Abstract
This article makes a case for Euler as the first discoverer of what has come to be known as d’Alembert’s paradox. Suppose a body is immersed in an unbounded fluid and moves with constant velocity relative to the fluid, which is otherwise undisturbed: d’Alembert’s paradox asserts that, contrary to experimental evidence, the fluid exerts no drag force on the body (in the direction opposite to the body’s motion) if the fluid is inviscid and incompressible. Euler demonstrates this, for a two-dimensional body or an axisymmetric body whose axis aligns with its motion, in his extensive 1745 commentary on New Principles of Gunnery, a book published in 1742 by Benjamin Robins. After a rigorous analysis, Euler recognizes that the absence of a drag force conflicts with experience for fluids like air and water, and he uses Robins’ experiments with musket balls to explain this anomaly as a consequence of greater fluid pressure fore of the body than aft of it, due to a corresponding fore–aft asymmetry in the density of the fluid. Essentially, he resolves the apparent paradox by removing the assumption of the fluid’s incompressibility.
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Notes
- 1.
I have used Euler’s (1745) commentary only to find the German original of some key terms translated by Brown and to use Euler’s original notation, as Brown uses Newton’s notation for fluxions and fluents.
- 2.
Euler’s convention for dealing with velocity in his work on ballistics requires an explanation: a body starting at rest and falling freely from a height H acquires a speed \(V=\sqrt {2gH}\), where g is the acceleration of gravity; ignoring coefficients, as Euler often does, or taking \(g=\frac {1}{2}\), gives \(V=\sqrt {v}\) when H = v; sometimes, he lets g = 1 or includes in g both gravity and the buoyancy of the fluid.
- 3.
Euler’s notation for roots and parenthesized expressions has been slightly modernized here.
- 4.
Differentiating V 2 = 2gv with respect to time t and using Equation (2) produces \(mac^2\frac {dV}{dt}=-2vnc^2g\).
- 5.
Like Newton, Euler realized that a body moving steadily in an otherwise quiescent fluid is equivalent to a uniform stream (far away from a body) impinging steadily on a stationary body. Figure 3 describes the latter situation.
- 6.
In deriving Equation (8), Euler lets the “weight of the [fluid] be expressed by [its] bulk” or volume.
- 7.
In the context of d’Alembert’s paradox for which a stationary body is immersed in a steady flow that tends to a uniform stream as the distance from the body to a point in the flow approaches infinity, a canal starting at a point A far upstream of the body must inevitably arrive at a point D far downstream of the body where its velocity is the same as it was at A, as both A and D lie in a uniform stream.
- 8.
From this, it follows that the arcs LD, DM (Fig. 13) will not be equal; for if they were, …the body would suffer no resistance from the fluid: this is contrary to experience.
- 9.
I have modernized d’Alembert’s notation here.
- 10.
By Euler’s reckoning, the height of Earth’s atmosphere (regarded as a homogenous air mass of constant density) is 29100 Rhenish feet; a Rhenish foot equals roughly 31.38 cm, cf. an English foot is about 30.48 cm. Euler finds the speed of 1348 Rhenish feet/s simply using \(\sqrt {2gh}\), where g is the acceleration of gravity; Truesdell (1954, p. XLI) points out that this “escape speed” is actually incorrect, as it fails to account for the heat capacity ratio of air.
- 11.
Truesdell (1954, p. XXXIX) points out that “it is Robins (1742) who first suggested cavitation, …as a partial explanation of resistance.” Indeed, Robins writes
that if a Globe sets out in a resisting medium, with a Velocity much exceeding that with which the Particles of the Medium would rush into a void Space, in consequence of their Compression, so that a Vacuum is necessarily left behind the Globe in its Motion, the Resistance of this Medium to the Globe will be near three times greater, in Proportion to its Velocity, than what we are sure, from Sir Isaac Newton, would take Place in a slower Motion. (Robins 1742, pp. 73–74)
- 12.
Titi ES (2018/01/10) The Navier-Stokes, Euler and Related Equations. Joint Mathematics Meetings, San Diego.
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Acknowledgements
The author gratefully acknowledges that all the images used herein were scanned by and are used at the courtesy of, the University of Calgary, Military Museums Library and Archives.
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Hackborn, W.W. (2018). Euler’s Discovery and Resolution of D’Alembert’s Paradox. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90983-7_3
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