Zigzag flow of water along a vertical glass window

I've observed this behavior many times. When it rains, the rainwater will form vertical channels along a glass window. The flow of water is mostly confined within these vertical channels and the channels are (more or less) stable.

But sometimes - and I suspect this happens when the flow intensity in one of the channels increases - the channel will switch from a vertical configuration into a zig-zag configuration. The zig-zag is composed of short segments running horizontally that are connected by semicircular (vertical) segments. The zig-zag is unstable and lasts only for 0.1 second or so. Then the channel reverts to its vertical configuration.

I have made photographs of this behavior but I cannot find them now.

I have seen similar patterns in the book "The self made tapestry" by Philip Ball, page 145. This shows growth instabilities in glass cracks. Is says "at higher speeds the crack becomes oscilatorry with a constant wavelength". This is what I see in the water flow. It feels counterintuitive.

There must be a good explanation for this behavior. Can you point me to it?

EDIT Here is a video .

• This is not so easy, because droplet sliding on previously dry glass is not described by continuum mechanics--- there is violation of no slip as the drop slides. This makes it stochastic and unpredictable. If you post a picture, maybe one can figure out what is going on, I don't know any literature on this. – Ron Maimon Jul 28 '12 at 21:24
• If I beleive this paper, continuum mechanics can be used. And it seems that unless you apply a random tangential force in eq 1, you get straight trajectory. amath.unc.edu/Faculty/mucha/Reprints/droplet.pdf – Shaktyai Jul 30 '12 at 22:13
• @Shaktyai: That's just false--- it's obvious you can't use continuum dynamics, you need violation of no-slip somewhere to get the drop to slide--- the air has to get out of the way. The paper uses a contact angle formalism which is not fundamental, but might be accurate, I don't know, but it doesn't invalidate what I was saying above. – Ron Maimon Sep 8 '12 at 3:11
• @Ron: A drop sliding on a glass leaves a trail of water which is pretty much in accordance with a no slip condition. I see the problem more like the upper part of the drop rolling over the one in direct contact with the glass (no slip there) and surface tension providing a restoring force that maintains the overall shape. Once a drop has scouted a trail, the water film it has left behind provides a sliding track for the other drops. I don't have time to look any deeper in it but I will try someday. – Shaktyai Sep 8 '12 at 5:27
• @Shaktyai: For the drop to roll and make contact, it must push the air entirely out of the way, which requires a no-slip violation on the air. Because it's air, no slip violations are much easier than for water, but they are forbidden in continuum mechanics just as much as any other. – Ron Maimon Sep 8 '12 at 5:30

This is a guess:

If the glass is pictured as a square where y is the vertical and x the horizontal, it is a random walk process in the x direction and gravity constrained in the y.

The randomness comes because of dust and other adhesions, even in the cleanest glass. Gravity would pull a single drop straight down,but on the way the drop randomly hits a discontinuity which breaks the surface tension at that point giving a change in direction.

When there are many drops, as in rain, I would expect that they follow a channel of discontinuities, and the randomness is overwhelmed by the gravitational vertical force, constrained to the channel you describe. Larger random discontinuities might build up within each channel which would give rise to the effect you describe.

Here are some papers adressing the problem of the motion of a drop on a surface. There seems to be two kinds of motion, depending on wether the fluid wetts the surface or not. Sliding and rolling. But contact line deformation and drop breaking seem also to be of importance. The subject is too wide to be simply explained (assuming I could) here.

http://fluids.snu.ac.kr/publication/sliding.pdf. http://www.pmmh.espci.fr/fr/gouttes/Publications_files/Rolling%20drops.pdf http://archives.njit.edu/vol01/etd/2000s/2008/njit-etd2008-076/njit-etd2008-076.pdf (page 25) http://stilton.tnw.utwente.nl/people/snoeijer/Papers/2011/WinkelsEPJST11.pdf

Could it be caused by microscopic faults/impurities in the glass itself? To the naked eye the glass may appear clear & faultless; for a flowing fluid such as glass the combination of the impurity and (in your scenario ) perhaps a sudden breeze could cause a momentary deviation in the path.

• Glass itself usually doesn't have many faults/impurities in itself, but surface contaminates are common. – Rick Feb 3 '15 at 18:34

TL;DR a droplet intersects the stream, temporarily diverting it.

The glass is not uniform due to surface contaminates like oil and dirt. This makes the binding energy between the water and the glass vary by location. A droplet that land on the glass with one edge on a high binding energy location and the opposite edge on the low binding energy location will move away from the low energy towards the higher energy.

Lets assume there is a uniform distribution of very small droplets hitting the vertical glass pane. At first all of the droplets will just move to maximize their binding energy and sliding down the glass would require losing more of the binding energy than they would gain from gravity. Thus all the droplets stick to the window.

As droplets continue to fall some land overlapping with droplets that were already there. These droplets reform to accommodate their new volume and mass. They might even slip down a little to bigger binding energy ledge. Their new binding energy loss/gain by sliding should be proportional to the diameter as it's only the perimeter of the droplet that changes. However, the energy gain from gravity is proportional to the diameter cubed. This means that as the droplets get bigger gravity will dominate and pull the droplet down the glass.

If you look near the top of a pane of glass where no water is pouring down this is pretty much all you see, droplets getting larger and larger until they slide off down the pane. If there are especially high and low binding energy regions you'll even see that the droplets tend to grow in the same locations.

However, for long panes of glass these sliding droplets will collide with other droplets, getting even bigger. Whenever these sliding droplets hit a droplet that is slightly to the side, it will be pulled towards that droplet slightly diverting its path. Additionally the surface binding energies will still pull the droplet so that it tends to follow a high binding energy path.

Eventually the droplet becomes so big and is moving so fast that it starts growing a tail where the water still wants to stick to the glass and the reduced surface tension of the large curvature isn't enough to rip it off. Another droplet coming along that happens to hit the tail will immediately divert to it as the already wetted surface acts like an extremely high binding energy region. This process is what accumulates droplets into streams.

These streams tend to go straight down, buy divert slightly to take advantage of higher binding energy regions. The cause of the sudden side track of a stream is a temporary high binding energy region in the form of a droplet. While all of the streams are making their way down the glass, the regions in between the streams are still accumulating ever growing droplets. When one of these droplets grows to the point it runs into a stream, the stream is diverted.

In the video it looks like the droplets landing on the window may not all be tiny, and thus this process is accelerated by big droplets appearing randomly, but the same effects can be observed (perhaps more clearly) by spraying a fine mist on a piece of glass.

I'm pretty confident that this is a phenomenon called Hydrodynamic Instability. This is an ubiquitous phenomenon in fluid mechanical processes. Your case could be the Kelvin-Helmholtz instability, but don't nail me down on it.