Some Philosophy of Fluid Mechanics

“Hydrodynamics and magnetohydrodynamics are fundamental, regardless of popular opinion.”
- Eugene N. Parker (1927-2022)
- Quoted in the retrospective by Stuart D. Bale, Science, 376: 461 (29 Apr 2022).
“The leitmotif, the ever recurring melody, is that two things are indispensable in any reasoning, in any description we shape of a segment of reality: to submit to experience and to face the language that is used, with unceasing logical criticism.”
- Richard von Mises (1883-1953)
- Mathematical Theory of Compressible Fluid Flow (New York: Academic Press, 1958; reprinted by Dover, 2004), epigraph.
“To begin with, we should bear in mind the inherent limitations of all mathematical analyses of physical events. First, although our aim is to formulate general rules of conduct for fluid in motion, it is the fluid itself and not our analysis which finally decides whether it will conform to those rules; the justification of an analysis, therefore, lies not in its apparent rigour nor in its elegance–though these things count for much–but in its agreement with practical observations. Next, such analysis can account only for those properties of a physical event whose existence our limited knowledge can conceive and which we may hope to observe and measure; its value will depend on the discernment with which we single out certain properties for special attention and on their natural significance. Last, the claim has never been made that the equations, which we construct to express the mutual dependence of these properties, exactly represent physical fact, and that any deviations from observed fact are due to defects in the observations; nor, on the other hand, can the infallibility which we thus deny to theory be claimed by experimental observation.”
- The authors of Incompressible Aerodynamics: An Account of the Theory and Observation of the Steady Flow of Incompressible Fluid past Aerofoils, Wings, and Other Bodies, edited by Bryan Thwaites (Oxford University Press, 1960; reprinted by Dover, 1987), Chapter 1.
- The contributors are listed as D. G. Ainley, L. F. Crabtree, M. B. Glauert, D. Kuchemann, M. J. Lighthill, W. A. Mair, E. C. Maskell, T. R. F. Nonweiler, R. C. Pankhurst, J. H. Preston, A. G. Smith, D. A. Spence, B. Thwaites, J. Weber, L. G. Whitehead, and L. C. Woods.
“One should not forget that the fluid equations do not have a fundamental nature, i.e., they are ultimately phenomenological equations and for this reason one ‘cannot ask too much from them’. In order to pose appropriate questions it is necessary to concentrate on the heuristic and phenomenological aspects of the theory….

“…the lack of existence and uniqueness theorems (in three space dimensions) had little or no practical consequences on research. Fearless engineers write gigantic codes that are supposed to produce solutions to the equations: they do not care the least (when they are conscious of the problem, which unfortunately seldom seems to be the case) that what they study are not the Navier-Stokes equations, but just the informatic code they produced. No one is, to date, capable of writing an algorithm that, in an a priori known time and within a prefixed approximation, will produce the calculation of any property of the equations solution following an initial datum and forces which are not ‘very small’ or ‘very special.’ Statements to the contrary are not rare, and they may even appear on the news: but they are wrong.

“It should not be concluded from this that engineers or physicists that work out impressive amounts of papers (or build airliners or reentry vehicles) on the ‘solutions’ of the Navier-Stokes equations are dedicating themselves to a useless, or risible, job. On the contrary, their work is necessary, difficult and highly qualified. It is, however, important to try to understand in which sense their work can be situated in the Galilean vision that wishes that the book of Nature be written in geometrical and mathematical characters.”
- Giovanni Gallavotti (b. 1941)
- Foundations of Fluid Dynamics (Berin: Springer, 2002), preface.
“In all specializations of mechanics, the foundations, that is, a precise definition of the relevant class of systems and the formulation of the physical principles and equations generally controlling the behavior, were known at an early stage. However, our ability to develop the consequences of these foundations in a fruitful manner has grown gradually from their inception to the present. This discrepancy between foundations and applicability is extreme in the case of fluid mechanics. In this field, the fundamental equations and principles were already known in the 1750s for inviscid fluids, and in the 1820s for viscous fluids. In the eighteenth and nineteenth century, a few problems of practical interest could be solved….

“However, two of the practically most important problems, fluid retardation (in pipes) and fluid resistance (to the motion of immersed bodies) remained essentially unsolved in the most common case of slightly viscous fluids (high Reynolds number). Progress in this domain awaited the boundary-layer theory….After generalization to turbulent boundary layers, it produced the long awaited, universal solutions of the retardation and resistance problems encountered by hydraulicians and aeronautic engineers.”

“The moral is that the Navier-Stokes equations, although they implicitly cover most cases of flow encountered in nature, remain useless for an enormous variety of concrete problems unless proper conceptual furniture is brought to them through specializtion, approximation, regionalization, and asymptotics. There are evident reasons for this complexity and richness: the infinite number of degrees of freedom, the drastic non-linearity of the fundamental equations and the ensuing instability of simple solutions. Physicists have gradually learned how to harness this complexity, although there are still many open questions, especially about turbulence. It was much easier to discover the fundamental equations–no offense to D’Alembert and Euler–than to discover their physical content in the application game.”
- Olivier Darrigol (b. 1955)
- “Geometry, mechanics, and experience: a historico-philosophical musing” (2022). European Journal for Philosophy of Science, 12:60.
“Fluid dynamics, including mechanics of liquids and gases, is a part of physics. We can divide the processes of any physical theory into four steps:
1. Placing observed data into general physical laws, assuming the uniformity of nature.
2. Transferring the physical laws into mathematical form, thus obtaining a logically consistent system of axioms which usually consists of a system of differential equations.
3. Drawing conclusions from these differential equations. This is a purely mathematical step.
4. Check on the results obtained in (3) by experiments.
“It is often assumed that the first two steps are completed for all branches of mechanics and that, therefore, mechanics at present offers only mathematical difficulties. This, however, is not so for certain facts in fluid dynamics (and elsewhere).

“The first two steps above have been carried out, as far as fluids are concerned, in two forms. First a theory of ‘perfect fluids’ has been developed, and then it was supplemented by a theory of ‘viscous fluids’. However, these theories do not explain all the known facts in fluid dynamics. They cover the explanation of so-called laminar flow. Here the particles travel along smooth curves, or the jet emanating from the opening of a tank. On the other hand, motion in larger channels, like a river, may look smooth as a whole, but each particle by itself has a complicated irregular motion. It fluctuates violently and may exhibit chaotic behavior. Such a motion is called turbulent.

“We may say that the first two steps of the physical investigation have been completed for laminar motion; the theory of perfect fluids was given by Euler about 1760, that for viscous fluids by Navier and by Stokes about 1850. In both cases, we have a definite system of partial differential equations. Now the third step consists of the solution of certain boundary value problems of these partial differential equations. This mathematical investigation is still going on. However, in the case of turbulent motion hardly the first step has been completed. Some general rules have been derived, but at present there is no complete and consistent set of axioms (principal equations) which would explain all particular problems of turbulent motion.”
- Richard von Mises (1883-1953) and Kurt O. Friedrichs (1901-1982)
- Fluid Dynamics (New York: Springer-Verlag, 1971), Introduction.

In the spirit of “unceasing logical criticism”, may I add that the above account by two mathematicians is grossly oversimplified to the point of mockery; a much-needed alternative perspective is given by an experimental physicist below.

“One believes that the basic physical laws governing the behaviour of fluids in motion are the usual well-known laws of mechanics–essentially conservation of mass and Newton’s laws of motion, together with, for some problems, the laws of thermodynamics. The complexities arise in the consequences of these laws. The way in which observed flow patterns do derive from the governing laws is often by no means apparent. A large theoretical and conceptual structure, built, on the one hand, on the basic laws and, on the other hand, on experimental observations, is needed to make the connection….

“Thus, for the most part, fluid dynamical problems are concerned with the behaviour, subject to known laws, of a fluid of specified properties in a specified configuration….Ideally one would like to be able to solve such problems through an appropriate mathematical formulation of the governing laws; the role of experiment would be to check that solutions so obtained do correspond to reality. In fact the mathematical difficulties are such that a formal theory often has to be supplemented or replaced by experimental observations and measurements. Even in cases where a fairly full mathematical description of a flow has been obtained, this has often been possible only after experiments have indicated the type of theory needed. The subject involves an interplay between theory and experiment. The proportion each contributes to our understanding of flow behaviour varies greatly from topic to topic.”
- David J. Tritton (1935-1998)
- Physical Fluid Dynamics, second edition (New York: Oxford University Press, 1988), Chapter 1.

These views are not contradicted by a pair of theoretical physicists:

“Fluid dynamics is an experimental discipline; much of our current understanding has come in response to laboratory investigations….Relatively few problems in fluid dynamics admit complete, closed-form, analytic solutions, so progress in describing fluid flows has usually come from introducing clever physical models and using judicious mathematical approximations. Semi-empirical scaling laws are also common, especially for engineering applications. In more recent years, numerical fluid dynamics has come of age, and in many areas of fluid dynamics, computer simulations are complementing and even supplanting laboratory experiments and measurements.”
- Kip S. Thorne (b. 1940) and Roger D. Blandford (b. 1949)
- Elasticity and Fluid Dynamics (Princeton University Press, 2021), introduction to Part V.

The last word belongs to a mechanical engineer-turned-oceanographer:

“As in other fields, our mathematical ability is too limited to tackle the complex problems of real fluid flows. Whether we are primarily interested in understanding the physics or in the applications, we must depend heavily on experimental observations to test our analyses and to develop insights into the nature of the phenomenon. Fluid dynamicists cannot afford to think like pure mathematicians.”
- Pijush K. Kundu (1941-1994)
- Fluid Mechanics, first edition (San Diego: Academic Press, 1990), Chapter 1.

Finally, I soundly reject the account given by one of the founders of theoretical hydrodynamics, as expressed below:

“From a single experiment on the pressure of fluids, we derive all the laws of their equilibrium and their movement.”
- Jean le Rond D’Alembert (1717-1783)
- Preliminary Discourse to the Encyclopedia of Diderot, Part 1 (Paris, 1751). Translated by Richard N. Schwab (University of Chicago Press, 1995).

Why do we believe the Navier-Stokes equations are relevant?

In light of OpenAI’s Sept. 8, 2026, discovery of a counterexample to the conjectured existence and smoothness of solutions to the 3-dimensional, incompressible Navier-Stokes equations, the very relevance of these equations to actual fluid dynamics, observationally and experimentally, could be questioned. Some age-old insights include the following.

“The theory of viscous flow is based upon a set of general equations of motion for an incompressible viscous fluid first given by Claud Louis Marie Henri Navier in a paper read before the French Academy of Sciences on March 18, 1822. These equations were derived later by S. D. Poisson, B. de Saint Venant and G. G. Stokes from different assumptions and were extended so as to be applicable to a compressible fluid. They were used by these writers to solve a number of problems. The general acceptance of these equations is due largely to the agreement with experiment of the theoretical laws of flow in capillary tubes and the motion of a fluid in the space between two circular cylinders, one of which rotates with uniform velocity. These two types of motion give two independent methods of determining a measure of the viscosity of a fluid and the fact that values determined by the two methods agree very closely may be regarded as a first quantitative check on the validity of the general equations.”

Harry Bateman (1882-1946)
“General physical properties of a viscous fluid”, (1932), Report of the Committee on Hydrodynamics, National Research Council of the National Academy of Sciences, Division of Physical Sciences, Bulletin 84 (Washington, D.C.), Part I, Chapter 3. Reprinted by Dover as Hydrodynamics by Hugh L. Dryden, Francis P. Murnaghan, and Harry Bateman.

Writing more generally about the theory of continuum mechanics (CM), of which the Navier-Stokes equations is a special case:

“As if to refute positivism, classical CM, such as it grew during the 18th and 19th centuries, was believed on reason rather than on experiment. In fact only a few exact solutions of the basic equations were known, chiefly because of the nonlinearities – a good example of poor testability and even poorer actual tests due to the underdeveloped state of the theory (due partly to the nonexistence of a general theory of nonlinear equations) rather than to the lack of experimental facilities….Then, even when exact solutions were available, the corresponding experimental data were often irrelevant to them: they concerned, say, the turbulent flow of a viscous liquid (usually water) endowed with some memory, whereas the formulas concerned the lamellar flow of a nonviscous and forgetful fluid….No wonder that CM was insufficiently confirmed by experiment. Finally the available measuring instruments were and still are too few and too coarse: the field did not attract experimental physicists. Yet people believed CM. Why?

“CM was believed to be true not only because some of its theorems had been confirmed but mainly because it was assumed (a) that CM was just an application of [particle mechanics] – a theory in turned believed to be completely true at least in the classical approximation, because it was endorsed by astronomy, and/or (b) that, being thoroughly mathematical in form, CM was analytic – rational – and consequently in no need of experimental test. We now know that the first belief is false; and a semantical analysis of CM shows that CM has an intended physical referent – an arbitrary body – whence it must have some factual truth value, even if not adequately known, in addition to being mathematically correct.

“The testability of CM has improved since the mid-1940’s with its own rapid maturation – growth of the body of theorems and deepening of the foundations – as well as with its connection with other fields of physics (e.g., thermodynamics) and with progress in experimental techniques. Yet the situation is still unsatisfactory not only because of the mathematical complexities of CM and the complexities of real situations – most of which must be ironed out in a theoretical treatment – but also because of some persistent misunderstandings concerning the functions of CM vis a vis experiments. The experimentalist – who in this field is often a research engineer – often thinks that CM is too far from reality for it does not cover all the details of the behavior of his pet material: he sometimes complains that CM is not specific enough and at other times that CM employs too many experimentally inaccessible (‘operationally meaningless’) functions….

“In recent years CM has grown both theoretically and experimentally to a point where its testability has become adequate….Yet it will probably turn out that its extension or domain of truth will be less than the one it claims. In fact CM does not seem to apply to an arbitrary body in an arbitrary state: for example it does not hold for superfluids such as liquid helium, which exhibits nonclassical properties – e.g. a quantized vorticity….Quantum fluids seem to require a quantum continuum mechanics and both this theory and CM can in principle, though not yet de facto, be based on atomic-statistical theories. Yet such a further foundation will not decrease the value of CM as a theory of bulk matter: CM will stay because it is largely insensitive to the particular atomic structure hypothesized to underlie it. Its weakness is its strength.”

Mario Bunge (1919-2020)
Foundations of Physics (New York: Springer, 1967), Chapter 3, Sec. 2.4.

Some Philosophy of Fluid Mechanics

An Invitation to Fluid Mechanics for Physicists

Some Philosophy of Fluid Mechanics

Why do we believe the Navier-Stokes equations are relevant?

Further reading

Disclaimers