Darrigol O., Frisch U. «From Newton's mechanics to Euler's equations» (1123933), страница 6
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This reasoning, and a similarone for closed canals, establish d’Alembert’s new principle ofequilibrium.40Applying this principle to an infinitesimal loop, d’Alembertobtained (the Cartesian-coordinate form of) the differentialcondition∇ × f = 0,(24)as Clairaut had already done. Combining it with his principleof dynamics, and confining himself to the steady motion(∂v/∂t = 0, so that γ = (v · ∇)v) of an incompressible fluid,he obtained the two-dimensional, Cartesian-coordinate versionof∇ × [(v · ∇)v] = 0,(25)which means that the fluid must formally be in equilibrium withrespect to the convective acceleration.
D’Alembert then showedthat this condition was met whenever ∇ × v = 0. Confusing asufficient condition with a necessary one, he concluded that thelatter property of the flow held generally.41Fig. 7. Flow around a solid body according to D’Alembert (1752: plate 13).This property nonetheless holds in the special case of motioninvestigated by d’Alembert, that is, the stationary flow of anincompressible fluid around a solid body when the flow isuniform far away from the body (Fig. 7).
In this limited case,d’Alembert gave a correct proof of which a modernized versionfollows.42Consider two neighboring lines of flow beginning in theuniform region of the flow and ending in any other part ofthe flow, and connect the extremities through a small segment.According to d’Alembert’s Hprinciple together with the principleof equilibrium, the integral (v·∇)v·dr vanishes over this loop.Using the identity1 2(v · ∇)v = ∇v − v × (∇ × v),(26)2Hthis implies that the integral (∇ × v) · (v × dr) also vanishes.The only part of the loop that contributes to this integral is thatcorresponding to the little segment joining the end points ofthe two lines of flow.
Since the orientation of this segment isarbitrary, ∇ × v must vanish.D’Alembert thus derived the condition∇ ×v=039 D’Alembert, 1752: 14–17. On the early history of theories of the figure ofthe Earth, cf. Todhunter, 1873. On Clairaut, cf. Passeron, 1995. On Clairaut’sprinciple and Newton’s and MacLaurin’s partial anticipations, cf. Truesdell,1954: XIV–XXII.40 As is obvious to the modern reader, this principle is equivalent to theexistence of a single-valued function (P) of which f is the gradient and whichhas a constant value on the free surface of the fluid. The canal equilibriumresults from the principle of solidification, the history of which is discussed inCasey, 1992.41 D’Alembert, 1752: art. 78.
The modern hydrodynamicist recognizes in Eq.(25) a particular case of the vorticity equation. The condition ∇ × v = 0 is thatof irrotational flow.(27)from his dynamical principle. In addition, he obtained the(incompressibility) condition∇ ·v=0(28)by considering the deformation of a small parallelepiped offluid during an infinitesimal time interval. More exactly, he42 For a more literal rendering of d’Alembert’s proof, cf. Grimberg,1998: 43–48.O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869Fig.
8. D’Alembert’s drawing for a first proof of the incompressibilitycondition. He takes an infinitesimal prismatic volume NBDCC’N’B’D’ (upperfigure). The faces NBDC and N’B’D’C’ are rectangles in planes passingthrough the axis of symmetry AP; after an infinitesimal time dt the pointsNBDC have moved to nbdc (lower figure). Expressing the conservationof volume and neglecting higher-order infinitesimals, he obtains Eq. (29).From the 1749 manuscript in the Berlin-Brandeburgische Akademie derWissenschaften; courtesy Wolfgang Knobloch and Gérard Grimberg.obtained the special expressions of these two conditions in thetwo-dimensional case and in the axially-symmetric case. In thelatter case, he wrote the incompressibility condition as:dppdq+= ,dxdzz(29)where z and x are the radial and axial coordinates and p andq the corresponding components of the velocity.
D’Alembert’s1749 derivation (repeated in his 1752 book) is illustrated by ageometrical construction (Fig. 8).43In order to solve the system Eqs. (27) and (28) in the twodimensional case, d’Alembert noted that the two conditionsmeant that the forms udx + vdy and vdx − udy were exactdifferentials (u and v denote the velocity components alongthe orthogonal axes O x and O y). This property holds, heingeniously noted, if and only if (u − iv)(dx + idy) is anexact differential. This means that u and −v are the real andimaginary parts of a (holomorphic) function of the complexvariable x + iy.
They must also be such that the velocityis uniform at infinity and at a tangent to the body alongits surface. D’Alembert struggled to meet these boundaryconditions through power-series developments, to little avail.4443 It thus would seem appropriate to use “d’Alembert’s condition” whenreferring to the condition of incompressibility, written as a partial differentialequation.44 D’Alembert, 1752: 60–62. D’Alembert here discovered theCauchy–Riemann condition for u and −v to be the real and imaginarycomponents of an analytic function in the complex plane, as well as apowerful method to solve Laplace’s equation 1u = 0 in two dimensions.
In1761: 139, d’Alembert introduced the complex potential ϕ + iψ such that1865The ultimate goal of this calculation was to determine theforce exerted by the fluid on the solid, which is the same as theresistance offered by the fluid to the motion of a body with avelocity opposite to that of the asymptotic flow.45 D’Alembertexpressed this force as the integral of the fluid’s pressure overthe whole surface of the body.
The pressure is itself givenby the line integral of −dv/dt from infinity to the wall, inconformity with d’Alembert’s earlier derivation of Bernoulli’slaw. This law still holds in the present case, because −dv/dt =−(v · ∇)v = −∇(v2 /2). Hence the resistance could bedetermined, if only the flow around the body was known.46D’Alembert was not able to solve his equations and totruly answer the resistance question. Yet, he had achievedmuch on the way: through his dynamical principle and hisequilibrium principle, he had obtained hydrodynamic equationsfor the steady flow of an incompressible axisymmetrical flowthat we may retrospectively identify as the incompressibilitycondition, the condition of irrotational flow, and Bernoulli’slaw. The modern reader may wonder why he did not tryto write general equations of fluid motion in Cartesiancoordinate form.
The answer is plain: he was following anolder tradition of mathematical physics according to whichgeneral principles, rather than general equations, were appliedto specific problems.D’Alembert obtained his basic equations without recourseto the concept of pressure. Yet, he had a concept of internalpressure, which he used to derive Bernoulli’s law. Curiously,he did not pursue the other approach sketched in his theory ofwinds, that is, the application of Newton’s second law to a fluidelement subjected to a pressure gradient. Plausibly, he favoreda derivation that was based on his own principle of dynamicsand thus avoided the kind of internal forces he judged obscure.It was certainly well known to d’Alembert that hisequilibrium principle was nothing but the condition of uniformintegrability (potentiality) for the force density f.
If one thenintroduces the integral, say P, one obtains the equilibriumequation f = ∇ P that makes P the internal pressure! Withd’Alembert’s own dynamical principle, one then reaches theequation of motionf−ρdv= ∇ P,dt(30)(u − iv)(dx + idy) = d(ϕ + iψ). The real part ϕ of this potential is the velocitypotential introduced by Euler in 1752; its imaginary part ψ is the so-calledstream function, which is a constant on any line of current, as d’Alembertnoted.45 D’Alembert gave a proof of this equivalence, which he did not regard asobvious.46 D’Alembert had already discussed fluid resistance in part III of his treatiseof 1744. There, he used a molecular model in which momentum was transferredby impact from the moving body to a layer of hard molecules.
He believed,however, that this molecular process would be negligible if the fluid moleculeswere too close to each other – for instance when fluid was forced through thenarrow space between the body and a containing cylinder. In this case (1744:205–206), he assumed a parallel-slice flow and computed the fluid pressure onthe body through Bernoulli’s law. For a head-tail symmetric body, this pressuredoes not contribute to the resistance if the flow has the same symmetry. Afternoting this difficulty, d’Alembert invoked the observed stagnancy of the fluidbehind the body to retain only the Bernoulli pressure on the prow.1866O.
Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869which is nothing but Euler’s second equation. But d’Alembertdid not proceed along these lines, and rather wrote equations ofmotion not involving internal pressure.47We finally turn to Euler himself, for whom we shall besomewhat briefer than we have been with the Bernoullis andd’Alembert (whose papers are not easily accessible to theuntrained modern reader; not so with Euler). “Lisez Euler, lisezEuler, c’est notre maı̂tre à tous” (Read Euler, read Euler, he isthe master of us all) as Pierre-Simon Laplace used to say.48of rivers written around 1750–1751.
There he analyzed steadytwo-dimensional flow into fillets and described the fluid motionthrough the Cartesian coordinates of a fluid particle expressedas functions of time and of a fillet-labeling parameter (a partialanticipation of the so-called Lagrangian picture). He wrotepartial differential equations expressing the incompressibilitycondition and his new principle of continuum dynamics.Through a clever combination of these equations, he obtainedfor the first time the Bernoulli law along the stream lines of anarbitrary steady incompressible flow.
Yet he himself judged thathe had reached a dead end, for he could not solve any realisticproblem of river flow in this manner.514.1. Pressure4.2. The Latin memoirAfter Euler’s arrival in Berlin, he wrote a few articleson hydraulic problems, one of which was motivated by hisparticipation in the design of the fountains of Frederick’ssummer residence Sanssouci.
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