Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 7
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From these threelaws and some further recourse to the principle of sufficient reason, d'Alembert believedhe could derive a complete system of dynamics without recourse to the older, obscureassumed from the start a generalization ofNewton's resistance formula for the impact of fluid on a solid segment.It did not involve any dynamics either, for he computed the deflection of the solid on the basis of Newton's secondlaw. Cf. Academie Royale des Sciences, Proces-verbaux 60 (1741) pp. 369-404, 424-38; 61 (1742) pp. 126-33, 34956 (text of the memoirs on refraction, also in Part Ill of d'Alembert [1744] with little change); Proces-verbaux 61(1742) p.
424 (mention that D'Alembert has·read a memoir on a new principle of dynamics). D'Alembert was notthe only one to feel the need of a systematization of mechanics at that time; in the Memoires of the same year,([1742] pp. 1-52), Clairaut published his own 'general and direct principle' of dynamics (pp. 21-2), based on theintroduction of internal forces (such as thread tension).23D'Aiembert [1743] pp.
69-70.24Cf. Vilain [2000] pp. 456-9. D'Alembert [1743] reproduced and criticized Johann Bernoulli's derivation onp. 71.250n Jacob Bernoulli as a source, cf. Lagrange [1788] pp. 176-7, 179-80, Dugas [1950] pp. 233-4, Vilain [2000]pp. 444-8. Jacob Hermann's treatment of the compound pendulum in his Phoronomia (1716) and Euler's earlytreatment of the same problem (1734) read like convoluted statements of Jacob Bernoulli's method.THE DYNAMICAL EQUATIONS13concept of force as the cause of motion.
He defined force as the motion impressed on abody, that is, the motion that a body would take if this force were acting alone without anyimpediment. The third law then implies that two contiguous bodies subjected to oppositeforces are in equilibrium. Consequently, d'Alembert regarded statics as a particular case ofdynamics in which the various motions impressed on the parts of the system mutuallycancel each other.26Based on this concept, d'Alembert derived the principle of virtual velocities, accordingto which a connected system subjected to various forces remains in equilibrium if the workof these forces vanishes for any infinitesimal motion of the system that is compatible withthe connections.
27 As for the principle of dynamics, he regarded it as a self-evidentconsequence of his dynamic concept of equilibrium. In general, the effect of the connections in a connected system is to destroy part of the motion that is impressed on itscomponents by means of external agencies. The rules of this destruction should be thesame whether the destruction is total or partial. Hence, equilibrium should hold for thatpart of the impressed motions that is destroyed through the constraints.
This is d'Alembert's principle of dynamics.Stripped of d'Alembert's philosophy of motion, this principle stipulates that a connected system in motion should be, at any time, in equilibrium with respect to the fictitiousforces sf - my, where f denotes the force applied on the mass point m of the system, and'Y the acceleration of this mass point. As a simple example, consider two masses mA and mshanging on the two sides of a massless pulley by an inextensible, massless thread (seeFig. 1 .6). According to d'Alembert's principle, the forces mAg - mA "YA and msg - ms"Ysshould be in equilibrium, and therefore should be equal.
This condition, together with thekinematic condition -yA + "YB = 0, yields the equation of motion of the system. Comparedto other treatments of the same problem, the essential advantage of d' Alembert's methodBAFig. 1 .6.Simple connected system for illristrating d'Alembert's principle.26D'Alembert [1743] pp. xiv-xv, 3. Cf. Hankins [1968], Fraser [1985].27The principle of virtual velocities was first stated generally by Johann Bernoulli and thus named by Lagrange[1788] pp. 8-1 1.
Cf. Dugas [1950] pp. 221-3, 320. The term work is of course anachronistic.14WORLDS OF FLOWis that it does not require the introduction of the subtle (and obscure for d' Alembert)concept ofthe tension of the thread. It directly gives the equations of motion of the systemif only the conditions of equilibrium are known.1 .3.2 Ejjlux revisitedAt the end of his treatise on dynamics, d'Alembert considered the hydraulic problem ofefflux through the vessel of Fig. 1 .2. His first task was to determine the condition ofequilibrium of the fluid when subjected to an altitude-dependent gravity g(z).
For thispurpose he considered an intermediate slice of the fluid, and required the pressure from thefluid above this slice to be equal and opposite to the pressure from the fluid below thisslice. According to a slight generalization of Stevin's hydrostatic law, these two pressuresare given by the integral of the variable gravity g(z) over the relevant range of elevation.Hence the equilibrium condition reads:28I dz = -S(?) I g(z) dz,(ZJzo(S(?) g(z)orI g(z) dz = 0.(1.12)ZJ(1. 13)zoAccording to d'Alembert's principle, the motion of the fluid under a constant gravity gmust be such that the fluid is in equilibrium under the fictitious gravity g(z) = g - dvjdt,where dvjdt is the acceleration of the fluid slice at the elevation z.
Hence follows theequation of motion(1.14)which is the same as Johann Bernoulli's eqn (1 .7).D'Alembert further proved that this equation implied the conservation of live forces inDaniel Bernoulli's form. To this end, he inserted the product Sv, which does not depend on,,z, in the above equation. This givesI V�� s dz = I gvS dz.Z!Ztzozo(1.15)As the two integrals can be regarded as sums over moving slices of fluid, this equation isequivalent to28D'A1embert [1 743] pp. 183-6.THE DYNAMICAL EQUATIONS15dt J 2 v S dz = dt J gzS dz,dZl12dzoZ1(1.16)zowhich is the differential version of Daniel Bernoulli's eqn(1 .4).D'Alembert intended his new solution of the efflux problem to illustrate the power of hisprinciple of dynamics.
He clearly relied on the long-known analogy with a connectedsystem of solids. Yet he believed this analogy to be imperfect. Whereas in the case of solidsthe condition of equilibrium was derived from the principle ofvirtual velocities, in the caseof fluids d'Alembert believed that only experiments could determine the condition ofequilibrium. As he explained in his treatise ofI 744 on theequilibrium and motion offluids, the interplay between the various molecules of a fluid was too complex to allow fora derivation based on the only a priori known dynamics, that of individual molecules.29In this second treatise, d'Alembert provided a similar treatment of efflux, including hisearlier derivations of the equation of motion and the conservation of live forces, with aslight variant: he now derived the equilibrium condition(1.13)by setting the pressureacting on the bottom slice of the fluid to zero.30 Presumably, he did not want to basethe equations of equilibrium and motion on the concept of internal pressure, in conformance with his general avoidance of internal contact forces in his dynamics.
His statement ofthe general conditions of equilibrium of a fluid, as found at the beginning of his treatise,only required the concept of wall pressure. Yet, in a later section of his treatise, d'Alembertintroduced 'the pressure at a given height'(1.17)just as Johann Bernoulli had done, and for the same purpose of deriving the velocitydependence of wall pressure.31In the rest of his treatise, d'Alembert solved problems similar to those in Daniel Bernoulli'sHydrodynamica, with nearly identical results. The only important difference concerned cases involving the sudden impact oftwo layers of fluids.
Whereas Daniel Bernoullistill applied the conservation oflive forces in such cases (save for eventual dissipation intoturbulent motion), d'Alembert's principle of dynamics there implied a destruction of liveforce. Daniel Bernoulli disagreed with these and a few other changes.
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