The Status of Stokes’ Equations

All real fluids are viscous, but not all viscous fluids are real. A mathematician may construct a model of a fluid which makes provision for viscosity, but it remains an open question whether any real fluid satisfies the specifications of the model. Lamb was very clear on this matter. He raised it in connection with the derivation of Stokes’ equations. All such derivations must start from assumptions, and these typically involve simplifications. A few years before Stokes’ work, Navier in France had arrived at these same equations and so they are often known as the Navier-Stokes equations.23 Navier, however, worked from assumptions about the supposed forces operating between the particles that made up fluids. Stokes is generally considered to have improved on this account by finding a way to avoid speculating about the ultimate particles of a fluid. He treated a fluid as a continuum and confined himself to considering the tangential stresses and shear forces on the sides of a fluid element. This modification avoided Navier’s assumptions but inevitably introduced others. What laws were obeyed by the stresses and forces? Stokes made the assump­tion that there was a linear relation between the shear force and the rate of shear. Lamb was careful to point this out to the reader.24 The assumption of linearity, he said, was exactly that—an assumption. He hastened to add that the assumption was plausible and the success of the equations, where they had been tested empirically, gave every reason to believe it was correct.

It will be noticed that the hypothesis made above that the stresses. . . are linear functions of the rate of stress. . . is of a purely tentative character, and that although there is considerable a priori probability that it will represent the facts accurately in the case of infinitely small motions, we have so far no assurance that it will hold generally. It was however pointed out by Reynolds that the equations based on this hypothesis have been put to a very severe test in the experiments of Poiseuille and others. . . . Considering the very wide range of values over which these experiments extend, we can hardly hesitate to accept the equations in question as a complete statement of the laws of viscosity. (571)

Assumptions had been made, but it turned out that the assumptions were correct. The Stokes equations were not approximations in competition with other approximations. Evidence, said Lamb, shows that the equations are to be accepted as a “complete statement” of the laws of the real-world phenom­enon of viscosity. In a word, the Stokes equations were true.

Given the immense authority behind this judgment, it can be difficult to realize that it was not necessitated by the facts. Lamb could have drawn a different conclusion. He was making a methodological choice and did not have to choose as he did. Others adopted a different stance toward Stokes’ equations and the experimental evidence that Lamb cited.25 I illustrate this point by reference to the work of the applied mathematician Richard von Mises (fig. 5.4). As well as his broad literary and philosophical interests, von Mises made important contributions to aerodynamic theory by generaliz­ing the mathematical technique for creating aerofoil shapes by conformal

figure 5.4. Richard von Mises (1883-1953). A leading applied mathematician who worked extensively in fluid dynamics and aerodynamics, von Mises adopted an empiricist or “positivist” stance toward the equations of fluid dynamics and treated both the Euler and the Stokes equations as abstractions.

transformations.26 He also corresponded extensively with Prandtl about fluid dynamics and contributed to boundary-layer theory.27 Von Mises had lec­tured on aerodynamics to military aircrew in Berlin as early as 1913 and had himself learned to fly at Adlershof. Before the war von Mises held a chair at the University of Strassburg, where he published Elemente der technischen Hydromechanik (The elements of technical hydrodynamics).28 On the title page von Mises was styled as a Maschinenbau-Ingenieur, or “mechanical en­gineer.” During the Great War he returned to Vienna, served as a pilot and an instructor, and then worked on the design of a giant aircraft for which he had provided the wing profile.29 Toward the end of the war he published his military lectures in the form of a textbook, Fluglehre.30 The little book was warmly welcomed by Prandtl because it was written by someone who could handle both the scientific and the technical sides of the aeronautics.31

In 1909, in an article on the problems of technical hydromechanics, von Mises had made a proposal that was designed to rationalize the relation be­tween perfect fluid theory and the theory of viscous flow.32 He called it the “hydraulic hypothesis” and claimed that it was implicit in many of the practi­cal applications of hydrodynamics, even if it was not usually made explicit. Rather than emphasizing the fundamental difference between viscous and inviscid theory (for example, by saying that one referred to something real while the other referred to something unreal), the hydraulic hypothesis em­bodied the view that they were intimately connected. Von Mises still used the hypothesis many years later in his advanced textbook on aerodynamics, the Theory of Flight, first published in English in 1945.33

According to von Mises, so-called ideal fluids represent a process of av­eraging out the statistical fluctuations always present within real fluids. The implied relation between the ideal fluid and real fluid may be illustrated by an analogy. An element of an ideal fluid stands to the elements of a real fluid, that is, the molecules, in roughly the way that, say, the average taxpayer stands to the array of real taxpayers. The behavior of an element of perfect fluid mathematically encodes real information about a specified collection of real things, without itself constituting a further item in the collection. The only fluids are real fluids, just as the only taxpayers are real taxpayers. The concept of an ideal fluid is an instrument by which we talk about, reason about, and refer to real fluids. Indirectly, equations that involve ideal fluids have a real reference, just as statistical data about taxpayers have a real reference. The Euler equations capture the mean values of a statistically fluctuating reality.34 In one formulation von Mises put it like this:

the flow around an aerofoil in a wind tunnel is doubtless a turbulent flow of a viscous fluid. But if the small oscillations are disregarded, the remaining steady velocity values agree very well with those computed from the theory of perfect fluids. . . . The hydraulic hypothesis does not contend that the viscosity effects are negligible. On the contrary. . . the viscosity is responsible for the continual fluctuations or for the turbulent character of the motion. It is left undecided whether the instantaneous (fluctuating) velocities of the real fluid follow the Navier-Stokes equations or not. The hydraulic hypothesis states only that the mean velocity values satisfy, to a certain extent, the perfect-fluid equations. (84-85)

This wording comes from Theory of Flight and therefore dates from 1945, but it is entirely consistent with the original formulation of the hypothesis.

What, on this view, is the relation between Euler’s equations and Stokes’ equations? Von Mises’ answer is interesting. He insists that both are idealiza­tions. In neither case do their concepts have objects that are to be simply or directly identified with real fluids. Both have an indirect relationship. “It should be kept in mind that the ‘viscous fluid’ as well as the ‘perfect fluid’ are idealizations. In introducing the viscous fluid the presence of shearing stresses is admitted, and thus a broader hypothesis is used, which can be ex­pected to give a better approximation to reality. However, we are not entitled to call ‘real fluid’ what is still only an idealization” (76-77). The term “real fluid,” said von Mises, should only be used, “when reference is made to ob­served facts” (77). Real fluids are encountered in experiments and practical engineering. They always stand in contrast to the equations of the mathe­matician—a point that holds whether the equations describe a perfect or a viscous fluid. Both are idealizations, approximations, and constructions, and neither can be identified with reality.35 This was a very different position from that adopted by Lamb and his colleagues. Although Lamb acknowledged the idealization that entered into the construction of Stokes’ equations, he con­cluded that experiment had confirmed their truth. Such confirmation lifted the equations out of the realm of conjecture and put the stamp of reality on them. The idea that, formally, Stokes’ equations stood in the same relation­ship to real fluids as the equations of Euler would have blurred the funda­mental distinction that Lamb and his British colleagues wanted to make.

For certain purposes, some idealizations may be better than others. Von Mises acknowledged that, by taking into account the shearing stresses in the fluid, Stokes had offered a “broader hypothesis” and a “better approxima­tion” than that provided by a perfect fluid. This point must be handled with care. It is surely correct but it does not follow that Stokes’ equations will al­ways give a more accurate answer than that given by the Euler equations. It does not follow that a viscous fluid idealization will always outperform an inviscid idealization. Calling one a “better approximation” than the other may create a certain presumption to that effect, but, given that they are both idealizations, this should not be taken for granted. Von Mises’ own discus­sion of Poiseuille’s results provides a salutary reminder.

Poiseuille’s experiments concerned the uniform flow of a viscous fluid down a straight tube of circular cross section. The fluid will travel more quickly along the middle of the tube than it will closer to the perimeter, and it will be stationary on the walls of the tube itself. Of course, the fluid will have an average velocity, and this will depend on the pressure gradient. The velocity vector will always be parallel to the axis, so the flow is “laminar.” In these simple conditions it can be deduced from Stokes’ equations that the velocity will be distributed over the diameter of the tube in the form of a parabola. The shape of the parabola is determined by the result that the maxi­mum velocity, on the axis, turns out to be exactly twice that of the average velocity. Poiseuille established these facts experimentally, and it was Stokes’ ability to deduce them theoretically that Lamb cited as the grounds for the truth of his equations. But the deductions only hold good if the velocity of the flow is below a certain critical speed. Von Mises reported that, for air in a one-inch pipe, the critical speed is a little below 4 feet per second. Above that speed the analysis fails because the flow ceases to be laminar and becomes turbulent.

In turbulent flow the velocity distribution in the cross section of the pipe alters markedly. Instead of the parabolic distribution, a much flatter distri­bution prevails where the maximum is only a few percent higher than the average. Did this directly contradict Stokes’ equations? It remained unclear whether this behavior contradicted them or not. The relation to the equations could not be determined, and no one could predict the pattern of turbulent flow from them. But while the equations of viscous flow were no help, it was evident that the flat distribution looked strikingly similar to that predicted on the assumption of an inviscid fluid. A frictionless fluid would not adhere to the sides of the pipe, so the fluid there would not be retarded relative to that near the center. There would be no parabolic distribution of velocities but a uniform march forward on a straight front. And this is very nearly what happens in turbulent viscous flow. As von Mises put it in the Theory of Flight. “This uniform velocity distribution of the perfect fluid flow agrees much better with observations under turbulent conditions than the veloc­ity distribution of a laminar viscous flow” (83). The perfect fluid provided a better approximation to the complicated case of turbulent flow than did the equations of viscous laminar flow.

Nor did this superiority hold just for the case of a fluid in a pipe. Von Mises argued that it applied to other practically interesting flows such as those through curved channels and those with varying cross sections, and, as we have seen, to the flow in wind tunnels. “If the small fluctuations are disregarded and attention is given only to the average values at each point, there appears a marked resemblance to the irrotational flow pattern of a per­fect fluid. The mean values of the velocity are distributed very much like the instantaneous velocities in a perfect fluid” (84).

The hydraulic hypothesis was a particular expression of a more general view that von Mises adopted toward the state of mechanics in the early de­cades of the twentieth century. His understanding of both the Euler equations and the Stokes equations brought them into line with his views on probability theory and his understanding of the modern scientific picture of the world— “das naturwissenschaftliche Weltbild der Gegenwart.” He was impressed by current developments in quantum theory and understood them to mean that behind the differential equations of classical mechanics there lay a reality governed by statistical rather than causal laws.36

Whatever one makes of the hydraulic hypothesis and the ultimate indeter­minism of physical laws, the essential point that von Mises was making about the Stokes equations still holds good. Even if von Mises’ statistical interpreta­tion of the Euler equations were to be rejected, his claim that both the Euler and Stokes equations were idealizations would not be directly threatened and could be defended on independent grounds. The point was forcefully made by the American mathematician Garrett Birkhoff. In his book Hydrodynam­ics: A Study in Logic, Fact and Similitude,31 Birkhoff lists molecular dissocia­tion and ionization at hypersonic speeds, chemical kinetics, and sound atten­uation as some of the physical effects not covered by the equations. He also noted that “the first supersonic wind-tunnels were plagued by condensation shocks due to water vapor in the air—another ‘hidden variable’ ignored by the metaphysics of Navier and Stokes” (31). Impressive though they are, the Stokes equations are not a complete statement of the laws of viscosity. They should be seen, as von Mises saw them, as idealizations covering a very in­complete range of phenomena in a very partial manner.

I now put these divergent responses to perfect fluids, d’Alembert’s para­dox, and Stokes’ equations into context. I identify two divergent traditions of mathematical work, one British, the other German. The tradition with the strong boundary between ideal and real fluids might be called “Cambridge – style mathematical physics.” The other, with the weaker boundary between the real and the ideal, is the tradition of technical mechanics as it was devel­oped in the German system of technical colleges. I start with a characteriza­tion of the Cambridge approach and, again, take Horace Lamb as my refer­ence point.