From Euler to Navier–Stokes: A Spatial Analysis of Conceptual Changes in Nineteenth-century Fluid Dynamics

Abstract

This article provides a spatial analysis of the conceptual framework of fluid dynamics during the nineteenth century, focusing on the transition from the Euler equation to the Navier–Stokes equation. A dynamic version of Peter Gärdenfors's theory of conceptual spaces is applied which distinguishes changes of five types: addition and deletion of special laws; change of metric; change in importance; change in separability; addition and deletion of dimensions. The case instantiates all types but the deletion of dimensions. We also provide a new view upon limiting case reduction at the conceptual level that clarifies the relation between the predecessor and successor conceptual framework. The nineteenth-century development of fluid dynamics is argued to be an instance of normal science development.

Acknowledgements

We consider this joint work, our names being listed in alphabetical order. We thank Peter Gärdenfors and Ulrich Gähde, three anonymous reviewers and the journal's editor, James W. McAllister, for comments that improved this manuscript. Graciana Petersen acknowledges funding from the Environmental Wind Tunnel Laboratory, University of Hamburg, and Frank Zenker from the Swedish Research Council (VR).

Notes

[1] Our conjectures are based on several observations. First, conceptual spaces have already proven useful for the description of scientific theories in other cases (Gärdenfors and Zenker 2011, 2013; Zenker and Gärdenfors 2014, forthcoming). Further, conceptual spaces are already used—alas, largely unconsciously—in many real-world applications; consultancies, for instance, apply a simplified version of conceptual spaces as a pragmatic tool to describe and solve complex problems, thus in theory choice. Finally, conceptual spaces benefit from their proximity to mathematics, thus enriching scientific communication and contributing to the learning landscape.

[2] Known as one of the millennium problems, the challenge is to prove the existence and smoothness of the NSE (Fefferman 2014).

[3] A practical criterion for identifying domains can thus be connected to the measurement procedures for the domains, for instance, in experimental research. But domains may also be identified mathematically, if only to ask later whether such pure distinctions can be supplied with feasible measurement methods.

[4] Strictly, forces need not be introduced separately; they ‘fall out’ according to Newton's F = ma. So forces do not form a domain in the sense of section 2.2.

[5] For Sneed, a T-theoretical dimension is one whose value cannot be determined without applying the theory T. Mass and force thus become T-theoretical dimensions in Newtonian mechanics (Sneed 1971).

[6] Standardly, the unification of microscopic particle dynamics with macroscopic fluid dynamics proceeds via a limiting case demonstration (assuming infinitely many particles), and has the microscopic theory collapse into the macroscopic one. So density is never completely unimportant, but under a macroscopic view one can disregard its complexity.

[7] Microscopic particle dynamics can be accounted for by the Boltzmann equation (Hirschfelder, Curtiss, and Bird 1965); also, the Einstein kinetic model or other stochastic processes may be considered for molecular motions (Einstein 1905; von Smoluchowski 1906; Erdös 2012). Moreover, a linear Boltzmann equation can be formulated that couples Boltzmann's particle distribution function with the so-called random jump process on the sphere of velocities. A key difficulty pertains to the combination of microscopic and macroscopic fluid scales in the collision of reversible and irreversible processes, since Hamiltonian mechanics (i.e. classic mechanics) is reversible in time and deterministic, while the Boltzmann equation is irreversible. The result is an inevitable loss of information at a macroscopic scale (Erdös 2012, 7).

[8] Such a reduction of spatial dimensionality, here from three dimensions to one, can simplify a problem and is often useful in computational modelling.

[9] Definition: let f(x) be a real-valued function of a real variable. Then f is odd if −f(x) = f(−x) for all x in R.

[10] Abbreviated: . Further, ‘x is an element in ℝ³’ means that x = (x₁, x₂, x₃) is a vector in ℝ³, i.e. x₁ is a real number, and so are x₂ and x₃.

[11] The link between conceptual spaces and manifolds is as follows: provided that G is diffeomorphic to ℝ¹¹, formulated within (x, t, ρ, F, Σ) cut out manifolds in ℝ¹¹.

[12] Recall that volume forces and surface forces act on average on the volume element. Here, averaging does not lead to a loss of accuracy since, according to the continuum hypothesis, the location of x was already only ‘somewhere in the volume’, but not at a specific (or fixed) location.

[13] Metaphorically, Lagrange sits in a boat following the motion of particles in the fluid, tracing their dynamic history. Euler stands in the river, at a fixed reference point, observing the velocity of particles passing through this point. The coordinate system, for instance, may be the riverbed.

[14] The term surface force may potentially mislead, since viscosity effects can be physically interpreted as effects of internal friction.

[15] To illustrate what is thus excluded: ‘[S]uspensions and solutions containing very long chain-like molecules may exhibit some directional preferences owing to alignment of these molecules in a manner which depends on the past history of the motion’ (Batchelor 1970, 143).

[16] Technically, one can add the boundary conditions of the NSE as domains to G_NSE since the former determine the solutions, should such exist. Also pressure, p, can be added as a quality dimension.

[17] With respect to their impact on a fluid volume, all are treated as long-range forces. Modern textbook treatments tend not to explain this change, but merely restate it when declaring such forces to act equally on all matter contained in a (given small) fluid volume, and to be proportional to the size of the volume element and the density of the fluid (see section 3.2).