Grimberg G., Pauls W., Frisch U. Genesis of d'Alembert's paradox and analytical elaboration of the drag problem (794387), страница 3
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Grimberg et al. / Physica D 237 (2008) 1878–1886being that the force caused by the deflection in the portion MDis not directed toward the body:supplemented. He observed that, in order to determine the dragon the body, one must first determineThe other part DM produces a force which is opposite to the first, andwould cause the body to move back in the direction BA. Now, as onlya true pressure [a positive one] can set a body into motion, the latterforce can only act on the body insofar as the pressure of the fluid matterfrom behind is strong enough to move the body forwards .27. .
. the pressure of the fillet of Fluid which glides immediately on thesurface of the body. For this it is necessary to know the velocity of theparticles of the fillet. 32Hence he departed from strict dynamical reasoning to follow adubious intuition of the transfer of force through the fluid.28To sum up, Euler performed a real tour de force by derivingthe correct expression for the force on a fillet of fluid withouthaving the equations of motion but practically he was notable to reach much beyond Newton’s impact theory whenconsidering the global interaction between the fluid and thebody.4.
d’Alembert and the treatise on the resistance of fluids(1749)In a treatise29 written for the prize of the Berlin Academyof 1749 whose subject was the determination of the drag aflow exerts upon a body, d’Alembert gives a description of themotion of the fluid analogous to that of Euler. It is not clearif d’Alembert knew about Euler’s “Commentary on Gunnery”.As noted by Truesdell,30 some figures in d’Alembert’s treatiseare rather similar to those found in the Gunnery but thereare also arguments in the Gunnery which would have allowedd’Alembert, had he been aware of them, to extend his 1768paradox to cases not possessing the head–tail symmetry hehad to assume.
Anyway, d’Alembert was fully aware of D.Bernoulli’s work on jet impact in which, as we already pointedout, a similar figure is found.In the treatise d’Alembert described the motion of anincompressible fluid in uniform motion at large distance,interacting with a localized axisymmetric body. He observedthat the streamlines and the velocity of the fluid at each pointin space are time-independent. The velocity a of the fluid farupstream of the body is directed along the axis of symmetry(which he takes for the abscissa); the other axis is chosen to beperpendicular to this direction. In this frame a point M of thefluid is characterized by the cylindrical coordinates (x, z) andthe corresponding velocity has the components avx and avz .31D’Alembert’s first aim is to derive the partial differentialequations which determine the motion of the fluid, and theappropriate boundary conditions with which they must be27 Euler, 1745: 268.
Aus dem andern Theil DM aber ensteht eine Kraft,welche jener entgegen ist, und von welcher des Körper nach der DirectionBA zurück gezogen werden sollte. Da nun kein Körper anders, als durch einenwürklichen Druck in Bewegung gesetzt werden kann, so kann auch die letztereKraft nur in so ferne auf den Körper würken, als der Druck der flüßigen Materievon hinten stark genug ist, den Körper vorwärts zu stossen.28 Darrigol, private communication, 2007.29 D’Alembert, [1749], 1752.30 Truesdell, 1954: LII.31 D’Alembert uses a similarity argument to prove that the velocity fieldaround a body of a given shape is proportional to the incoming velocity a(D’Alembert, [1749]: §42–43, 1752: §39).By considering the motion of fluid particles during aninfinitesimal time interval, d’Alembert is able to find theexpressions of the two components of the force acting on anelement of fluid:∂vz∂vz2γz = a −vx− vz,(10)∂x∂zand∂vx∂vxγx = a −vx.− vz∂x∂z2(11)From this d’Alembert derived for the first time the partialdifferential equations for axisymmetric, steady, incompressibleand irrotational flow, but he does not use such equations inconsidering the problem of “fluid resistance”.33How does d’Alembert calculate the drag? From anassumption about the continuity of the velocity he infers,contrary to Euler, that there must be a zone of stagnating fluidin front of the body and behind it, bordered by the streamlineTFMDLa which attaches to the body at M and detaches at L(see Fig.
4).34In his calculation of the drag d’Alembert used an approachwhich differed from that of Euler in the Gunnery: insteadof calculating the balance of forces acting on the fluid heconsidered the pressure force exerted on the body by the fluidfillet in immediate contact with it. D’Alembert noted first that,for each surface element of the body, the force exerted by thefluid particles is perpendicular to this surface, because of thevanishing of the tangential forces, characterizing the flow of anideal fluid.35In conformity with Bernoulli’s law, d’Alembert expressedthe pressure along the body as a 2 (1−vx2 −vz2 ). With ds denotingthe element of curvilinear length along the sections of the bodyby an axial plane such as that of Fig. 4, the infinitesimal elementof surface of revolution of the body upon which this pressure isacting is 2π zds.
The component along the axis of the pressureforce exerted is2πa 2 (1 − vx2 − vz2 )zdz.(12)Further integration along the profile AMDLC yields the verticalcomponent of the drag.Then came a very important remark. D’Alembert noted thatin the case of a body which is not only axisymmetric but has32 D’Alembert, 1752: xxxi.
. . . la pression du filet de Fluide qui glisseimmédiatement sur la surface du corps. Pour cela il est nécessaire de connoı̂trela vitesse des particules de ce filet.33 Cf. Truesdell, 1954: LIII, Grimberg, 1998: 44–46, Darrigol, 2005: 20–21.34 D’Alembert, [1749]: §39, 1752: §36.35 D’Alembert, [1749]: §40, 1752: §37. This vanishing, as we know,characterizes an ideal fluid; d’Alembert did not relate it to the nature of thefluid.G. Grimberg et al. / Physica D 237 (2008) 1878–18861883one year after Euler established his famous equations in theirfinal form.40 In his review of previous efforts to understand thedrag problem for incompressible fluids, Saint-Venant41 writesthe following about the Dilucidationes:And it is obvious that, when the flow is assumed indefinite or verybroad, the theory of the Dilucidationes can only be and actually is justa return to the vulgar theory, .
. . . 42Here, Saint-Venant understands by “vulgar theory” theimpact theory which goes back to the seventeenth century.Actually, in 1756 Euler was rather pessimistic regarding theapplicability of his equations to the drag problem:But the results which I have presented in several previous memoirson the motion of fluids do not help much here. Because, even thoughI have succeeded in reducing everything that concerns the motionof fluids to analytical equations, the analysis has not reached thesufficient degree of completion which is necessary for the solving ofsuch equations.43Fig.
4. Figure 14 of D’Alembert, [1749] redrawn. Not all elements shown hereare used in our arguments.a head–tail symmetry,36 the contributions to the drag from twosymmetrically located points would be equal and of oppositesign and thus cancel.37 In order to avoid the vanishing of thedrag, he assumed that the attachment point M and the separationpoint L are not symmetrically located:From there it follows that the arcs LD and DM cannot be equal;Rbecause, if they were, the quantity— 2π ydy( p 2 +q 2 ) would be equalto zero so that the body would not experience any force from the fluid:which is contrary to experiments.38This stress on “experiments”, already present in the 1749manuscript and which will not reappear in d’Alembert’s 1768paradox paper, seems to reflect just common sense.
It cannot beexplained by d’Alembert’s hypothetical desire to adhere to laterecommendations by the Berlin Academy which emphasizedcomparisons with experiments for the 1750 prize on resistanceof fluids. D’Alembert did not seem pleased with such latechanges and these recommendations were probably formulatedonly in May 1750.39D’Alembert’s new idea, compared to Euler, is to considerthe drag as the resultant of the pressure forces directed alongthe normal to the surface of the body over its entirety.
But ford’Alembert it is still unimaginable to obtain a vanishing drag.Truesdell discusses the Dilucidationes in detail.44 Actuallythis paper is quite famous because of a remark Eulermade on the cavitation that arises from negative pressure inincompressible fluids. Truesdell is also rightly impressed byEuler’s success in doing something non-trivial with his equationfor flows around a parabolic cylinder; for this Euler uses asystem of curvilinear coordinates based on the streamlines andtheir orthogonal trajectories.The Dilucidationes are however not contributing much toour understanding of drag.
In Section 15, Euler expresses hisdoubts regarding the applicability of his 1745 calculation toboth the front and the back of a body (which would result invanishing drag):. . . the boat would be slowed down at the prow as much as it would bepushed at the poop . . . .45We must mention here that, because of a possible nonvanishing transfer of kinetic energy to infinity, the moderntheory of the d’Alembert paradox does not apply to flow with afree surface, such as a boat on the sea.Thus, in the Dilucidationes we find a first attempt tointroduce a new analytical treatment of streamlines unrelatedto the previous theories and coming closer to the moderndescription of a fluid flow.
Nevertheless, Euler does not succeedin using his 1755 equations to improve our understanding of thedrag problem.5. Euler and the ‘Dilucidationes’ (1756)The Dilucidationes de resistentia fluidorum (Enlightenmentregarding the resistance of fluids) have been written in 1756,36 In d’Alembert [1752] this additional symmetry is explicitly assumed;in d’Alembert, [1749] the language used only suggests such a symmetry.37 D’Alembert, [1749]: §62, 1752: §70.38 D’Alembert, 1752: §70.
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