Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 9
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pp.1 12-13), he used the perpendicularity off + g - 'Y to the free surface of the fluid.In38The elimination of 1J leads to the easily integrable equation(gh - R2w2) dv, + gh d(sine)j sin e - R2wKsin 9d(sin9) = 0.THE DYNAMICAL EQUATIONS.
(&v"'.)-1..Ov<f>-1..e .+ vo sm'+'smv<P = w 8e cos '+' 8</> tan ()sin </>•19(1 .23)For the same reasons as before, d'Alembert identified these derivatives with the accelerations Ye and 'Y<t> · He then applied his dynamic principle to obtainLastly, he obtained the continuity condition'Y<t> = -gR sin e 8</>(1 .24)(1 .25)in which the modern reader may recognize the expression of a divergence in sphericalcoordinates.39D'Alembert judged the resolution of this system to be beyond his capability.
Thepurpose of this section of his memoir was to illustrate the power and generality 'of hismethod for deriving hydrodynamic equations. For the first time, he gave the completeequations of motion of an incompressible fluid in a genuinely bidimensional case. Thusemerged the velocity field and the corresponding partial derivatives with respect to twoindependent spatial coordinates. D' Alembert pioneered the application to the dynamics ofcontinuous media of the earlier calculus of differential forms by Alexis Fontaine andLeonhard Eu1er. His notation of course differed from the modern one. Where we nowwrite 8f/8x (following Gustav Kirchhoff), Fontaine wrote df/dx, and d'Alembert wroteA, with df = A dx + B dy + . .
. .1 .3.4 The resistance offluidsIn 1 749, d'Alembert competed for another Berlin prize on the resistance of fluids, andfailed: the Academy judged that none of the competitors had reached the point ofcomparing his theoretical results with experiments. D'Alembert did not deny the importance of this comparison for the improvement of ship design. However, he judged that therelevant equations cou1d not be solved in the near future, and that his memoir deservedconsideration for its methodological innovations. In 1752, he published the Iatin text andan augmented translation as a book.40Compared with the earlier treatise on the equilibrium and motion of fluids, the firstimportant difference was a new formu1ation of the laws of hydrostatics.
In 1 744, d'Alembert started with the uniform and isotropic transmissibility of pressure by any fluid (from39D'Alembert [1747] pp. 1 1 1-14 (equations E, F, G, H, I). To complete the correspondence given infootnote 35, take </> __, A, v.; __, 7), 'Yo __, 'IT, 'Y.; __, <p, gfR __, p, 8TJ/89 __, -p, 87]/84> __, -u, &u0/89 __, r,&uo/84> __, .\., &u.; 89 __, -y, &u.;/84> --+ {3.
D'Alembert has the ratio of two sines instead of the product in thelast term in each of eqns (1.22) and (1 .23). An easy, modem way to obtain these equations is to rewrite eqn.(1.21) as v = [("' x r) · 'V]v + "' x v, with v = (0, v0, v.;), r = (R, 0, 0), "' = w(sin 9 sin </>, cos 9 sin </>, cos 4>), and'il = (8" 8o/R, 8.;/Rsin9) in the local basis.40D'Alembert [1752] p. xxxvili. For an insightful study of d'Alembert's work on fluid resistance, cf. Grimberg[1998]. See also Calero [1996] Chap.
8.20WORLDS OF FLOWone part of its surface to another). He then derived the standard laws of this science, suchas the horizontality of the free surface and the depth dependence of wall pressure, byqualitative or geometrical reasoning. In contrast, in his new memoir he relied on amathematical principle borrowed from Alexis-Claude Clairaut's memoir of 1743 on theshape of the Earth. According to this principle, a fluid mass subjected to a force density fisin equilibrium if and only if the integral J f · di vanishes over any closed loop within thefluid and over any path whose ends belong to the free surface of the fluid.4 1D'Alembert regarded this principle as a mathematical expression of his earlier principleof the uniform transmissibility of pressure.
If the fluid is globally in equilibrium, hereasoned, then it must also be in equilibrium within any narrow canal of section sbelonging to the fluid mass. For a canal beginning and ending on the free surface of thefluid, the pressure exerted by the fluid on each of the extremities of the canal must vanish.According to the principle ofuniform transmissibility of pressure, the force f acting on thefluid within the length di of the canal exerts a pressure sf di that is transmitted to bothends of the canal (with opposite signs). As the sum of these pressures must vanish, so doesthe integral J f · dl.
This reasoning and a similar one for closed canals establish d'Alembert's new principle of equilibrium.42Applying this principle to an infinitesimal loop, d' Alembert obtained (the Cartesiancoordinate form of) the differential condition·\1Xf = 0,(1 .26)as Clairaut had already done. Combining it with his principle of dynamics, and confininghimself to the steady motion (&v/8t = 0, so that 'Y = (v \J)v) of an incompressible fluid,he obtained the two-dimensional, Cartesian coordinate version of·\1x [(v · \J)v] = 0,(1.27)which means that the fluid must formally be in equilibrium with respect to the convectiveacceleration.
D'Alembert then showed that this condition was met whenever \1 x v = 0.Confusing a sufficient condition with a necessary one, he concluded that the latterproperty of the flow held generally.43This property nonetheless holds in the special case of motion investigated by d'Alembert, that is, the stationary flow of an incompressible fluid around a solid body when theflow is uniform far away from the body (see Fig. 1 .8). In this limited case, d'Alembert gavea correct proof, of which a modernized version follows.44Consider two neighboring lines of flow beginning in the uniform region of the flow andending in any other part of the flow, and connect the extremities through a small segment.41D'Alembert [I 752] pp. 14-17.
On the figure of the Earth, cf. Todhunter [1873]. On Clairaut, cf. Passeron [1995].On Newton's and MacLaurin's partial anticipations ofC!airaut's principle, cf. Truesdell [1954] pp. XIV-XXII.42As is obvious to the modern reader, this principle is equivalent to the existence of a single-valued function (P)of which f is the gradient and which has a constant value on the free surface of the fluid.
The canal equilibriumresults from the principle of solidification, the history of which is discussed in Casey [1992].43D'Alembert [1752] art. 78. The modern hydrodynamicist may recognize in eqn (1 .27) a particular case of thevorticity equation. The condition \7 x v = 0 is that of irrotational flow.44For a more literal rendering of d'Alembert's proof, cf. Grimberg [1998] pp. 43-8.THE DYNAMICAL EQUATIONSTFig. 1.8.21pFlow around a solid body according to d'Alembert ([1752] plate).According to d'Alembert's principle together with the principle of equilibrium, the integralf (v · 'V)v · dr vanishes over this loop. Using the identity(v 'V)v = \7·G)v2 -vx (\7 x v),(1.28)this implies that the integral f (\7 x v) · (v x dr) also vanishes.
The only part of the loopthat contributes to this integral is that corresponding to the small segment joining the endpoints of the two lines of flow. Since the orientation of this segment is arbitrary, \7 x vmust vanish.D'Alembert thus derived the condition'V x v = O(1.29)from his dynamic principle . He also obtained the continuity condition'V · v = O(1.30)by requiring the constancy of the volume .of a given element of fluid during its motion.More exactly, he obtained the special expressions of these two conditions in the cylindrically-symmetric case and in the two-dimensional case .
In order to solve this system of twopartial differential equations in the two-dimensional case, he noted that the two conditionsmeant that the forms u dx + v dy and v dx-u dy were exact differentials. This propertyholds, he ingeniously noted, if and only if (u - iv) (dx + i dy) is an exact differential. Thismeans that u and -v are the real and imaginary parts of a ( holomorphic) function of thecomplex variable x + iy.
They must also be such that the velocity is uniform at infinity andWORLDS OF FLOW22tangent to the body along its surface. D'Alembert struggled to meet these boundaryconditions through power-series developments, to little avaiJ.45The ultimate goal of this calculation was to determine the force exerted by the fluid onthe solid, which is the same as the resistance offered by the fluid to the motion of a bodywith a velocity opposite to that of the asymptotic flow.46 D'Alembert expressed this forceas the integral of the fluid's pressure over the whole surface of the body.
The pressure isitself given by the line integral of - dvj dt from infinity to the wall, in conformance withd'Alembert's earlier derivation of Bemoulli's Jaw. This law still holds in the present case,because - dvjdt = -:-(v 'i7)v = - 'ii'( � ) Hence the resistance could be determined, if onlythe flow around the body was known.47In 1 749, d'Alembert did not know enough about this flow to reach definite conclusionson the resistance. A few years later, he realized that, for a head-tail symmetric body, asolution of his differential equations was possible in which the fluid velocity was the sameat the front and at the rear of the body (up to a sign change).
Bemoulli's Jaw gives zeroresistance for this solution, since the head pressure exactly balances the tail pressure. Asd'Alembert knew, his equations only admit one solution. Therefore, the flow is unique andsymmetric, and the resistance must vanish. D'Alembert concluded:48·if.Thus I do not see, I admit, how one can satisfactorily explain by theory the resistanceof fluids.
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