Non-Linear Art. (Should Actually Be: Random Thoughts on Fluid Dynamics)

In my favorite ancient classical mechanics textbook I found an unexpected statement. I think 1960s textbooks weren’t expected to be garnished with geek humor or philosophical references as much as seems to be the default today – therefore Feynman’s books were so refreshing.

Natural phenomena featured by visual artists are typically those described by non-linear differential equations . Those equations allow for the playful interactions of clouds and water waves of ever changing shapes.

So fluid dynamics is more appealing to the artist than boring electromagnetic waves.

Is there an easy way to explain this without too much math? Most likely not but I try anyway.

I try to zoom in on a small piece of material, an incredibly small cube of water in a flow at a certain point of time. I imagine this cube as decorated by color. This cube will change its shape quickly and turn into some irregular shape – there are forces pulling and pushing – e.g. gravity.

This transformation is governed by two principles:

  • First, mass cannot vanish. This is classical physics, no need to consider the generation of new particles from the energy of collisions. Mass is conserved locally, that is if some material suddenly shows up at some point in space, it had to have been travelling to that point from adjacent places.
  • Second, Newton’s law is at play: Forces are equal to a change momentum. If we know the force acting at time t and point (x,y,z), we know how much momentum will change in a short period of time.

Typically any course in classical mechanics starts from point particles such as cannon balls or planets – masses that happen to be concentrated in a single point in space. Knowing the force at a point of time at the position of the ball we know the acceleration and we can calculate the velocity in the next moment of time.

This also holds for our colored little cube of fluid – but we usually don’t follow decorated lumps of mass individually. The behavior of the fluid is described perfectly if we know the mass density and the velocity at any point of time and space. Think little arrows attached to each point in space, probably changing with time, too.

Digesting that difference between a particle’s trajectory and an anonymous velocity field is a big conceptual leap in my point of view. Sometimes I wonder if it would be better not to learn about the point approach in the first place because it is so hard to unlearn later. Point particle mechanics is included as a special case in fluid mechanics – the flowing cannon ball is represented by a field that has a non-zero value only at positions equivalent to the trajectory. Using the field-style description we would say that part of the cannon ball vanishes behind it and re-appears “before” it, along the trajectory.

Pushing the cube also moves it to another place where the velocity field differs. Properties of that very decorated little cube can change at the spot where it is – this is called an explicit dependence on time. But it can also change indirectly because parts of it are moved with the flow. It changes with time due to moving in space over a certain distance. That distance is again governed by the velocity – distance is velocity times period of time.

Thus for one spatial dimension the change of velocity dv associated with dt elapsed is also related to a spatial shift dx = vdt. Starting from a mean velocity of our decorated cube v(x,t) we end up with v(x + vdt, t+dt) after dt has elapsed and the cube has been moved by vdt. For the cannon ball we could have described this simply as v(t + dt) as v was not a field.

And this is where non-linearity sneaks in: The indirect contribution via moving with the flow, also called convective acceleration, is quadratic in v – the spatial change of v is multiplied by v again. If you then allow for friction you get even more nasty non-linearities in the parts of the Navier-Stokes equations describing the forces.

My point here is that even if we neglect dissipation (describing what is called dry water tongue-in-cheek) there is already non-linearity. The canonical example for wavy motions – water waves – is actually rather difficult to describe due to that, and you need to resort to considering small fluctuations of the water surface even if you start from the simplest assumptions.


14 Comments Add yours

  1. I found this post fascinating. Since my artwork includes fluid dynamics, always, it would be interesting to find ways to make the forces involved more visible to the viewer. My mind is already crackling wtih ideas to show this. Thanks!

    1. elkement says:

      Thanks a lot – I am always happy if artists do appreciate my sciencey posts :-)

  2. Paint is a fluid and the painting process flows along. The preference might be inevitable.

    1. elkement says:

      You are the second commentator referring to the flow of paint – I admit I haven’t thought about it that way! My line of reasoning was more like: Artists like the intriguing shapes of clouds and waves (due to non-linear effects) ;-)

  3. So … non-linear differential equations … allow for the playful interactions of clouds and water waves of ever changing shape. Are you telling me then that the stochasticity and unpredictability of nature can be predicated by equations, given enough data? Where is the role of chaos in all of this? When you talk this way and tell me that all can be described with equations you channel Kepler in his role as Celestial Mechanic! D

    1. elkement says:

      No – the behavior of water and clouds cannot be predicted. I didn’t say this or didn’t intend to do so. I might be guilty us using some physicist’s shorthand in talking about physics in natural language though.
      To state something is “described” (in the sense of “governed”) by certain equations does not at all imply that these equations can always be solved in a way that is equivalent with prediction. Depending on the parameters used in the equations you can say if a system turns out chaotic or not – in hydrodynamics e.g. the so-called Reynolds number (dependent on three parameters, flow velocity, viscosity and characteristic length scale) determines if the flow in a tube becomes chaotic. “Non-linear differential equations” is usually nearly equivalent with “chaotic”.

      But I cannot resist nitpicking as the unpredictability of classical mechanics is of different sort than in quantum mechanics. In contrast to QM you could actually predict the behavior of fluids given nearly infinite computational power – it is practical, not fundamental. You solve these classical equations by calculating step-by-step, one time-slot after the other and computational errors increase. The funny thing is that the fundamental limit to the calculations of classical physics is actually related to QM again. I remember an example of “predicting” the collisions of billard balls (Nassim Taleb has used in The Black Swan to illustrate uncertainty). Depending on how many collisions you want to plan ahead you would need to take into account more and more impacts of the environment as e.g. the body heat of people, somebody opening the door to the room… and on trying to predict more than X collisions you end up with the requirement on considering what any atom in the universe does…. and thus quantum effects.

      1. Joseph Nebus says:

        And there’s some wonderful variations. My particular field of research involved the statistical mechanics of fluid flows, so that I never cared what any particle in the fluid was doing and was just looking for equilibriums, where the overall pattern stays constant. This, for my work, was done entirely in Monte Carlo methods, starting from a randomly drawn approximate solution and using randomness to come to a better solution. It’s kind of a lot to get used to but it’s uncanny how well it does work.

        1. elkement says:

          Thanks, Joseph – I remember your great post on that, adding the link here for future readers:!

          Yes, I did not refer to Lagrange and Hamiltonians in this post :-) (… based on the reactions on my related post on the “jelly” ;-))

  4. especially when I return from the night shift, and wait until the shops open so I can buy food, which is also when I am the tiredest possible, I can actually follow physics and math the very best. Which is now. Oh, yes, add hungry. The light bulb burns brightest before it burns out. Well, I’m not actually burning out.

    1. elkement says:

      Being very tired is a similar state of mind as being drunk (the experts say – so you should not drive when you are tired). As Maurice once said in a comment my posts interact in unexpected ways with mind-altering substances – so obviously you have experiences the same effect :-)

      1. I have lots of experience with mind-altering drugs and other states of mind ;)

  5. Fascinating. Two very powerful ideas I took away: 1–you are absolutely correct in the assertion that over-learning particle based mechanics makes it exceedingly to grasp on to fields. I never realized it until you said it! 2–what an interesting way to look at graphics. Before this I had never thought at all how DI’s could model the brush strokes, swirls and gradients of both thickness and tone. Art just got a whole lot bigger.
    You also caused me to think, spontaneously, on how this might be applied to the later analysis of the types of images created by machines. I am thinking along the lines of those created by CAT scans and MRIs. Right now the majority of the diagnosis is done (at least think) by visual inspection. As such the analysis is prone to confirmation bias and such. If, instead models were fitted to the data using some of the math you suggested perhaps we would be much ore along the way of recognizing the pathology that may be otherwise missed. Sort of like “optical character recognition” but now, instead of looking for known symbols we are instead looking for pathologies..

    1. elkement says:

      Thanks a lot Maurice – I rather figured this post is pretty boring ;-)
      Your comment on automation the analysis of images is interesting. Nicholas Carr (whose upcoming book on automation I await eagerly) is wary of software supporting medical doctors – I quote from his essay

      “Many radiologists today use analytical software to highlight suspicious areas on mammograms. Usually, the highlights aid in the discovery of disease. But they can also have the opposite effect. Biased by the software’s suggestions, radiologists may give cursory attention to the areas of an image that haven’t been highlighted, sometimes overlooking an early-stage tumor. Most of us have experienced complacency when at a computer. In using e-mail or word-processing software, we become less proficient proofreaders when we know that a spell-checker is at work.”

      1. That’s a very good point too. I suppose there’s no ideal, just a realisation that we need to apply our best thinking to the problem at hand, regardless.

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