The derivation of Poiseuille’s law: heuristic and explanatory considerations

Abstract

This paper illustrates how an experimental discovery can prompt the search for a theoretical explanation and also how obtaining such an explanation can provide heuristic benefits for further experimental discoveries. The case considered begins with the discovery of Poiseuille’s law for steady fluid flow through pipes. The law was originally supported by careful experiments, and was only later explained through a derivation from the more basic Navier–Stokes equations. However, this derivation employed a controversial boundary condition and also relied on a contentious approach to viscosity. By comparing two editions of Lamb’s famous Hydrodynamics textbook, I argue that explanatory considerations were central to Lamb’s claims about this sort of fluid flow. In addition, I argue that this treatment of Poiseuille’s law played a heuristic role in Reynolds’ treatment of turbulent flows, where Poiseuille’s law fails to apply.

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Notes

  1. 1.

    This law is sometimes called the Hagen–Poiseuille law due to Hagen’s priority in discovering it. See Sect. 3 for some discussion.

  2. 2.

    Another episode of this sort is emphasized by Heidelberger: early in the twentieth century Prandtl “found a way to bring together the purely empirical engineering tradition of hydraulics and the purely theoretical mathematical tradition of rational mechanics as it had developed in the 18th century” (Heidelberger 2006, p. 50). See also (Darrigol 2005, ch. 7) for more on Prandtl as well as Darrigol (2008) for Darrigol’s discussion of the philosophical import of this history.

  3. 3.

    Lamb (1879), pp. 2–3. See Kundu and Cohen (2008), pp. 9–10 for a more thorough contemporary treatment.

  4. 4.

    Lamb (1879), p. 5.

  5. 5.

    See Darrigol (2005), pp. 135–140 for discussion of Stokes.

  6. 6.

    See Kundu and Cohen (2008), pp. 6–8, pp. 100–104 for some discussion.

  7. 7.

    See, e.g., the discussion of Regnault in Chang (2004, pp. 96–102).

  8. 8.

    See Kundu and Cohen (2008, pp. 302–303), Milnor (1989, pp. 11–17).

  9. 9.

    All page references to these papers are from the reprinted versions in Reynolds (1901).

  10. 10.

    See Langhaar (1951, p. 24) for a modern reconstruction of Reynolds’ reasoning. However Darrigol discusses the gap between modern dimensional analysis and Reynolds’ discussion, noting that the “legend” of Reynolds’ reasoning began with Stokes (Darrigol 2005, pp. 255–258).

  11. 11.

    Reynolds varied the viscosity by varying the temperature of the fluid.

  12. 12.

    This shift in Lamb’s treatment of the no slip boundary condition is noted by Day (1990).

  13. 13.

    Lamb also here notes the work by Maxwell on the kinetic theory of gases that informed the note to the 1879 edition that I discuss below in Sect. 5. This note, however, does not appear in the 1895 edition.

  14. 14.

    I am grateful to an anonymous referee for urging me to make clearer the links between this case and the accounts of explanation offered by Hempel and Kitcher.

  15. 15.

    See Hempel (1965, ch. 10, 12) for classic discussion as well as Salmon (1989) for some now classic objections.

  16. 16.

    See Bangu (2020) for another recent contribution to this debate.

  17. 17.

    The second sentence and the note are removed in the 1895 edition.

  18. 18.

    Lamb here notes that he is following Maxwell’s 1867 paper “On the Dynamical Theory of Gases”, which is reprinted in Maxwell (1890).

  19. 19.

    For some more general discussion of how idealized models can explain, see Bokulich (2017), Potochnik (2017) and Pincock (2021).

  20. 20.

    See Launder (2014) for some discussion of Lamb’s reactions to Reynolds’ theoretical innovations in an 1895 paper. Lamb emphasizes that the transition to turbulence is “the chief outstanding difficulty of our subject” (Lamb 1895, p. 572), and notes Reynolds’ proposal without endorsing it (Lamb 1895, p. 579). See Darrigol (2005, pp. 260–262) for some discussion.

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Acknowledgements

This case was central to a paper presented during a symposium at the Pacific Division Meeting of the American Philosophical Association in March 2016 and the Philosophy Colloquium at the Leibniz Hannover University in June 2016. I am grateful to both audiences for their feedback, especially Alisa Bokulich, Uljana Feest, Mathias Frisch and Michael Strevens. This paper was substantially revised for this special issue. I am very much indebted to the two anonoymous referees and the editors for their help in developing the paper for publication.

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This article belongs to the topical collection ”Explanatory and Heuristic Power of Mathematics”, edited by Sorin Bangu, Emiliano Ippoliti, and Marianna Antonutti.

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Pincock, C. The derivation of Poiseuille’s law: heuristic and explanatory considerations. Synthese (2021). https://doi.org/10.1007/s11229-021-03306-1

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Keywords

  • Fluid dynamics
  • Explanation
  • Idealization
  • Model