5 Five Men, One Equation
5.1 The List
The Navier-Stokes equations were derived five times, independently, between 1822 and 1845:
| Year | Author | Country | Method |
|---|---|---|---|
| 1822 | Claude-Louis Navier | France | Molecular forces |
| 1828 | Augustin-Louis Cauchy | France | Stress tensor (general theory) |
| 1829 | Siméon Denis Poisson | France | Molecular model, different from Navier’s |
| 1843 | Adhémar Jean Claude Barré de Saint-Venant | France | Continuum, rate of strain |
| 1845 | George Gabriel Stokes | England | Continuum, stress tensor |
The first four are French. The fifth is Irish-born, English-educated, working in England. The time span is twenty-three years. None of the later authors knew the work of all the earlier ones. Saint-Venant in 1843 knew about Navier but disputed his molecular approach; Stokes in 1845 knew about Navier, Cauchy, and Poisson, and cited them.
The five derivations are not minor variations on a theme. They use different mathematical frameworks, different physical starting points, different notations. They arrive at the same equations.
5.2 Why This Happens
Multiple independent discovery is not as unusual in science as the mythology of individual genius suggests. Newton and Leibniz both invented calculus. Darwin and Wallace both developed natural selection. Oxygen was isolated by Scheele in 1772 and by Priestley in 1774, and the theoretical significance was grasped by neither until Lavoisier in 1777. The radio was developed by Marconi, Tesla, and Lodge in overlapping and disputed sequence.
The pattern is explained by what the historian of science Derek de Solla Price called “the logic of scientific development.” When the tools are available and the problem is pressing, multiple people will find the answer. The tools in 1820 were: Newtonian mechanics, the calculus of variations, Euler’s equations, and a community of trained mathematicians and engineers who knew all of these. The problem was pressing: industry needed viscous flow theory. The answer was constrained: there are only a few mathematical forms that make physical sense.
What makes the Navier-Stokes case remarkable is the number of independent discoverers — five is high — and the speed with which they appeared once Navier had opened the field. Cauchy came six years after Navier; Poisson one year after Cauchy; Saint-Venant and Stokes within two years of each other, both more than a decade after Poisson. The burst in the 1820s reflects the concentration of mathematical talent in Paris; the final pair in the 1840s reflects the maturation of continuum mechanics as a framework.
5.3 Cauchy and Poisson
Augustin-Louis Cauchy is one of the founding figures of mathematical rigor. He produced the first systematic treatment of what we now call real analysis, defining limits and continuity in terms that still appear in undergraduate textbooks. He also, in 1828, published a derivation of the equations of motion for elastic solids and viscous fluids that used the stress tensor as its fundamental object. Cauchy’s derivation was more general than Navier’s — it applied to elastic as well as fluid media — and it placed viscosity in the correct mathematical context.
Siméon Denis Poisson, a year later, used a molecular model similar to Navier’s but more carefully constructed. Where Navier had assumed a specific form for the intermolecular force, Poisson derived more generally what form was consistent with the observed macroscopic behavior of fluids. He arrived at the same viscous term, with additional subtleties about compressibility that were not fully resolved until Stokes.
Both Cauchy and Poisson knew of Navier’s work. Both disputed it on physical grounds while confirming it on mathematical ones. This is the peculiar situation: the equations were confirmed correct by people who disagreed about why they were correct.
5.4 Saint-Venant’s Complaint
Of all five derivers, the one with the strongest claim to historical neglect is Adhémar Jean Claude Barré de Saint-Venant.
Saint-Venant was born in 1797 and lived to 1886, spanning nearly the entire nineteenth century. He was an engineer, not a pure mathematician, trained at the École Polytechnique and employed for most of his career in the French Corps of Bridges and Roads — the same institution that trained Navier. He made fundamental contributions to elasticity theory, to the theory of torsion in beams, and to the mechanics of plasticity. Any structural engineer who has used Saint-Venant’s principle (that the effect of a localized load becomes uniform at distances large compared to the loading region) has used his work.
In 1843, Saint-Venant published a derivation of the viscous flow equations based on the rate of strain — the rate at which neighboring fluid elements are deforming relative to one another. His approach was, in retrospect, the clearest physical argument of all five: viscous stress is proportional to the rate at which the fluid is shearing, and the coefficient of proportionality is the viscosity. This is exactly the statement that defines a Newtonian fluid, and Saint-Venant made it two years before Stokes.
Saint-Venant spent decades arguing, with characteristic persistence, that the equations should be called the Navier-Saint-Venant equations, or at minimum that his contribution should be recognized alongside Stokes’s. He had a reasonable case. He had the continuum derivation before Stokes, and his physical argument was sound. The mathematical community, to the extent it engaged with the question at all, acknowledged the claim and continued to say “Navier-Stokes.”
Names in science, like names on buildings, do not always go to those who did the most.
5.5 What the Convergence Means
The five independent derivations are not merely a curiosity. They carry mathematical and physical weight.
When one person derives an equation from one set of assumptions, the result might be an artifact of those assumptions. When five people, using different assumptions and different methods, arrive at the same equation, the likelihood that the result is a mere artifact becomes negligible. The convergence is evidence that the equations are not derived so much as revealed — that they are the inevitable mathematical description of viscous fluid motion under the constraint of Newtonian mechanics.
This kind of convergence is one of the strongest arguments in science. It does not prove the equations correct in any absolute sense — experiment does that — but it argues strongly that any correct equations for viscous flow would have to look like these. The structure is not arbitrary. The viscous term \(\mu \nabla^2 \mathbf{u}\) is the only linear, isotropic, second-order correction to the Euler equations that respects the basic symmetries of space and time. Once you decide that viscous effects are linear in the velocity gradient, and that space is isotropic, and that the equations are second-order in space, you have no choice.
Five men found the same door because there was only one door.
5.6 Summary
Between 1822 and 1845, five scientists independently derived the equations now called Navier-Stokes. The French quartet — Navier, Cauchy, Poisson, Saint-Venant — worked within the tradition of French rational mechanics. Stokes brought the Cambridge mathematical tradition to bear two years after Saint-Venant. Multiple independent discovery reflects the maturity of the intellectual tools and the pressure of industrial need. Saint-Venant’s claim to recognition was substantively reasonable and historically unsuccessful. The convergence of five independent derivations on the same mathematical structure is itself evidence that the equations are not an artifact of any particular derivation but a fundamental description of viscous flow.
5.7 Further Reading
- Darrigol, O. Worlds of Flow, Chapter 5. Covers all five derivations and the priority disputes in detail.
- Stokes, G.G. “On the theories of the internal friction of fluids in motion.” 1845. Contains Stokes’s own account of his predecessors.
- de Solla Price, D. Science Since Babylon. Yale University Press, 1961. The classic treatment of multiple independent discovery and the sociology of scientific progress.