×

From Euler to Navier-Stokes: a spatial analysis of conceptual changes in nineteenth-century fluid dynamics. (English) Zbl 1361.01006

The article offers a rational reconstruction of the development of fluid dynamics in the 19th century, studying in particular the transition from the Euler equation (the core of fluid mechanics) to the Navier-Stokes equation (the core of fluid dynamics). The reconstructive tools employed in the article are a type of meta-framework called “conceptual space”. Broadly speaking, every empirical theory is formulated in the background of a conceptual framework, which can then modelled as a conceptual space. Conceptual spaces have their own topology: they are sets endowed with dimensions which in their turn possess a geometrical structure. Conceptual spaces can therefore be studied as mathematical objects (phase spaces). For instance, the conceptual space in classical mechanics is formed by mass, space and time (three dimensions), while force constitutes an “integral domain” (i.e., it is a dimension which depends on variations within other dimensions). Moreover, mass and time differ in their geometrical structure: the former varies over non-negative real numbers, whereas the latter can vary over the full real-number line. The authors then distinguish five types of changes of scientific theories which can be modelled as changes within or of conceptual spaces: (i) addition and deletion of special laws; (ii) change of scale or metric; (iii) change in the importance of dimensions; (iv) change in the separability of dimensions; (v) addition or deletion of dimensions. Only the last change can be identified with a Kuhnian change of paradigm. After these theoretical preliminaries, the authors study how the conceptual space of fluid dynamics changed in the transition from Euler to Navier-Stokes. Their study reveal that the shift from Euler fluid dynamics to Navier-Stokes fluid dynamics contains types of change (i)–(iv) but does not contain any type change (v). Hence, this development is “conservative in the strong sense”: in Kuhnian language, it does not involve any change of paradigm. In conclusion: “The transition from the Euler equation to the Navier-Stokes equation may be viewed as an instance of normal science”. The paper is interesting especially insofar it uses tools from mathematics to study philosophical or historical problems, like the continuity or the change between scientific theories. On these grounds, it also gives a classification of possible changes within normal science. It will be interesting to complement this article with a more historically-oriented study, in order to see whether the change from the Euler equation to the Navier-Stokes equation can be also classified, from a historical point of view, as an example of change within normal science.

MSC:

01A55 History of mathematics in the 19th century
00A30 Philosophy of mathematics
76-03 History of fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Batchelor G., An Introduction to Fluid Dynamics (1970) · Zbl 0958.76001
[2] Batterman R., The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence (2002) · Zbl 1092.00006
[3] DOI: 10.1093/0195171276.001.0001 · doi:10.1093/0195171276.001.0001
[4] Darrigol O., Worlds of Flow: A History of Hydrodynamics from Bernoulli to Prandtl (2005) · Zbl 1094.76002
[5] DOI: 10.1002/andp.19053220806 · JFM 36.0975.01 · doi:10.1002/andp.19053220806
[6] DOI: 10.1093/acprof:oso/9780199652495.003.0001 · doi:10.1093/acprof:oso/9780199652495.003.0001
[7] DOI: 10.1007/978-94-017-1185-2 · doi:10.1007/978-94-017-1185-2
[8] Gärdenfors P., Conceptual Spaces: The Geometry of Thought (2000)
[9] Gärdenfors P., Philosophy of Science Meets Belief Revision Theory pp 137– (2011)
[10] DOI: 10.1007/s11229-011-0060-0 · Zbl 1284.03054 · doi:10.1007/s11229-011-0060-0
[11] DOI: 10.1016/S1355-2198(01)00038-7 · doi:10.1016/S1355-2198(01)00038-7
[12] Hirschfelder J. O., Molecular Theory of Gases and Liquids (1965) · Zbl 0057.23402
[13] Krantz D. H., Foundations of Measurement (1971)
[14] Kuhn T. S., The Structure of Scientific Revolutions, 2. ed. (1970)
[15] DOI: 10.1017/CBO9780511621123 · Zbl 0373.02002 · doi:10.1017/CBO9780511621123
[16] DOI: 10.1016/0016-0032(71)90160-8 · doi:10.1016/0016-0032(71)90160-8
[17] Maddox W. T., Multidimensional Models of Perception and Cognition pp 147– (1992)
[18] DOI: 10.1016/0016-0032(81)90475-0 · Zbl 0461.01005 · doi:10.1016/0016-0032(81)90475-0
[19] DOI: 10.1016/S0166-4115(08)61782-3 · doi:10.1016/S0166-4115(08)61782-3
[20] DOI: 10.1016/0039-3681(71)90042-2 · doi:10.1016/0039-3681(71)90042-2
[21] Roche J., The Mathematics of Measurement: A Critical History (1998) · Zbl 0913.00012
[22] DOI: 10.1002/andp.19063261405 · JFM 37.0814.03 · doi:10.1002/andp.19063261405
[23] DOI: 10.1007/978-94-010-3066-3 · doi:10.1007/978-94-010-3066-3
[24] Stegmüller W., The Structuralist View of Theories (1976) · Zbl 0411.03002
[25] DOI: 10.1126/science.103.2684.677 · Zbl 1226.91050 · doi:10.1126/science.103.2684.677
[26] Suppes P., Representation and Invariance of Scientific Structures (2002) · Zbl 1007.03003
[27] Tokaty G. A., A History and Philosophy of Fluid Mechanics (1971) · Zbl 0853.76001
[28] DOI: 10.1007/s10670-013-9582-9 · Zbl 06536024 · doi:10.1007/s10670-013-9582-9
[29] Zenker F., Applications of Geometric Knowledge Representation · Zbl 1318.68041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.