Grimberg G., Pauls W., Frisch U. Genesis of d'Alembert's paradox and analytical elaboration of the drag problem (794387)
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Physica D 237 (2008) 1878–1886www.elsevier.com/locate/physdGenesis of d’Alembert’s paradox and analytical elaboration of the dragproblemG. Grimberg a , W. Pauls b,c , U. Frisch b,∗a Instituto de Matemática, Universidade Federal do Rio de Janeiro (IM-UFRJ), Brazilb Labor. Cassiopée, UNSA, CNRS, OCA, BP 4229, 06304 Nice Cedex 4, Francec Fakultät für Physik, Universität Bielefeld, Universitätsstraße 25, 33615 Bielefeld, GermanyAvailable online 20 January 2008AbstractWe show that the issue of the drag exerted by an incompressible fluid on a body in uniform motion has played a major role in the earlydevelopment of fluid dynamics. In 1745 Euler came close, technically, to proving the vanishing of the drag for a body of arbitrary shape; for thishe exploited and significantly extended the existing ideas on decomposing the flow into thin fillets; he did not however have a correct picture ofthe global structure of the flow around a body.
Borda in 1766 showed that the principle of live forces implied the vanishing of the drag and shouldthus be inapplicable to the problem. After having at first refused the possibility of a vanishing drag, d’Alembert in 1768 established the paradox,but only for bodies with a head–tail symmetry. A full understanding of the paradox, as due to the neglect of viscous forces, had to wait until thework of Saint-Venant in 1846.c 2008 Elsevier B.V. All rights reserved.PACS: 47.10.A-; 47.15.kiKeywords: History of science; Fluid dynamics; D’Alembert’s paradox1.
IntroductionThe first hint of d’Alembert’s paradox – the vanishing of thedrag for a solid body surrounded by a steadily moving idealincompressible fluid – had appeared even before the analyticaldescription of the flow of a “perfect liquid”1 was solidlyestablished. Leonhard Euler in 1745, Jean le Rond d’Alembertin 1749 and Jean-Charles Borda in 1766 came actually veryclose to formulating the paradox, using momentum balance(in an implicit way) or energy conservation arguments, whichactually predate its modern proofs.2 D’Alembert in 1768 wasthe first to recognize the paradox as such, although in asomewhat special case. Similarly to Euler and Borda, hisreasoning did not employ the equations of motion directly,∗ Corresponding author.
Tel.: +33 4 92003035; fax: +33 4 92003058.E-mail addresses: gerard.emile@terra.com.br (G. Grimberg),uriel@obs-nice.fr (U. Frisch).1 Kelvin’s name of an incompressible inviscid fluid.2 See, e.g. Serrin, 1959 and Landau and Lifshitz, 1987.c 2008 Elsevier B.V. All rights reserved.0167-2789/$ - see front matter doi:10.1016/j.physd.2008.01.015but nevertheless used a fully constituted formulation of thelaws of hydrodynamics, and exploited the symmetries hehad assumed for the problem. A general formulation ofd’Alembert’s paradox for bodies of an arbitrary shape wasgiven in 1846 by Adhémar Barré de Saint-Venant, who pointedout that the vanishing of the drag can be due to not taking intoaccount viscosity.
Other explanations of the paradox involveunsteady solutions, presenting for example a wake, as discussedby Birkhoff.3Since the early derivations of the paradox did not rely onEuler’s equation of ideal fluid flow, it was not immediatelyrecognized that the idealized notion of an inviscid fluid motionwas here conflicting with the physical reality.
The difficultiesencountered in the theoretical treatment of the drag problemwere attributed to the lack of appropriate analytical tools ratherthan to any hypothetical flaws in the theory. In spite of the greatachievements of Daniel and Johann Bernoulli, of d’Alembert3 Euler, 1745; D’ Alembert, [1749]; Borda, 1766; Saint-Venant, 1846, 1847;Birkhoff, 1950: Chap. 1, §9.1879G. Grimberg et al.
/ Physica D 237 (2008) 1878–1886and of Euler4 the theory of hydrodynamics seemed beset withinsurmountable technical difficulties; to the contemporaries itthus appeared of little help, as far as practical applications wereconcerned. There was a dichotomy between, on the one hand,experiments and the everyday experience and, on the other handthe eighteenth century’s limited understanding of the natureof fluids and of the theory of fluid motion. This dichotomy isone of the reasons why neither Euler nor Borda nor the earlyd’Alembert were able to recognize and to accept the possibilityof a paradox.We shall also see, how the problem setting became moreand more elaborated in the course of time.
Euler, in his earlywork on the drag problem appeals to several physical examplesof quite different nature, such as that of ships navigatingat sea and of bullets flying through the air. D’Alembert’s1768 formulation of the drag paradox is concrete, preciseand much more mathematical (in the modern sense of theword) than Euler’s early work. This is how d’Alembert wasable to show – with much disregard for what experiments or(sometimes irrelevant) physical intuition might suggest – thatthe framework of inviscid fluid motion necessarily leads to aparadox.For the convenience of the reader we begin, in Section 2, byrecalling the modern proofs of d’Alembert’s paradox: one proof– somewhat reminiscent of the arguments in Euler’s 1745 work– relies on the calculation of the momentum balance, the otherone – connected with Borda’s 1766 paper – uses conservationof energy.
In Section 3 we describe Euler’s first attempt, in1745, to calculate the drag acting on a body in a steady flowusing a modification of a method previously introduced byD. Bernoulli.5 In Section 4 we discuss d’Alembert’s 1749analysis of the resistance of fluids. In Section 5 we reviewEuler’s contributions to the drag problem made after he hadestablished the equations of motions for ideal fluid flow.Section 6 is devoted to Borda’s arguments against the use ofa live-force (energy conservation) argument for this problem.In Sections 7 and 8 we discuss d’Alembert’s and SaintVenant’s formulation of the paradox.
In Section 9 we give theconclusions.Finally, we mention here something which would hardly benecessary if we were publishing in a journal specialized in thehistory of science: the material we are covering has alreadybeen discussed several times, in particular by such toweringfigures as Saint-Venant and Truesdell.6 Our contributions canonly be considered incremental, even if, occasionally, wedisagree with our predecessors.2.
Modern approaches to d’Alembert’s paradoxLet us consider a solid body K in a steady potential flowwith uniform velocity U at infinity. In the standard derivationof the vanishing of the drag7 one proceeds as follows: Let4 See, e.g., Darrigol, 2005; Darrigol and Frisch, 2008.5 Bernoulli, 1736.6 Truesdell, 1954; Saint- Venant [1888].7 See, e.g., Serrin, 1959.Ω be the domain bounded in the interior by the body Kand in the exterior by a sphere S with radius R (eventually,R → ∞).
The force acting upon K is calculated by writinga momentum balance, starting from the steady incompressible3D Euler equationv · ∇v = −∇ p,∇ · v = 0.(1)The contribution of the pressure term gives the sum of the forceacting on the body K and of the force exerted by the pressureon the sphere S. It may be shown, using the potential characterof the velocity field, that the latter force vanishes in the limitR → ∞. The contribution of the advection term can be writtenas the flux of momentum through the surface of the domainΩ : the flux through the boundary of K vanishes because of theboundary condition v · n = 0; the flux through the surface ofS vanishes because the velocity field is asymptotically uniform(v ' U for R → ∞). From all this it follows that the force onthe body vanishes.
This approach proves the vanishing of boththe drag and the lift.8Alternatively, one can use energy conservation to show thevanishing of the drag.9 Roughly, the argument is that the workof the drag force, due to the motion with velocity U , should bebalanced by either a dissipation of kinetic energy (impossible inideal flow when it is sufficiently smooth) or by a flow to infinityof kinetic energy, which is also ruled out for potential flow. Thisargument shows only the vanishing of the drag.A more detailed presentation of such arguments may befound in the book by Darrigol.10In the following we shall see that many technical aspects ofthese two modern approaches were actually discovered aroundthe middle of the eighteenth century.3. Euler and the new principles of Gunnery (1745)In 1745 Euler published a German translation of Robins’book “New Principles of Gunnery” supplemented by a seriesof remarks whose total amount actually makes up the double ofthe original volume.
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