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 at Conjecture.
“Worlds of Flow,” a history of 19th and early 20th-century hydrodynamics by Oliver Darrigol, concludes:
What distinguishes the history of hydrodynamics from that of other physical theories is not so much the tremendous effect of challenges from phenomenal worlds, but rather it is the slowness with which these challenges were successfully met. Nearly two centuries elapsed between the first formulation of the fundamental equations of the theory and the deductions of laws of fluid resistance in the most important case of large Reynolds numbers…The reasons for this extraordinary delay are easily identified a posteriori. They are the infinite number of degrees of freedom and the nonlinear character of the fundamental equations, both of which present formidable obstacles to obtaining solutions in concrete cases. Moreover, instability often deprives the few known exact solutions of any physical relevance…
These difficulties have barred progress along purely mathematical lines. They have also made physical intuition a poor guide, and a source of numerous paradoxes. Hydrodynamicists therefore sought inspiration in concrete phenomena. Engagement with and challenges from the real worlds of flow were essential to the development of the above-mentioned strategies. The challenged theorists strove to find new solutions and to develop new methods of approximation. Experience indicated some general properties of the motion, such as the existence of boundary layers, the random character of turbulence, the sudden character of the Reynolds transition, or the formation of trailing vortices…Altogether, there were many ways in which practical concerns oriented theorists in the conceptual maze of fluid dynamics. The evolution from a paper theory to an engineering tool thus depended on transgressions of the limits between academic hydrodynamics and applied hydrodynamics.
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.
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 is turbulent or laminar. Reynold’s number could have potentially been hypothesized as a consequence of Navier-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 Reynolds first noticed hints of this behavior from looking at eddies in the wake of the propellers of steamers:
(Worlds of Flow, page 247)
As British seamen had learned at their expense, when the rotational velocity of the propeller becomes too large, its propelling action as well as its counteracting torque on the engine's axis suddenly diminish. A damaging racing of the engine follows. Reynolds explained this behavior by a clever analogy with efflux from a vase. The velocity of the water expelled by the propeller in its rotation, he reasoned, cannot exceed the velocity of efflux through an opening of the same breadth as its own. For a velocity higher than this critical velocity, a vacuum should be created around the propeller, or air should be sucked in if the propeller breaks the water surface. According to this theory, a deeper immersion of the propeller should retard the racing (for the efflux velocity depends on the head of water) and the injection of air next to it should lower the critical velocity (for the efflux velocity into air is smaller than that into a vacuum). While verifying the second prediction, Reynolds found out that air did not rise in bubbles from the screw, but followed it in a long horizontal tail.
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 emphasized that "imagination or reason had failed to show" such forms of fluid motion. Everyone knew the impotence of rational hydrodynamics, but "it would seem that a certain pride in mathematics has prevented those engaged in these investigations from availing themselves of methods which might reflect on the infallibility of reason." Only with hints from colored water could mathematicians proceed further: “Now that we can see what we are about, mathematics can be most usefully applied.”
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 Laplace missing the existence of standing wavefronts because 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 got ironclad 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.
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)
There is, however, a puzzling contrast between the conciseness and ease of the modern treatment of [wave equations], and the long, difficult struggles of nineteenth-century physicists with them. For example, a modern reader of Poisson's old memoir on waves finds a bewildering accumulation of complex calculations where he would expect some rather elementary analysis. The reason for this difference is not any weakness of early nineteenth-century mathematicians, but our overestimation of the physico-mathematical tools that were available in their times. It would seem, for instance, that all that Poisson needed to solve his particular wave problem was Fourier analysis, which Joseph Fourier had introduced a few years earlier. In reality, Poisson only knew a raw, algebraic version of Fourier analysis, whereas modern physicists have unconsciously assimilated a physically 'dressed' Fourier analysis, replete with metaphors and intuitions borrowed from the concrete wave phenomena of optics, acoustics, and hydrodynamics.
In our minds, a Fourier component is no longer a mere coefficient in an algebraic development, it is a periodic wave that may interfere with other waves in a manner we can easily imagine. The transition from a dry mathematical analysis to a genuinely physico-mathematical analysis occurred gradually in the nineteenth century, through reversible analogies between different domains of physics. It concerned not only Fourier analysis, but also the theory of ordinary differential equations, potential theory, perturbative methods, Cauchy's method of residues, etc. The modern recourse to such mathematical techniques involves a great deal of implicit knowledge that only becomes apparent in comparisons with older usage. The motivation for the introduction of more powerful tools of analysis was mainly experimental. Most water-wave phenomena were known well before they could be explained. In most cases, they were discovered in connection with navigation problems…”
This is a problem beyond just hydrodynamics. For example, the original version of Maxwell’s work on electromagnetism contained 20 equations spanning pages of derivations. The modern versions of Maxwell’s equations, which are much easier to understand, come from Heavyside. 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 Rayleigh had an 1877 paper describing the rotation of a tennis ball in flight which apparently could have been used to derive wing 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 Lanchester was able to develop the idea of an airfoil by using Newtonian mechanics to imagine how air would react when hitting the wing of a bird or glider:
(Worlds of Flow, page 306)
In 1892, [Lanchester] imagined a singular theory of what he called an 'aerofoil', that is, the organ of sustentation of airplanes and birds. As he had 'very little acquaintance with classical hydrodynamics', he reasoned by direct application of the laws of mechanics to the particles of the fluid. In the first, Newtonian approximation, the fluid particles hit the aerofoil independently of each other, which leads to a resistance proportional to the squared sine of the inclination.
This was very difficult for anyone to understand:
(Worlds of Flow, page 308)
As Prandtl later put it, 'Lanchester's treatment is difficult to follow, since it makes a very great demand on the reader's intuitive perceptions.' Only a reader who would have known the results to be essentially correct would have bothered penetrating the car maker's odd reasoning.
And so Lanchester taught himself more formal methods to communicate:
(Worlds of Flow, page 308)
Lanchester must have become aware of this communication problem, since he immersed himself in Lamb's Treatise and sought more academically acceptable justifications of his intuition of the flow around an aerofoil.
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.
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.