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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 .

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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

4 Answers 4

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

Weird behaviour: http://journals.cambridge.org/action/displayAbstract;jsessionid=585FC36C71A59060B2028AD639A8D40A.journals?fromPage=online&aid=391658

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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.

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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.

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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.

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