Bernoullian influences on Leonhard Euler’s early fluid mechanics

Abstract

Today, Leonhard Euler receives most of the credit for developing the basic theory of fluids, and modern fluid mechanics is indeed descended from his original insights. However, fluid mechanics as originally conceived by Euler is strikingly different in appearance from today’s treatment of the subject. Here, we examine two of Euler’s earliest works in fluid mechanics – first, an extract from a letter to Johann Bernoulli, and second, an early manuscript published many years after it was written. From these works, we see the influence that Daniel and Johann Bernoulli had on the development of Euler’s ideas.

Acknowledgements

The author would like to thank Stacy Langton for his insights on the mathematics and Latin language from Euler’s 1739 letter to Johann Bernoulli.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Erik R. Tou http://orcid.org/0000-0001-6185-3857

Notes

1 The forthcoming geometric approach does not appear in Bernoulli’s 1727 paper on the subject (Bernoulli 1729).

2 The Latin word amplitudinum is rendered here simply as ‘amplitude’, though more precisely it can be thought of as cross-sectional area, or in simple cases, as the width of a layer.

3 Good summaries of this derivation can be found in (Mikhailov 2002, 51–55) or (Levi 1995, 257–263).

4 In particular, Bistafa (2015, 181–185) provides a good analysis of Johann Bernoulli’s developing theory of fluids and the problem of efflux as it appears in the Hydraulica, along with other key moments in early fluid theory.

5 In the eighteenth century, corresponding heights were often used as a way to represent quantities such as velocity and pressure.

6 This derivation, of which Euler only gives the final result, hinges on the fact that $1-2\mathrm{d}y/y$ is a first-order series approximation of $(1/\left(1+\mathrm{d}y/y\right){)}^2$, or in other words, $1-2\mathrm{d}y/y$ is obtained by eliminating higher-order differentials.

7 This work, an excerpt from a letter of Euler’s, is included in Carmody and Kobus’s English translation of the Hydraulica (Bernoulli 1743).

8 D’Alembert also expresses this opinion in his article ‘Hydrodynamique’ in the Encyclopédie.

9 Euler refers here to the same momentum principle that has already been discussed in the present work. An outworking of this principle can be found in his paper ‘Découverte d’un nouveau principe de méchanique’ (Euler 1752), which Euler presented to the Academy in September 1750, and which appeared in the Berlin Mémoires for 1752.

10 While the notion of incompressibility had been used in the science of fluids before this point (it was well known that a stratum’s speed would be inversely proportional to its amplitude), Euler’s use of a two-dimensional fluid packet makes this explicit.

11 Fontaine’s notation (for example, $\left(\mathrm{d}u/\mathrm{d}x\right)$) comes from a 1738 paper presented to the Paris Academy, and was discussed by Euler, Daniel Bernoulli, and Clairaut in their correspondence.

12 Truesdell (1964, XXXVIII) makes the further claim that neither Daniel Bernoulli (in the Hydrodynamica) nor d’Alembert (in the Traité) possessed a concept of internal pressure.

13 According to Eneström, this paper was presented to the Berlin Academy on 31 August 1752. The interested reader should look to Walter Pauls’ excellent English translation (see Euler 1761) for details.

14 The specific publication details for these papers: Principes généraux de l’état d’équilibre des fluides (E225, presented to the Berlin Academy on 11 October 1753), Principes généraux du mouvement des fluides (E226, presented to the Berlin Academy on 4 September 1755), and Continuation des recherches sur la théorie du mouvement des fluides (E227, presented to the Berlin Academy on 2 October 1755).

15 In (Euler 1761), Euler credits Fontaine for the new notation, though Fontaine’s 1738 paper did not appear in print until 1764.

16 The original notation in this paper is a blend of new and old; for example, l implicitly denotes $\left(\mathrm{\partial }u/\mathrm{\partial }y\right)$ while L is given explicitly as $\left(\mathrm{d}u/\mathrm{d}y\right)$. Euler himself did not use the partial derivative symbol $\mathrm{\partial }$ here; this was only introduced by Legendre in 1786.

17 Here, $R=\left(\mathrm{\partial }p/\mathrm{\partial }x\right)$, $r=\left(\mathrm{\partial }p/\mathrm{\partial }y\right)$, and $\mathrm{\Re }=\left(\mathrm{\partial }p/\mathrm{\partial }t\right)$. Note that Euler does not assume pressure will be constant over time.