Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 6
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Although he had been the most ardent supporter of Leibniz'sprinciple of live forces, he now regarded this principle as an indirect consequence of morefundamental laws of mechanics. His aim was to base hydraulics on an incontrovertible,Newtonian expression of these laws. To this end he adapted a method that he had inventedin1 714 to solve the paradigmatic problem of the compound pendulum.1 .2.1 TranslationConsider again the pendulum of Fig. 1.1. According to Johann Bernoulli, the gravitationalforce msg acting on B is equivalent to a force(b/a)msg acting on A, because, according toFig.
1 .5.Effects of the velocity dependenceof pressure according to Daniel Bernoulli([1738] plate).15D. Bernoulli [1738] pp. 12 (eddies), 124 (enlarged conduit), 13 (imperfect fluid).10WORLDS O F FLOWthe law of levers, two forces that have the same moment have the same effect. Similarly, the'accelerating force' msM of the mass B is equivalent to an accelerating force(b/a)msM = ms(b/aiaiJ at A.
Consequently, the compound pendulum is equivalent toa simple pendulum with a mass mA + (b/a?ms located on A and subjected to the effectivevertical force mAJ5 + (b/a)msg. It is also equivalent to a simple pendulum of length(crmA + b2ms)/(amA + bms) oscillating in the gravity g, in conformance with Huygens'result. In summary, Johann Bemoulli reached the equation of motion by applying Newton's second law to a fictitious system obtained by replacing the forces and the momentumvariations at any point of the system with equivalent forces and momentum variations atone point of the system. This replacement, based on the laws of equilibrium of the system,is what Bemoulli called 'translation' in the introduction to his Hydraulica.
1 6Now consider the canonical problem of water flowing by parallel slices through avertical vessel of varying section (see Fig. 1.2). Johann Bemoulli 'translates' the weightgSdz of the slice dz of the water to the location z 1 of the frontal section of the fluid. Thisgives the effective weight S1gdz, because, according to a well-known law of hydrostatics, apressure applied at any point of the surface of a confmed fluid is uniformly transmitted toany other part of the surface of the fluid. Similarly, Bemoulli translates the 'acceleratingforce' (momentum variation) (dv/dt)Sdz of the slice dz to the frontal section of the fluid,with the result(dv/dt)SJ dz . He then obtains the equation of motion by equating the totaltranslated weight to the total translated accelerating force:(1.7)zozo1 .2.2 GorgeFor Johann Bemoulli the crucial point was the determination of the acceleration dv/dt.Previous authors, he contended, had failed to derive correct equations of motion from thegeneral laws of mechanics because they were only aware of one contribution to theacceleration of the fluid slices, namely, that which corresponds to the instantaneouschange of velocity at a given height z, or &v/8t in modem terms.
They ignored theacceleration due to the broadening or to the narrowing of the section of the vessel,which Bemoulli called gurges (gorge). In modem terms, he identified the convectivecomponent v&v/ 8z of the acceleration. Note that his use of partial derivatives was onlyimplicit: due to the relation v =(So/S)vo, he could split v into a time-dependent factor voand a z -dependent factor So/S, and thus express the total acceleration as(So/S)(dvo/dt) -(ifoSff/S3)(dS/dz ).1716J.
Bernoulli [1714], [1742] p. 395. In modem terms, Johann Bemoulli's procedure amounts to equating thesum ofmoments of the applied forces to the sum ofmoments of the accelerating forces (whlch is the time derivativeof the total angular momentum). Cf. Vilain [2000] pp. 448-50.17J. Bernoulli [1742] pp. 432-7. He misleadingly called the two parts of the acceleration the 'hydraulic' and the'hydrostatic' components. Truesdell's translation of gurges ([1954] p. XXXlll) as 'eddy' seems inadequate(although it does have this meaning in classical Iatin), because Bernoulli only meant the velocity differencebetween successive layers of the fluid. In his treatise on the equilibrium and motion of fluids ([1744] p. 1 57),d'Alembert interpreted J.
Bernoulli's expression of the acceleration in terms of two partial differentials.11THE DYNAMICAL EQUATIONSThanks to thegurges, Johann Bernoulli successfully applied eqn (I.7 ) to various cases ofefflux and retrieved his son's results.18 He also offered a novel approach to the pressure ofa moving fluid on the sides of its container. This pressure, he asserted, was simply thepressure, orvis immaterialis, that contiguous fluid parts exerted on one another, just astwo solids in contact act on each other:19The force that acts on the side of the channel through which the liquid flows ...
isnothing but the force that originates in the force of compression through whichcontiguous parts of the fluid act on one another.Accordingly, Bernoulli divided the flowing mass of water into two parts separated bythe sectionz={Following the general idea of 'translation', the pressure that the upperpart exerts on the lower part is(1 .8)More explicitly, this is!:'zozoII- -8t I vdz.8t = g(t' - zo) --2 v2(C) + -if(zo)28v8z IOv-dzI dz- I v -dz!:P(() =gzof)!:(1 .9)zoJohann Bernoulli thus obtained (in a widely different notation) a generalization of hisson's law to unsteady parallel-slice flow.Z0Johann Bernoulli interpreted the relevant pressure as aninternal pressure analogous tothe tension of a thread or the mutual action of contiguous solids in connected systems.
Yethe did not rely on this new concept of pressure to establish the equation of motion(1. 7). Heonly introduced this concept as a shortcut to the velocity dependence of wall pressure.Z11.3. D'Alembert's fluid dynamics1.3.1 The principle ofdynamics1 743, the French geometer and philosopher Jean le Rand d'Alembert published hisinfluential Traite de dynamique, which subsumed the dynamics of connected systems underIna few general principles.22 The first illustration he gave of his approach was Huygens'18D'Alembert later explained this agreement, see pp. 14-15.19J.
Bernoulli [1742] p. 442.201. Bernoulli [1 742] p. 444. His notation for the internal pressure was 11'. In the first section of his Hydraulica,which he communicated to Euler in 1839, he only treated the steady flow in a suddenly enlarged tube. In hisenthusiastic reply (5 May 1739, in Euler [ 1998] pp. 287-95), Euler treated the vertical, accelerated effiux from avase of arbitrary shape with the same method of'translation', not with the later method of balancing gravity withthe internal pressure gradient, contrary to Truesdell's claim ([1954] p. XXXIII). Bernoulli subsequently wrote hissecond part, adding only obliqueness of the vessel to Euler's treatment.21For a different view, cf. Truesdell [1954] p. XXVI, Calero [1996] pp.
460-74.22D'Alembert began to read a memoir on the same theme at the Academic des Sciences on 24 Nov. 1 742. A fewmonths earlier, he had read two memoirs on the refraction of solids moving in fluids of variable density (a thenclassical approach to the refraction of light). His treatment did not involve any explicit fluid mechanics, for heWORLDS OF FLOW12compound pendulum.23 As we saw, Johann Bernoulli's solution to this problem leads tothe equation of motion(1.10)which may be rewritten as(1. I I)This last equation is the condition of equilibrium of the pendulum under the action of theforcesmAg - mA'YA and msg - ms'Ys acting, respectively, on A and B.
In d'Alembert'smAg and msg are the motions impressed (per unit time) on theterminology, the productsbodies A and B under the sole effect of gravitation (without any constraint). The productsmA'YA and ms'Ys are the actual changes of their (quantity of) motion (per unit time). Thedifferences mAg - mA 'YA and msg - ms'Ys are the parts of the impressed motions that aredestroyed by the rigid connection of the two masses through the freely-rotating rod.Accordingly, d'Alembert saw in eqn( 1 . 1 1) a consequence of a general dynamic principlefrom which the motions destroyed by the connections should be in equilibrium.24How d'Alembert arrived at this principle is not known.
In1703Jacob Bemoulli, theelder brother of Johann, had derived the center of oscillation of the compound pendulumthrough the same method. D'Alembert did not refer to this source. Perhaps he found hisinspiration while meditating on the compound pendulum. Or he could have deduced theprinciple from a new philosophy of motion, as is suggested by the presentation given in theTraite de dynamique.25D'Alembert based dynamics on three laws, which he regarded as necessary consequences of the principle of sufficient reason. The first law is that of inertia, according towhich a freely-moving body moves with a constant velocity in a constant direction. Thesecond law stipulates the vector superposition of motions impressed on a given body.According to the third Jaw, two (ideally rigid) bodies come to rest after a head-on collisionif and only if their velocities are inversely proportional to their masses.
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