Book Review: Worlds of Flow

16th Jan 2023

3ryan_b

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31a3orn

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I am beginning to think that histories of mathematical struggle and failure are my favorite kind. One that is similarly a tale of challenging and repeated failures on an unintuitive subject is thermodynamics, and an amazing book on this subject is *The Tragicomical History of Thermodynamics, 1822–1854 *by Clifford Truesdell, himself a mathematical physicist most famous for continuum mechanics.

tangentially related: one of the most incredible channels on youtube, steve brunton's lectures on everything from dynamical systems to fluid dynamics: https://www.youtube.com/@Eigensteve

Thanks for this, this was a fun review of a topic that is both intrinsically and instrumentally interesting to me!

This work was written atConjecture.“Worlds of Flow,” a history of 19th and early 20th-century hydrodynamics by Oliver Darrigol, concludes:

This quote captures one of the most significant lessons in the book: the study of concrete phenomena was critical in overcoming many of the difficulties in hydrodynamics. Roughly two upstream problems required interacting with concrete phenomena to solve. The first is summarized nicely by the quote above: theorizing and abstract thinking alone was not enough to solve the problems posed by hydrodynamics. The second is a subtler point: early mathematical and theoretical tools weren’t adapted to understanding hydrodynamics. Much of the necessary mathematical machinery existed quite early in the 1800s, but people hadn’t built the physico-mathematical tools or intuitions to tell us what they physically meant. Contact with reality forced scientists to confront the inadequacies of their theories while guiding the adaption of physico-mathematical tools. While the book is organized along rough problems that hydrodynamics faced (waves, viscosity, vortices, instability, etc.), this review will focus on broader scientific lessons. First on the two big themes I think are most important, then briefly on the other themes at the end.

## Practice and theory in hydrodynamics

The first problem was that abstract thinking and theorizing proved unable to solve many of the problems of hydrodynamics. A great example of this comes from the discovery of

Reynold’s number, which predicts whether flow isturbulentorlaminar. Reynold’s number could have potentially been hypothesized as a consequence ofNavier-Stokes, which describes viscous flow behavior. But Navier-Stokes is not analytically solvable, so Reynold’s number doesn’t come automatically. Making this more difficult is that turbulent flow, such as after submerged propellers, is generally invisible. Instead of reasoning his way there from first principles,Osborne Reynoldsfirst noticed hints of this behavior from looking at eddies in the wake of the propellers of steamers:(Worlds of Flow, page 247)

Following his discovery of the tail of bubbles behind the screw, he hypothesized that invisible currents played a much more significant role than earlier scientists, who only hinted at this behavior, believed. By injecting dye instead of air, he observed

complex vortex patterns. A year later, going off of stories from sailers that rain calms the seas, he dropped water into a wave tank with a thin layer of dye on top and observed a vortex ring forming at the surface and then falling downwards. This observation led him to conclude that this downwards motion could move turbulence at the surface downwards, smoothing the surface of the water. From these observations that demonstrated the importance of invisible flow, Reynolds created his theories on the behavior of turbulent flows. He was scathing about purely rational attempts at understanding hydrodynamics:(Worlds of Flow, page 247)

Reynolds' work is a striking example of a larger pattern of largely theoretical approaches proving inadequate in hydrodynamics. There were many other examples, such as

Laplacemissing the existence ofstanding wavefrontsbecause he got multiple (sine and cosine) solutions to a differential equation and assumed one of them was correct, missing that they both had physical meaning. This pattern continues to more modern times: rogue waves were only accepted after we gotironclad evidence. Only after this did we begin to develop the mathematical machinery to describe them. If we work from abstractions, it’s challenging to know when our theories capture the breadth of reality corresponding to the phenomena they purport to describe. Accessing reality can give us the necessary information that our imagination misses.## Modifying mathematical tools to fit physical reality

The other significant part of the story is understanding and adapting mathematical tools, so we have intuitions about what they mean and how to use them. Darrigol makes a great point that applies more broadly to mathematics in the history of science: the form they are presented in now is often not their original form, and we benefit from a century or more of progress in understanding and presentation:

(Worlds of Flow, pages 31-32)

This is a problem beyond just hydrodynamics. For example,

the original version of Maxwell’s work on electromagnetism contained 20 equationsspanning pages of derivations. Themodern versionsof Maxwell’s equations, which are much easier to understand, come fromHeavyside. This simplification leads us to both underestimate how hard the originals were to devise and the work required to build concise equations with nice physical implications and intuitions.In the case of hydrodynamics, building those powerful tools came from experiments. The lack of understanding of the mathematical tools of hydrodynamics had further downstream consequences. As we saw above, the consequences of Navier-Stokes took a lot of work to understand. Beyond making it hard to discover phenomena related to viscous flow, it also made it hard for Navier-Stokes to stick. At least five people discovered Navier-Stokes, but each time, it wasn’t clear that it had the explanatory power required to describe viscous flow, so it wasn’t widely adopted. This pattern of the implications of mathematical results not being understood and so not entering widespread use would be repeated.

Lord Rayleighhad an 1877 paper describing the rotation of a tennis ball in flight which apparently could have been used to derivewing theory, but nobody realized the implications of it until much later.The lack of good physico-mathematical tools also made it harder to develop ideas based on intuitive understanding.

Frederick Lanchesterwas able to develop the idea of anairfoil byusing Newtonian mechanics to imagine how air would react when hitting the wing of a bird or glider:(Worlds of Flow, page 306)

This was very difficult for anyone to understand:

(Worlds of Flow, page 308)

And so Lanchester taught himself more formal methods to communicate:

(Worlds of Flow, page 308)

So the development of physico-mathematical tools allowed scientists to reason about the physical properties of the mathematics they developed and allowed them to communicate the intuitions they used to develop new ideas.

## Conclusion

These are just what I found to be the most important two of the threads woven throughout “Worlds of Flow”. There are other interesting ones as well. One of those is the persistent role of analogies in developing the physico-mathematical machinery of hydrodynamics, as well as the limits of these analogies. The multiple discoverers of Navier-Stokes got to the same place using different analogies. For example, Navier started with elastic solids and reasoned about the behavior of molecules, while Stokes started with the behavior of air around a pendulum. But while these analogies could help build the mathematics, they were inadequate when it came to building the physico-mathematical tools and intuitions that make Navier-Stokes so powerful. Liquids were too different from solids for Navier’s elasticity analogy to get him to where Reynolds in turbulent flow, which required physical experimentation.

Another interesting thread was the number of different phenomena that contributed to understanding hydrodynamics. Scientists pulled from looking at smoke leaving chimneys or the motion of air out of organs. One scientist was inspired by looking at the behavior of cloud fronts as they moved over the Alps. The behavior of waves in water, both natural and artificial is an obvious one, but even waves had a huge amount of diversity, from observing the behavior of boats in canals to listening to stories that sailors told. This both tells a story of human ingenuity in pulling traces of evidence that exist in the world around us and one of a horrifying lack of imagination as many brilliant scientists were stuck with inadequate abstractions while ignoring all of this.

The many interesting, amazing stories of the clever work of individual scientists scattered throughout the book were another thread. This is a little harder to summarize because the lessons here are diverse and not particularly coherent. For example, Lord Rayleigh developed a good understanding of the units involved in hydrodynamics. This is the type of tool that seems obvious in hindsight but had to be developed in a clear form to be this obvious first. He would show up throughout the story, using unit analysis to tell people that the shape of their proposed answers was wrong, the shape should look like

this, and then vanishing.Ludwig Pradantl, who helped develop wing theory with Lanchester, taught himself intuitions by observing behavior in wave tanks and tracing the unrolling of equations through them, which gave him the intuitive understanding necessary to realize the value of Lanchester’s work. There are many of these stories, and reading them is always one of my favorite parts of reading about the history of science.There were a few flaws with “Worlds of Flow”. Much of “Worlds of Flow” was a slog through various attempts at building the machinery of hydrodynamics. At times, the way that the math was presented felt unintuitive and hard to work through, although a lot of the issues there were probably on my end. Some of the narrative threads in the book were a little hard to follow-for example, the chapter on Navier-Stokes ends with a hint that it took longer than was covered in the chapter for Navier-Stokes to be better understood and adopted. This is touched on through the rest of the book in a scattered fashion but never really addressed in a straightforward, satisfying manner.

These flaws were not large enough to detract from the overall import of “Worlds of Flow.” The history of hydrodynamics is a great example of how reality can challenge theoretical intuitions, and mathematics can be adapted to fit reality. Theory alone was invariably inadequate, and physical intuition was often wrong. Forcing our theories to address the breadth and complexities of reality took a lot of hard work and experiments. It often required developing more powerful physico-mathematical tools to understand the implications of the theories better. And though even now, our understanding of hydrodynamics is incomplete, the level of understanding we do have did not come easily and has valuable lessons on the operation of science today.