Grimberg G., Pauls W., Frisch U. Genesis of d'Alembert's paradox and analytical elaboration of the drag problem (794387), страница 5
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Saint-Venant and the first precise formulation of theparadox (1846)In three notes published in 1846 and then in a memoirpublished in 1847, Saint-Venant gives for the first time ageneral formulation of the paradox. A detailed write-up, mostlydating from the same period, was published only posthumouslyin 1888 and contains also a very interesting discussion ofprevious work.54 Saint-Venant’s memoir marks the beginningof the modern theory of the d’Alembert paradox which wasto flourish, in particular with major contributions by LudwigPrandtl.55We here give only a very brief description of the keyresults of Saint-Venant. He first specified the properties ofthe incompressible fluid: the pressure force is normal to thesurface element on which it is acting and therefore equal inall directions.
The fluid moves steadily around a body at rest.He gives a derivation of the paradox, closely related to Borda’s.Indeed, it suffices to establish the equation for the live forcesacquired by the fluid to see that the live-force (energy) loss ofthe system is zero:If the motion has reached, as one always assumes, a steady state, thelive force acquired by the system at every instant is zero; the workperformed by the exterior pressures is also zero and the same applies tothe work of the interior actions of the fluid whose density is assumed tobe unchanging.
Thus, the work of the impulse of the fluid on the body,and, consequently, the impulse itself, is necessarily equal to zero. 5652 D’Alembert, 1780: 212; Birkhoff, 1950: 21–22.53 The idea of viscosity ripened only in the XIXth century, see e.g. Darrigol,2005; in the eighteenth century there was only a concept of tenaciousness,e.g. resistance to the introduction of a body into fluid, which was still a longway from actual viscosity.54 Saint-Venant, 1846, 1847, [1888].55 Cf., e.g. Darrigol, 2005: Chap. 7.56 Saint-Venant, 1847: 243–244.
Si le mouvement est arrivé, comme on lesuppose toujours, à l’état de permanence, la force vive, acquise à chaque instantpar le système, est nulle ; le travail des pressions extérieures est nul aussi, et ilen est de même du travail des actions intérieures du fluide dont nous supposonsque la densité ne change pas.
Donc le travail de l’impulsion du fluide sur lecorps, et, par conséquent, cette impulsion elle-même, est nécessairement zéro.1885He adds that the situation is different for a real fluid made ofmolecules in which there is friction at the contact between twoneighboring fluid elements:But one finds another result if, instead of an ideal fluid – object ofthe calculations of the geometers of the last century – one uses a realfluid, composed of a finite number of molecules and exerting in itsstate of motion unequal pressure forces or forces having componentstangential to the surface elements through which they act; componentsto which we refer as the friction of the fluid, a name which has beengiven to them since Descartes and Newton until Venturi.57Thus, d’Alembert’s paradox is explained by Saint-Venant forthe first time as a consequence of ignoring viscous forces.
Ofcourse, a precise formulation of the paradox would not havebeen possible without a clear distinction between ideal andviscous fluids.9. ConclusionThe problem of the resistance of bodies moving in fluidswas – and still is – of great practical importance. It was thusnaturally one of the first non-trivial problems tackled withinthe nascent eighteenth century hydrodynamics.
Euler, who wasnot only a great “Geometer” but a person acutely aware ofthe needs of gunnery and ship building, tried – as we haveseen – reaching beyond the old impact theory of Newton—and failed. He was lacking both the concept of viscous forcesand a deep understanding of the global aspects of the topologyof the flow around a body.
His “failure” – as is frequentlythe case with major scientists – was however very creative:born was the idea of analyzing a steady flow into a set offluid fillets of infinitesimal and non-uniform section; he alsomanaged to calculate the forces acting on such fillet severalyears before there was any representation of the dynamics interms of partial differential equations.
Borda, being both aGeometer and an experimentalist, felt compelled to qualify asnon-sensical a very simple live-force argument discovered byhimself and which predicted a vanishing drag for bodies ofarbitrary shape. D’Alembert, another brilliant Geometer, wasprobably less constrained by experimental considerations, anddared eventually to present the paradox known by his name. Hisproof reveals a very good understanding of the global topologyof the flow but otherwise is very simple and limited intrinsicallyto bodies with a head–tail symmetry.We must stress that the statement as a paradox is very muchtied to the type of analytical representation of an ideal flow.From this point of view, experiments on flow past bodies,be they real or thought experiments, have rather been anobstacle to grasping the distinction between an ideal fluid anda real one.
The same kind of epistemological obstacle has57 Saint-Venant, 1847: 244. Mais on trouve un autre résultat si, au lieu dufluide idéal, objet des calculs des géomètres du siècle dernier, on remet unfluide réel, composé de molécules en nombre fini, et exercant dans l’état dumouvement, des pressions inégales ou qui ont des composantes tangentiellesaux faces à travers desquelles elles agissent; composantes que nous désignonspar le nom de frottement du fluide, qui leur a été donné depuis Descartes etNewton jusqu’à Venturi.1886G. Grimberg et al. / Physica D 237 (2008) 1878–1886accompanied the earlier birth of the principle of inertia, whichno experiment could at that time truly reveal; it was necessary todistance oneself from real conditions and to find an appropriatemathematical representation.
Finding such representations forfluid dynamics was a painfully slow process: a full centuryelapsed between Euler’s fragmentary results on drag and SaintVenant’s full understanding of the d’Alembert paradox.AcknowledgmentsOlivier Darrigol has been a constant source of inspirationto us while we investigated the issues discussed here. We alsothank Gleb K. Mikhailov for numerous remarks and RafaelaHillerbrand and Andrei Sobolevskii for their help.ReferencesBernoulli Daniel, 1736 ‘De legibus quibusdam mechanicis, quas naturaconstanter affectat, nondum descriptis, earumque usu hydrodynamico, prodeterminanda vi venae aqueae contra planum incurrentis’, Commentariiacademiae scientarum imperialis Petropolitanae, 8, 99-127. Alsoin Bernoulli, 2002, 425–444.Bernoulli Daniel, 1738 Hydrodynamica, sive de viribus et motibus fluidorumcommentarii, Strasbourg, 425–444.Bernoulli Daniel, 2002 Die Werke von Daniel Bernoulli, vol.
5, ed. Gleb K.Mikhailov, Basel.Birkhoff Garrett, 1950 Hydrodynamics, Princeton.Borda Jean-Charles de, 1766 ‘Sur l’écoulement des fluides par les orifices desvases.’ Académie Royale des Sciences (Paris), Mémoires [printed in 1769],579–607.D’Alembert Jean le Rond, [1749] Theoria resistentiae quam patitur corpus influido motum, ex principiis omnino novis et simplissimis deducta, habitaratione tum velocitatis, figurae, et massae corporis moti, tum densitatis& compressionis partium fluidi; manuscript at Berlin-BrandenburgischeAkademie der Wissenschaften, Akademie-Archiv call number: I-M478.D’Alembert Jean le Rond, 1752 Essai d’une nouvelle théorie de la résistancedes fluides, Paris.D’Alembert Jean le Rond, 1768 ‘Paradoxe proposé aux géomètres sur larésistance des fluides,’ in Opuscules mathématiques, vol.
5 (Paris), MemoirXXXIV, Section I, 132–138.D’Alembert Jean le Rond, 1780 ‘Sur la Résistance des fluides,’ in Opusculesmathématiques, vol. 8 (Paris) Memoir LVII, Section 13, 210–230.Darrigol Olivier, 2005 Worlds of flow: A history of hydrodynamics from theBernoullis to Prandtl, Oxford.Darrigol Olivier, Frisch Uriel, 2008 ‘From Newton’s mechanics to Euler’s equations,’ these Proceedings. Also at www.oca.eu/etc7/EE250/texts/darrigolfrisch.pdf.Euler Leonhard, 1745 Neue Grundsätze der Artillerie, aus dem englischen desHerrn Benjamin Robins übersetzt und mit vielen Anmerkungen versehen[from B. Robins, New principles of gunnery (London,1742)], Berlin. Alsoin Opera ommia, ser.
2, 14, 1-409, [Eneström index] E77. This wasretranslated into English as The true principles of gunnery investigated andexplained. Comprehending translations of Professor Eulers observationsupon the new principles of gunnery, published by the late Mr. BenjaminRobins, and that celebrated authors Discourse upon the track describedby a body in a resisting medium, inserted in the memoirs of the Royalacademy of Berlin for the year 1753. To which are added, many necessaryexplanations and remarks, together with Tables calculated for practice, theuse of which is illustrated by proper examples; with the method of solvingthat capital problem, which requires the elevation for the greatest rangewith any given initial velocity, by Hugh Brown, London, printed for I.Nourse, Bookseller to His Majesty 1777, E77A.Euler Leonhard, 1755 ‘Principes généraux du mouvement des fluides,’Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires11 [printed in 1757], 274–315.
Also in Opera omnia, ser. 2, 12, 54–91,E226.Euler Leonhard, 1756 ‘Dilucidationes de resistentia fluidorum’ [written in1756]. Novi Commentarii academiae scientiarum imperialis Petropolitanae, 8 [contains the Proceedings for 1755 and 1756, printed in 1763],197–229. Also in Opera omnia, ser. 2, 12, 215–243, E276.Euler Leonhard, 1760 ‘Recherches sur le mouvement des rivières,’ [writtenaround 1750–1751] Académie Royale des Sciences et des Belles-Lettresde Berlin, Mémoires 16 [printed in 1767], 101–118. Also in Opera omnia,ser.
2, 12, 272–288, E332.Grimberg Gérard, 1998 D’Alembert et les équations aux dérivées partielles enhydrodynamique, Thèse. Université Paris 7.Landau Lev Davydovich, Lifshitz Evgeniı̆ Mikhailovich, 1987 FluidMechanics, 2nd Edition, Oxford.Saint-Venant Adhémar Barré de, 1846 Société Philomathique de Paris, Extraitsdes procès verbaux (1846), 25–29, 72–78, 120–121.Saint-Venant Adhémar Barré de, 1847 ‘Mémoire sur la théorie de la résistancedes fluides.
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