. Using the expression for the most unstable wavelenght given above, we

find that the radius of the fishing line should be about 3.8 mm. While I don't know the radius of the fishing line, I do

know that it is 20lb test and that typical line of this rating has a radius of 4 mm. Remarkably good agreement between

theory and experiment for such a crude setup!