Grimberg G., Pauls W., Frisch U. Genesis of d'Alembert's paradox and analytical elaboration of the drag problem (794387), страница 4

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Delà il s’ensuit que les arcs LD, DM ne sauroientRêtre égaux ; car s’ils l’étoient, alors la quantité − 2π ydy( p 2 +q 2 ) seroit égaleà zéro de manière que le corps ne souffriroit aucune pression de la part dufluide : ce qui est contre l’expérience.39 D’Alembert, 1752: xxxviii; Yushkevich and Taton, 1980: 312–314; Grimberg, 1998: 9.40 Euler, 1755, 1756.41 Saint-Venant, [1888], probably mostly written around 1846.42 Saint-Venant, [1888]: 35. Et il est évident que, lorsque le courant estsupposé indéfini ou très large, la théorie des Dilucidationes d’Euler ne peutêtre et n’est réellement qu’un retour pur et simple à la théorie vulgaire,43 Euler, 1760: 200. Quae ego etiam nuper in aliquot dissertationibus demotu fluidorum exposui, nullum subsidium huc afferunt.

Etiamsi enim omniamquae ad motum fluidorum pertinent, ad aequationes analyticas reduxi, tamenipsa Analysis minime adhuc ita est exculta, ut illis aequationibus resoluendissufficiat.44 Truesdell, 1954: C–CVII.45 Euler, 1760: 206 . . .

puppis nauis paecise tanta vi propelleretur, quantaprora repellitur. . . .1884G. Grimberg et al. / Physica D 237 (2008) 1878–18866. Borda’s memoir (1766)In his memoir Borda, a prominent French “Geometer” andexperimentalist, studies the loss of “live force” (energy) inincompressible flows, in particular in pipes whose section isabruptly enlarged.46 At the end of his memoir Borda givesan example of what would be, in his opinion, “a bad use” ofthe principle of conservation of live forces. This is preciselythe problem of determining the drag force that a moving fluidexerts upon a body at rest. The particles of the fluid in theneighborhood of the body “delineate curved lines or rathermove in small curved channels”; the pressure force actingupon the body has to be determined.

But the channels becomenarrower at certain locations similarly to a siphon, so that theprinciple of live forces cannot be used. To prove this point hethen presents the following argument for the vanishing of thedrag:. . . suppose that the body D moves uniformly through a quiescent fluid,driven by the action of the weight P. According to this principle [oflive forces], the difference of the live force of the fluid must be equal tothe difference of the actual descent of the weight; however, since themotion is supposed to have reached uniformity, the difference of thelive forces equals zero. Therefore, the difference of the actual descentis also zero, which cannot happen unless the weight P is itself zero. Asthe weight P measures the resistance of the fluid, the supposition of theprinciple [of live forces] necessarily leads to a vanishing resistance.47This constitutes the first derivation of the d’Alembertparadox using an energy dissipation argument.

Borda’sexplanation of why the live-force conservation argument isinapplicable rests on the aforementioned analogy with thesiphon problem. This is illustrated by a figure48 not reproducedhere because of its poor quality. There one sees a fillet offluid narrowing somewhat as it approaches the body. Themodern concept of dissipation in high-Reynolds-number flowbeing confined to regions with very strong velocity gradients isdefinitely not what Borda had in mind.Borda’s reasoning is correct, but like Euler in 1745 andd’Alembert in 1749, he does not formulate the vanishing ofthe drag as a paradox. In his remarks Borda addresses neitherthe question of the nature of the fluid, nor the consequences ofhaving stationary streamlines, nor the problem of the contactbetween the fluid and the body (absence of viscosity in thecase of ideal flow) which, as we know, are quite central to theunderstanding of the paradox.46 Borda, 1766.47 Borda, 1766: 604–605.

. . . supposons que le corps D se meuveuniformément dans un fluide tranquille, entraı̂né par l’action du poids P: onsait que suivant le principe, la différence de la force vive du fluide devra êtreégale à la descente actuelle du poids P; mais puisque le mouvement est censéparvenu à l’uniformité, la différence des forces vives = 0; donc la différence dela descente actuelle sera aussi = 0, ce qui ne se peut pas à moins que le poidsP ne soit lui-même = 0: or le poids P marque la résistance du fluide : donc lasupposition du principe dont il s’agit, donne toujours une résistance nulle.48 Borda, 1766: Figure 14, found at the end of the 1766 volume on p.

847.7. D’Alembert’s memoirs on the paradox (1768 and 1780)In Volume V of his “Opuscules” published in 1768, apart of a memoir is entitled “Paradox on the resistance offluids proposed to geometers.”49 D’Alembert considers againan axisymmetric body, but now with a head–tail symmetry.More precisely, he assumes a plane of symmetry perpendicularto the direction of the incompressible flow at large distance anddividing the body into two mirror-symmetric pieces. To avoidthe problem of possible separation of streamlines upstream anddownstream of the body, he assumes that the front part andthe rear part of the body have needle-like endings.

First of allhe asserts that the velocities at every location in the fluid areperfectly symmetric in front/rear of the body, and that. . . under this assumption the law of the equilibrium and theincompressibility of the fluid will be perfectly obeyed, because, therear part of the body being similar and equal to its front part, itis easy to see that the same values of p and q [i.e. the velocitycomponents] which will give at the first instant the equilibrium andincompressibility of the fluid at the front part will give the same resultsfor the rear part.

50This statement is directly related to the remark in Section 70of d’Alembert’s 1752 treatise. In fact, the assumption used byd’Alembert in 1749 and 1752 to avoid a paradox is here lifted,since no separation of streamlines occurs except at the needlelike end points. D’Alembert here assumes that the solution withmirror symmetry is the only one: “The fluid has only one wayto be moved by the encounter of the body.” The pressure forcesat the front and rear part of the body are then also axisymmetricand mirror symmetric. Hence they combine into a force ofresistance (drag) which vanishes. D’Alembert concluded:Thus I do not see, I admit, how one can satisfactorily explain bytheory the resistance of fluids. On the contrary, it seems to me that thetheory, developed in all possible rigor, gives, at least in several cases, astrictly vanishing resistance; a singular paradox which I leave to futureGeometers to elucidate.

51It is clear that d’Alembert’s argument is less general than thatof Borda, since he is restricting the formulation of the paradoxto bodies with a head–tail symmetry. Nevertheless, d’Alembertis the first one to seriously propose the vanishing of the drag as aparadox. Twelve years later in Volume VIII of his “Opuscules”d’Alembert revisits the paradox in the light of a letter receivedfrom “a very great Geometer” who is not named and whopoints out that, when considering the flow inside or around a49 D’Alembert, 1768. In the eighteenth century “Geometer” was frequentlyused to mean “mathematician” (pure or applied).50 D’Alembert, 1768: 133. .

. . dans cette supposition les loix de l’équilibre& de l’incompressibilité du fluide seront parfaitement observées; car la partiepostérieure étant (hyp.) semblable et égale à la partie antérieure, il est aisé devoir que les mêmes valeurs de p & de q; qui donneront au premier instantl’équilibre & l’incompressibilité du fluide à la partie antérieure, donneront lesmêmes résultats à la partie postérieure.51 D’Alembert, 1768: 138. Je ne vois donc pas, je l’avoue, comment on peutexpliquer par la théorie, d’une maniere satisfaisante, la résistance des fluides.ll me paroı̂t au contraire que cette théorie, traitée & approfondie avec toute larigueur possible, donne, au moins en plusieurs cas, la résistance absolumentnulle ; paradoxe singulier que je laisse à éclaircir aux Géometres [sic].G.

Grimberg et al. / Physica D 237 (2008) 1878–1886symmetric body, there may be, in addition to the symmetricsolution, another one which does not possess such symmetryand to which d’Alembert’s argument for the vanishing of theresistance does not apply.52 D’Alembert concurs and discussedthe issue at length. It should however be noted that a breaking ofthe symmetry was already assumed by him in his early work onthe resistance when he assumed that the (hypothetical) pointsof attachment and detachment of the streamline following thebody are not symmetrically located (see Section 4).Thus d’Alembert was definitely the first to formulate thevanishing of the drag as a paradox within the accepted modelof that time, namely incompressible fluid flow, implicitly takenas ideal.53 He was however formulating it only for bodies withhead–tail symmetry, not realizing that techniques introduced byEuler and Borda could have allowed him to obtain the paradoxfor bodies of arbitrary shapes.8.

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