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).

Some Philosophy of Fluid Mechanics

An Invitation to Fluid Mechanics for Physicists

Some Philosophy of Fluid Mechanics

Further reading

Disclaimers