Grimberg G., Pauls W., Frisch U. Genesis of d'Alembert's paradox and analytical elaboration of the drag problem (794387), страница 2
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In the third remark of the first proposition(Dritte Anmerkung zum ersten Satz) of the 2nd Chapter Eulerattempts to calculate the drag on a body at rest surrounded by asteadily moving incompressible fluid.11In 1745 the general equations governing ideal incompressible fluid flow were still unknown. Nevertheless, Euler managedthe remarkable feat of correctly calculating the force acting onan element of a 2D steady flow around a solid body. For this, aswe shall see, he borrowed and extended the results obtained byD.
Bernoulli a few years earlier.128 The lift need not vanish if there is circulation.9 See, e.g., Landau and Lifshitz, 1987: §11.10 Darrigol, 2005: Appendix A.11 For the German original of the third remark, cf. Euler, 1745: 259–270(of Opera omnia which we shall use for giving page references); anEnglish version, taken from Hugh Brown’s 1777 translation is available atwww.oca.eu/etc7/EE250/texts/euler1745.pdf. We shall sometimes use our owntranslations.12 Bernoulli, 1736, 1738.1880G. Grimberg et al. / Physica D 237 (2008) 1878–1886Fig. 1. Figure 14 of Euler, 1745: 263: this figure represents a fillet of fluidaAMm, deflected by the solid body, but the shape of the body is not fullyspecified.Euler begins by noting that instead of calculating the dragacting on a body moving in a fluid one can calculate the dragacting on a resting body immersed in a moving fluid.
Thus, heconsiders a fluid moving into the direction AB13 (cf. Figs. 1 and3), past a solid body CD.14 Then Euler continues by describingthe motion of fluid particles and establishes a relation betweenthe trajectory and velocity of each fluid particle and the forcewhich is acting on this particle. He observes that, instead ofdetermining the force on the body, one can evaluate the reactionon the fluid:But since all parts of the fluid, as they approach the body, are deflectedand change both their speed and direction [of motion], the body has toexperience a force of strength equal to that needed for this change inspeed as well as direction of the particles.15Thus, one has to determine the force which is applied ateach point of the fluid. Euler chooses a fillet16 AaMm of fluidwith an infinitesimal width and observes that the velocity17 vof the particles passing through the section Mm is inverselyproportional to its (infinitesimal) width Mm = δz; so thatv δz = v0 δz 0 , where δz 0 = Aa and v0 are the width of the filletand the velocity at the reference point A.18 For later referencelet us call this relation mass conservation.
Euler assumes thatthe particles passing through the section Aa follow the filletAaMm. This is equivalent to assuming that the velocity in eachsection Mm along the trajectory depends only on the location13 Here, contrary to the usage in Eulers’ memoir, all geometrical points willbe denoted by roman letters, leaving italics for algebraic quantities.14 These are Euler’s own words; examination of various of his figures and ofthe scientific context shows that the body extends below CD and, perhaps alsoabove.15 Euler, 1745: 263.
Weil aber alle Theile der flüßigen Materie, so bald sichdieselben dem Körper nahen, genöthiget werden auszuweichen, und so wohlihre Geschwindigkeit, als ihre Richtung zu verändern, so muß der Körper eineeben so große Kraft empfinden, als zu dieser Veränderung so wohl in derGeschwindigkeit, als der Richtung der Theilchen, erfordert wird.16 Euler uses the word “Canal” (channel).17 Following early eighteenth century notation, Euler represents a velocity bythe corresponding height of free fall to √achieve the given velocity, starting arest; in modern notation this would be 2gh. In the 1745 paper Euler takesmostly g = 1 – but occasionally g = 1/2 – and denotes the height by v.
Inorder not to confuse the reader, we shall here partially modernize the notationand in particular denote the velocity by v.18 Euler denotes our δz, δz and v by z, a and √2b, respectively.00Fig. 2. Figure 1 of Bernoulli, 1736. A centripetal argument is used to calculatethe normal force acting along a fillet of fluid represented here just by the curveBD (changes in width are ignored).of the point M and not on time, in modern terms a stationaryflow. Here the concepts of streamline and of stationarity in twodimensions appear for the first time explicitly.With the above assumption, Euler definesA P = x,p = dy/dx,P Q = dx,P M = y,O N = dy,qqM N = dx 2 + dy 2 = dx 1 + p 2 .(2)Since the force exerted by the body on the fluid is orientedupward, we prefer orienting the vertical axis upward.
Hencey and p will be negative in what follows. Otherwise we shallmostly follow Euler’s notation. Euler calculates the normal andtangential components, dFN and dFT , of the infinitesimal forceacting on the element of fluid fillet MNnm (see Fig. 1).19With the assumed unit density, the mass of fluid in MNnm isqδz × M N = δzdx 1 + p 2 .(3)The normal acceleration dFN in the direction MR iscalculated by Euler as a centripetal acceleration, i.e., given bythe product of the square of the velocity v 2 and the curvature3(1 + p 2 ) 2 d p/dx. Euler may here be following D. Bernoulli.20The latter, in a paper concerned among other things with jetsimpacting on a plane, had developed an analogy between anelement of fluid following a curved streamline and a point masson a curved trajectory (cf. Fig.
2). Multiplying the accelerationby the elementary mass and using mass conservation,21 Eulerthen obtainsdFN = v0 δz 0 vd p/(1 + p 2 ),(4)in which the velocity v along the fillet is considered to be afunction of the slope p.19 The notation dF and dF is ours.NT20 Bernoulli, 1736 and 1738: Section XIII, §13.21 Bernoulli, 1738: 287 assumed a fillet of uniform width (fistulamimplantatam esse uniformis quidem amplitudinis) and thus did not use massconservation to relate the varying width and velocity.G.
Grimberg et al. / Physica D 237 (2008) 1878–18861881To obtain the tangential force dFT in the direction mS on theelement of fillet, Euler writesqq22δzdx 1 + p d(v /2) = −dFT dx 1 + p 2 ,(5)and thusdFT = −δzd(v 2 /2) = −δz 0 (v0 /v)d(v 2 /2).(6)For the case of Fig. 1 the force is oriented in the direction mS,because the fluid is moving more slowly at N than at M. Eulerdoes not elaborate on how he derives (5) but this seems typicallya “live-force” argument of a kind frequently used at that time,for example by the Bernoullis.22 Indeed the l.h.s. is the variationof the live force (kinetic energy) and the r.h.s.
is what we wouldnow call the work of the tangential force per unit mass.So as to later determine the drag, that is the force on the bodyin the vertical direction, Euler adds these normal and tangentialelementary forces, projected onto the vertical axis oriented inthe direction BA. He thus obtains the following elementaryvertical force on the fluid:!vd ppdv.(7)dFBA = v0 δz 0+p31 + p2(1 + p 2 ) 2Here a “miracle” happens: the r.h.s. of (7) is the exactdifferential of!vp.(8)v0 δz 0 p1 + p2Finding the exact form of the function v( p), as we now know,requires the solution of a non-trivial boundary value problem.The exact form does however not matter for the integrabilityproperty and – from a modern perspective – can be related to theglobal momentum conservation property of the Euler equation.In 1745 Euler did not comment on the miracle.
It is worthstressing that it does not survive if any error is made regardingthe numerical factors appearing in the normal and tangentialforces.Euler is now able to exactly integrate the elementary forcealong a fillet from its starting point A, assumed to be farupstream p( p = −∞), to a point m with a finite slope p. Notingthat − p/ 1 + p 2 is the cosine of the angle MSB, he obtainsthe following force on the body, due to the fillet:v2FAB = −v0 δz 0 1 −cos MSB .(9)v0Note that this is a force from a given fillet of infinitesimalwidth which must still be integrated over a set of filletsencompassing the whole fluid.
More important here is whereto terminate the fillet. It is clear that the relevant fillets start farupstream in the vertical direction; but where do they lead afterhaving come close to the solid body? Euler considers variouspossibilities, such as a 90◦ deflection. He then envisages a veryinteresting case:22 Cf., e.g., Darrigol, 2005: Chap. 1.Fig.
3. Figure 15 of Euler, 1745: 268 from which he tries to explain that thedrag should be calculated using only the portion AM of the fillet.It remains therefore only to fix upon the point which is to be esteemedthe last of the canal. If we go so far that the fluid may pass by the body,and attain its first direction and velocity then shall δz = δz 0 , and theangle mSB vanish, and therefore its cosine = 1, then shall the forceacting on the body in the direction AB = −v02 δz 0 (1 − 1) = 0, and thebody suffers no resistance.
23From a technical point of view Euler’s 1745 derivation ofthe vanishing of the drag force has many features of the modernproof. However Euler refuses here to see a paradoxical propertyof the model of ideal fluid flow (for which the equations are noteven completely formulated). He accepts the possibility that thevanishing of the drag applies to certain exotic fluids which are“infinitely fluid . . . and also compressed by an infinite force”24such as the hypothetical ether surrounding celestial bodies(called by him “subtle heavenly material”), but he firmly rejectsit for water and air.
Indeed, immediately after the previouscitation he writes:Hence it appears, that for air or water, we are not to take the point of thecanal for last, where the motion behind the body corresponds exactlywith that at the beginning of the canal. 25Euler then explains why in his opinion the “last point”should not at all be taken far downstream, but rather near theinflection point M where the angle MSB achieves its maximumvalue, as shown in Fig. 3.26 As pointed out to us by OlivierDarrigol, in Euler’s opinion the portion AM of the canal AD isthe only one that exerts a force on the body, the alleged reason23 Euler, 1745: 267. Hier kömmt es also nur darauf an, wo das Ende desCanals angenommen werden soll. Geht man so weit, biß die flüßige Materie umden Körper völlig vorbey geflossen, und ihren vorigen Lauf wiederum erlangethat, so wird .
. . , und der Winkel mSB verschwindet, dahero der Cosinusdesselben = 1 wird. In diesem Fall würde also die auf den Körper nach derDirection AB würkende Kraft . . . und der Körper litte gar keinen Wiederstand.24 Euler, 1745: 268–269. . . . unendlich flüßig . . . und von einer unendlichenKraft zusammen gedruckt . . .25 Euler, 1745: 267. Woraus erhellet, daß man für Wasser und Luft nichtdenjenigen Punkt des Canals, wo die Bewegung hinter dem Körper mit derersten wiederum völlig übereinkommt, für den letzten annehmen könne.26 Truesdell, 1954: XL writes that “Euler supposes that the oncoming fluidturns away from the axis, leaving a dead-water region ahead of the body”;actually, Euler does not assume any dead-water region in his Third Remark.1882G.
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