Why Does Water Flow in a Wavy Pattern on Inclined Surfaces?

[Mr Peanut]

On an inclined sheet of clean glass (or Teflon, or a simonized car, etc.) a trickle of water flows, not in a straight line, but in a wavy ribbon. It does this rivers too. In fact, in rivers (around here anyway) it usually creates a meander pattern, bending about every 5-7 stream widths.

Why does water move this way? Is it a Coriolis effect, a molecular interaction, ...?

 

[Drakkith]

On a small scale, such as water flowing on glass in a small trickle, the effects of surface tension and other effects can cause a large difference in the flow.

On a large scale, such as a river, it is due more to the makeup of the original terrain that the river originally flowed through. And since that terrain isn't usually the same everywhere, the river flows around and loops and such.

 

[Mapes]

Agree with Drakkith that the mechanisms are likely very different. http://ask.metafilter.com/92478/Calling-All-Ecohydrologists" is a thread on Metafilter that has information on river meandering.

I would like to learn more about how smaller trickles move down a surface; http://www.csie.nctu.edu.tw/~roger/rain/references/animationWaterDroplet.pdf" an article on mimicking the motion by animation that does not go into the true controlling factors.

 

[Andy Resnick]

On an inclined sheet of clean glass (or Teflon, or a simonized car, etc.) a trickle of water flows, not in a straight line, but in a wavy ribbon. It does this rivers too. In fact, in rivers (around here anyway) it usually creates a meander pattern, bending about every 5-7 stream widths.

Why does water move this way? Is it a Coriolis effect, a molecular interaction, ...?

It's a highly nonlinear process called "fingering instability" or the Rayleigh-Taylor instability:

http://en.wikipedia.org/wiki/Rayleigh–Taylor_instability
http://prl.aps.org/abstract/PRL/v62/i13/p1496_1

It's an active area of research.

 

[boneh3ad]

This has nothing to do with the Rayleigh-Taylor instability.

The original answer was correct. Flowing on a surface it has to do with a combination of surface tension on the water and imperfections on the surface it is flowing down. That is why it will move straighter down glass than, say, sandpaper.

With rivers it has to do with the terrain.

 

[Andy Resnick]

This has nothing to do with the Rayleigh-Taylor instability.

The original answer was correct. Flowing on a surface it has to do with a combination of surface tension on the water and imperfections on the surface it is flowing down. That is why it will move straighter down glass than, say, sandpaper.

With rivers it has to do with the terrain.

If you say so...

http://math.arizona.edu/~shottovy/papers/fluidspaper.pdf

This guy calls it an instability due to Tolliman-Schlichting waves.

http://pof.aip.org/resource/1/pfldas/v10/i2/p308_s1

This guy calls it a generic Hopf bifurcation

http://www.math.uh.edu/~gio/MyWeb/papers/guiglowJMFM.pdf

Either way, this is completely different than erosion of a river bed, and has nothing to do with surface imperfections. It's driven by instabilities in the free surface.

Again, my central point is that it is a highly complex phenomenon and is an area of active research (i.e. not fully understood).

 

[boneh3ad]

I can also tell you it shouldn't be a result of Tollmien-Schlichting waves because they are a streamwise instability so they shouldn't cause it to wildly careen off course. Additionally, T-S waves take a while to develop so I wouldn't start right away. If you notice the stream getting more erratic as it moves further down the surface, then you may be observing a T-S effect.

I am not familiar enough with a Hopf bifurcation to comment on that, but I can tell you that neither of these paper are even addressing the same problem as this thread. Those are talking about a layer of liquid poured down a slope, not just a single little stream.

Certainly the math behind the instability causing th meandering of a small stream of water is complicated, but the root cause is very likely a combination of surface tension and surface roughness. The interplay between the two is the nature of the instability.

 

[Andy Resnick]

Those are talking about a layer of liquid poured down a slope, not just a single little stream.

Ok, rather than discuss the stability of the film, you are apparently instead focusing on the tip of the finger. That involves wetting, which is not understood either. Wetting is less understood than film instabilities.

http://www.maths.bris.ac.uk/~majge/PhysRevE_72_061605.pdf

http://www.google.com/url?sa=t&sour...sg=AFQjCNHZvhWuk-G7ax5xabmXIOTXBft4gw&cad=rja

http://www.mech.northwestern.edu/courses/438/webstuff/PDF_1/Kumar.pdf

 

[boneh3ad]

Yes it involves wetting, but the major factors involved will be surface tension an wall roughness. This is not a perfectly smooth surface so there will be an effect from the roughness. Surface tension has a huge part in wetting as well. Look at all th sources you cite. Capillary pressure is a function of surface tension.

 

[Andy Resnick]

Yes it involves wetting, but the major factors involved will be surface tension an wall roughness. This is not a perfectly smooth surface so there will be an effect from the roughness. Surface tension has a huge part in wetting as well. Look at all th sources you cite. Capillary pressure is a function of surface tension.

Wetting behavior, as a static phenomenon, does depend on the interfacial energy only. However, a moving contact line involves a lot more because mass flow is involved- inertial and viscous forces must be incorporated (via the capillary number and Weber number, for example). The Bond number plays a role when buoyancy effects are important (like flow down an inclined plane). If there is a thermal gradient present, the Marangoni number is important.

Then, if the solid surface is heterogeneous in any way (surface roughness, chemical heterogeneity, etc), or a surfactant (a surface-only fluid phase) is involved, further complications ensue.

Again, it's not a simple problem. It has not been solved.

 

[boneh3ad]

Wetting behavior, as a static phenomenon, does depend on the interfacial energy only. However, a moving contact line involves a lot more because mass flow is involved- inertial and viscous forces must be incorporated (via the capillary number and Weber number, for example). The Bond number plays a role when buoyancy effects are important (like flow down an inclined plane). If there is a thermal gradient present, the Marangoni number is important.

Then, if the solid surface is heterogeneous in any way (surface roughness, chemical heterogeneity, etc), or a surfactant (a surface-only fluid phase) is involved, further complications ensue.

Again, it's not a simple problem. It has not been solved.

But wait, let's look at this. The Weber number is the ratio of inertial forces compared to surface tension. The capillary number is the ratio of viscous forces to surface tension. The Bond number is the ratio of buoyant forces to surface tension. The Marangoni number has to do with thermal surface tension vs. viscous forces.

It is certainly not a simple problem, but by far the most important physical factor is the surface tension. Of course there are inertial and viscous forces involved, but that is true of any real fluid flow.

Moving back to the original post - a small trickle of water slides down a smooth surface and takes a wavy path to the bottom - most of that is unimportant because there is no meaningful thermal gradient, there is no bubble moving around in the water and so really the only really important of those numbers are the Weber and capillary numbers. Of course the mention of the simonized surface is different since it would be a hydrophobic surface, but on the whole, the most important mechanism is going to be surface tension as far as I can see. Surface roughness of the inclined plane will likely be important insofar as it would be the primary method for exciting waves that can be destabilized by the surface tension driven instability.

Of course, the answer that suffices for the OP is just that it has nothing to do with rivers and that the reason is largely surface tension related.

 

[Andy Resnick]

But wait, let's look at this. The Weber number is the ratio of inertial forces compared to surface tension. The capillary number is the ratio of viscous forces to surface tension. The Bond number is the ratio of buoyant forces to surface tension. The Marangoni number has to do with thermal surface tension vs. viscous forces.

It is certainly not a simple problem, but by far the most important physical factor is the surface tension. Of course there are inertial and viscous forces involved, but that is true of any real fluid flow.

Moving back to the original post - a small trickle of water slides down a smooth surface and takes a wavy path to the bottom - most of that is unimportant because there is no meaningful thermal gradient, there is no bubble moving around in the water and so really the only really important of those numbers are the Weber and capillary numbers. Of course the mention of the simonized surface is different since it would be a hydrophobic surface, but on the whole, the most important mechanism is going to be surface tension as far as I can see. Surface roughness of the inclined plane will likely be important insofar as it would be the primary method for exciting waves that can be destabilized by the surface tension driven instability.

Of course, the answer that suffices for the OP is just that it has nothing to do with rivers and that the reason is largely surface tension related.

You still don't understand. Even the most simple case- a pure Newtonian nonpolar fluid flowing down a perfectly smooth, flat, homogeneous surface in a dilute gas is an unsolved problem.

Even though the only relevant quantities are the capillary and Bond numbers (it's buoyant flow because the more dense nonpolar fluid is displacing less dense air- bubbles are not relevant) you cannot predict the most simple result- how fast will the fluid flow down the incline?

This cannot currently be predicted because, even if the physical properties of both fluids (interfacial energies, densities, etc) are known and the equilibrium contact angle is known, the moving contact line is beyond the reach of theory, and in fact, violates the most basic laws of mechanics (because the stress tensor is discontinuous). The dynamic contact angle is quite independent of the static contact angle, and in general is poorly defined at a moving contact line.

Even more, introducing a simple-minded model of slip (which has no theoretical justification at this time), a wide range of fluid behaviors can be introduced simply by varying the angle of incline- see for yourself. The interfacial energy is only one parameter of interest, not the most important.

 

[boneh3ad]

You still don't understand. Even the most simple case- a pure Newtonian nonpolar fluid flowing down a perfectly smooth, flat, homogeneous surface in a dilute gas is an unsolved problem.

Even though the only relevant quantities are the capillary and Bond numbers (it's buoyant flow because the more dense nonpolar fluid is displacing less dense air- bubbles are not relevant) you cannot predict the most simple result- how fast will the fluid flow down the incline?

This cannot currently be predicted because, even if the physical properties of both fluids (interfacial energies, densities, etc) are known and the equilibrium contact angle is known, the moving contact line is beyond the reach of theory, and in fact, violates the most basic laws of mechanics (because the stress tensor is discontinuous). The dynamic contact angle is quite independent of the static contact angle, and in general is poorly defined at a moving contact line.

Even more, introducing a simple-minded model of slip (which has no theoretical justification at this time), a wide range of fluid behaviors can be introduced simply by varying the angle of incline- see for yourself. The interfacial energy is only one parameter of interest, not the most important.

On the contrary, clearly you don't understand. Just because something is unsolved - a claim I have not once disputed - doesn't meant that it is impossible to recognize the important factors. The fact remains that every nondimensional number you have cited has been a relation between something that is present in all or most flows (or in special cases) and surface tension. It may not be a solvable problem right now, but it is pretty clear that surface tension is the most important physical property, both based on simple intuition and the examples you have cited.

I really don't understand what you are arguing about at this point because this last post seemingly has nothing to do with my previous post.

 

[AlephZero]

I think Andy R is taling about the OP, not speculations about unstable nonlinear dynamics by somebody who has said they are not famiiliar with Hopf bifurcations.

That's a bit like speculation about quantum mechanics from somebody who has said they are not familiar with wave functions, IMO.

 

[Andy Resnick]

I think Andy R is taling about the OP, not speculations about unstable nonlinear dynamics by somebody who has said they are not famiiliar with Hopf bifurcations.

That's a bit like speculation about quantum mechanics from somebody who has said they are not familiar with wave functions, IMO.

Yes- you got it.

 

[boneh3ad]

I think Andy R is taling about the OP

Seems like his comments were directed at me, and if not, his comments weren't directly addressing the OP's question.

not speculations about unstable nonlinear dynamics by somebody who has said they are not famiiliar with Hopf bifurcations.

Call it speculation all you want, but it isn't. One doesn't need intimate familiarity with Hopf bifurcations to understand basic hydrodynamic stability. Also notice that I didn't comment on whether or not it was a Hopf bifurcation, as I admittedly was not very familiar with them. Still, given what I do know about them, they would be the mathematical nature of mode growth for an instability in which they are important, but it says nothing about the physical nature. You don't need to have knowledge of them to understand what the Rayleigh-Taylor instability is and under what conditions it arises or what Tollmien-Schlichting waves are and how they arise. This system doesn't fit the profile of a system dominated by either of those two phenomena.

It seemed to me that his point all along is that it is a wetting phenomenon and that it is an ill-understood problem, which I don't disagree with. My point has been that while the mathematical mechanism for the stability characteristics of this system may be a Hopf bifurcation or anything else for that matter, the dominant physical mechanism is going to be an effect of surface tension. Wetting problems may be an largely unsolved set of mysteries at this point, but nearly all if not all of them have surface tension as one of the most important physical factors as evidenced by the appearance of surface tension in the associated nondimensional numbers.

I don't understand the hostility here, to be honest. Maybe I came off a bit callous in my original post after Rayleigh-Taylor was suggested and if that is the case, I apologize. However, based on that suggestion, it appeared that Andy just threw the name out there without knowing what he was talking about, which was further supported by the linking to papers that were unrelated to the current problem. If that assumption was incorrect, that is my mistake of course, and I can see that it may well have been his purpose to point out that there is general disagreement from source to source on the nature of the instability. Again, I am certainly likely at fault here for helping foster a tad of this air of hostility, but why the personal attack, AlephZero? You know nothing of me or my qualifications or even how relevant Hopf bifurcations are to my work, so why imply that I don't have a clue based solely on the fact that I didn't refute something that I didn't know enough about?

EDIT: I suppose I should let it be known, I did a little digging through my hydrodynamic stability literature, and it seems I know more about Hopf bifurcations than I originally thought. It seems that my stability class was taught without explicit use of the term "Hopf bifurcation", though the principles involved were used plenty. I realize this probably looks like a cop-out, but the book I paged through in response to AlephZero was "Introduction to Hydrodynamic Stability" by Drazin if that at least serves to hint that maybe I didn't just make this up to save face.