Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 38
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Cf. Saint-Venant [1864b] pp. clxi-dxvii,Arnold [1983] parts 6 and 8.60Navier [1829a], [1 829b]. Poisson also objected to Navier's occasional assumption that in the natural state ofthe body the forces between any two molecules vanished. Navier, however, did not regard this assumption asnecessary to his derivations.61 Physicists today regard the existence of the equilibrium state of a solid as a quantum property, but theynevertheless allow a classical treatment of small perturbations of this state.VISCOSITY125points subjected to central forces acting in pairs, cannot have rigidity. The subterfuge of anonzero limit of R'+f is unavailable, because that would imply the divergence of theintegral f R3fdR. More fundamentally, the lack of rigidity is an immediate consequenceof the symmetry properties of a central-force continuum.
Neither Cauchy nor Poisson sawthis fact, which is only evident to modem physicists trained to exploit symmetries. It wasSaint-Venant who first remarked that the lack of shear stress in a perfectly continuousbody resulted from the perfect invariance of the central forces acting in such a body for alarge class of internal, shearing deformations. As a simple example, take a global shift ofthe half of an (infinite) body situated on one side of a fixed plane. 62Another of Poisson's objections to Navier was that the method of moments, whichLagrange had su=ssfully used for continuous media, did not apply to molecular systems.This is a surprising statement, since the principle of virtual velocities does not presupposethe continuity of the material system to which it is applied.
Poisson probably meant thatNavier's estimate of the total moment did not properly include the contribution ofmolecules whose sphere of action intersects the surface of the body. Indeed, the momentsof the forces between such a molecule and all other molecules of the body do not sum up tothe full value given in eqn (3.4). Nevertheless, the contribution of these bordering molecules is negligible, because their moment is to the total moment what the radius of actionis to the average radius of the body. Although Navier never gave this justification, hisintuitive estimate of the total moment was correct.6 3Navier's methods were more coherent than Poisson believed, and they had considerableadvantages. They minimized assumptions concerning the nature of molecular forces, andthey provided a direct link between these assumptions and macroscopic properties.
Forthis reason, several modern commentators have seen in Navier's theory an anticipation ofGeorge Green's potential-based theory of elasticity of 1 837. Regarding the necessity ofpreserving discrete sums, Poisson was essentially correct. However, he exaggerated thedifficulty; in the isotropic case the substitution of integrals for sums does not affect thestructure of the equations of motion as long as the integration over distance is notexplicitly performed.643.4.4 Fluids as temporary solidsIn 1 829, Poisson, the champion of molecular rigor, had to correct several flaws in his 1 828memoir that Cauchy's memoir had made apparent.
He took this opportunity to offer atheory of fluid motion based on the following assumption: a fluid, like a solid, experiencesstresses during its motion, but these stresses spontaneously relax in a very short time. Inthis picture, the liquid goes through a rapid alternation of stressed and relaxed states.62Saint-Venant [1834] sect. 2, [1844]. The remark on the limit of R"J is mine. By varying Poisson's centralforces around equilibrium, Navier's elastic force q, is easily seen to be related to Poisson's f (in my notation) byq, = R-1! + R d(R-1f)jdR, which implies that the integral of K'q, and Navier's elastic constant vanish.
SaintVenant's argument may have been inspired by Fresnel's remark, in his molecular ether-model of 1821, thatresistance to the shift of a slice of ether required molecular constitution with intermolecular distances muchsmaller than this shift (Fresnel [1821] pp. 630-2).63Poisson [1829a] p. 400.64Reference to Green is found, e.g., in Dahan [1992].
One way to save Navier's procedure is to introduce afinite lower limit in his integrals, see Clausius [1849] pp. 56-8.WORLDS OF FLOW126Poisson further assumed that the average stress system of the fluid was to the fluid's rate ofdeformation what the stress system of an isotropic solid was to its strain. This hypothesisleads to the Navier-Stokes equation, with some additions to the pressure gradient termthat depend on the compressibility of the fluid. 65Poisson did not refer to Navier's memoir on fluid motion, which he must have judgedincompatible with sound Laplacian reasoning. Nor did he mention Cauchy's 'perfectlyinelastic solid', despite the similarity between his and Cauchy's ways of relating the fluidstresses to those in an isotropic elastic solid.3.5 Saint-Venant: slides and shears3.5.1Le Pant des InvalidesNavier's and other Polytechnicians' efforts to reconcile theoretical and applied mechanicshad no clear effect on French engineering practice.
Industry prospered much faster inBritain, despite the lesser mathematical training of its engineers. Some of Navier's colleagues saw this and ridiculed the use of transcendental mathematics in concrete problemsof construction. 66 In the mid-1820s, a spectacular incident apparently justified theirdisdain. Navier's chef-d'oeuvre, a magnificent suspended bridge at the Invalides, had tobe dismantled in the fmal stage of its construction.Navier had learnt the newer technique of suspension during official missions to Englandand Scotland in 1820 and 1823. At the end of his ministerial report, he argued in favor of anew suspended bridge of unprecedented scale across the River Seine and facing theInvalides (see Fig. 3.4).
In Prony's and Saint-Venant's well-informed opinion, Navier'sinnovative design was based on sound experience and calculation. Yet, as the bridge wasnearly finished, an accidental flood caused the displacement of one of the rounded stoneson which the suspending chains changed direction before anchoring (see Fig. 3.4(b) ).
AsSaint-Venant later explained, Navier had mis-estimated the direction of the force exertedby the chain on the stone-a kind of oversight that frequently occurs in engineeringconstruction and that is easily corrected on the spot. Hostile municipal authorities nevertheless obtained the dismantlement of Navier's bridge. 67According to Saint-Venant, the incident meant more than a local administrativedeficiency:·At that time there already was a surge of the spirit of denigration, not only of thesavants, but also of science, disparaged under the name of theory opposed to practice;'65Pois�on [1831a] pp. 139-74.
Stokes showed that, for small compressions, Poisson's additional gradient termis (.)J,/3)\lC:V · v), as in Stokes's own molecular fluid model.66Cf. Belhoste (1994] pp. 24-5. Belhoste explains how this state of affairs prompted reforms at the EcolePolytechnique and at the Ecoles d'applications.67Navier (1823d), (1830]. Cf. Prony [1864] pp. xlv-xlvii, Saint-Venant (1864a] pp. lxv-lxix, Grattan-Guinness[1990] pp. 994-1000, Picon [1992] pp. 372-84, Kranalds [1997], Cannone and Friedlander [2003].
The popuiarperception of this event differed from Saint-Venant's, as shown by this extract from Honore de Balzac's Le cure devillage: 'All France knew of the disaster which happened in the heart of Paris to the first suspension bridge built byan engineer, a member of the Academy of Sciences; a melancholy collapse caused by blunders such as none of theancient engineers-the man who cut the canal at Briare in Henri IV's time, or the monk who built the PantRoyal-wonld have made; but our administration consoled its engineer for his blunder by making him a memberof the Council-general' (transl. by K.
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