Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 33
Текст из файла (страница 33)
307).17/bid. pp. 315-21, 328-29; Girard [1817] p. 235. In thls second memoir, Girard used and praised the graphicmethod that Prony had used for channel flow; he found that the linear term did not exist for mercury, as heexpected from the fact that mercury does not wet glass.18Girard [1817] p. 259.VISCOSITY109adhesion with friction, and therefore did not appreciate the circumstances that determinethe velocity profile.
Girard nevertheless obtained the linear behavior in HIL for thedischarge through narrow tubes, which is as well known today as it was surprising tocontemporary hydraulicians.3.1.4 The rational and the practicalIn summary, at the beginning of the nineteenth century no one expected rational fluiddynamics to explain the practically important phenomena of fluid resistance and flowretardation. Most knowledge of these phenomena was empirical and derived from theobservations and measurements accumulated by hydraulic engineers. Although somenotion of internal friction had been available since Newton, and although Du Buat,Venturi, Coulomb, and Girard somewhat revived it at the turn of the century, there wasno attempt to apply this insight to the mathematical determination of fluid motion.It may seem surprising that no one before Navier tried to insert new terms into Euler'shydrodynamic equations.
A first explanation is that the new hydrodynamics was part of arational mechanics that valued clarity, formal generality, and rigor above empiricaladequacy. Another is that Euler's equations were complex enough to saturate contemporary mathematical capability. They were among the first partial differential equations everwritten, and they involved the nonlinearity that has troubled mathematical physicists tothis day. Even if someone had been willing to modify Euler's equations, he would havelacked empirical clues about the structure of the new terms, because the concept of internalfriction was as yet immature.Last, but perhaps most important, the French mathematicians who were the mostcompetent at inventing new partial differential equations all accepted d'Alembert's fundamental principle of dynamics, according to which the equations of motion of a mechanical system can be obtained from the equilibrium condition between impressed forcesand inertial forces.
From this point of view, the hydrodynamic equations should resultdirectly from the laws of hydrostatics. Since the latter were solidly established, Euler'sequations seemed unavoidable.193.2 Navier: molecular mechanics of solids and fluids3.2. 1 X+PontsIn the jargon of the Grandes Ecoles, Claude-Louis Navier was an 'X+Ponts', that is, anengineer trained first at the Ecole Polytechnique and then at the Ecole des Ponts etChaussees. He embodied a new style of engineering that combined the analytical skillsacquired at the Polytechnique with the practical bent of the Ecoles d'application. Throughhis theoretical research and his teaching he contributed to a renewal of the science ofmechanics that made it fit much better to the needs of engineers.
Navier famouslypromoted considerations of 'live force' (kinetic energy) and 'quantity ·of action' (work)in the theory of machines, thus following Lazare Carnot's pioneering treatise and facilitating Gaspard Coriolis's and Jean-Victor Poncelet's later developments.2019Cournot expressed this view in his comment on Navier's equation, discussed later on p. 1 18.2°Cf. McKeon [1974] pp.
2-5. On the new style of engineering, cf. Belhoste [1994], Picon [1992] chaps 8-10. Onthe concept of work, cf. Grattan-Guinness [1984].1 10WORLDS OF FLOWOrphaned at fourteen, Navier was educated by his uncle Emiland Gauthey, a renownedengineer of bridges and canals. He later expressed his gratitude through a careful edition ofGauthey's works, published in 1 809-18 16. His competence in hydraulic architecture thenled him to edit Bemard Forest de Belidor's voluminous treatise, which had been acanonical reference on this subject since its first publication in 1737.
In this new edition,published in 1819, Navier left Belidor's text intact but denounced numerous theoreticalmisconceptions that still affected engineering practice in France and elsewhere. Hisfootnotes and appendices constituted a virtual book within the book, including a newpresentation of mechanics and a theory of machines based on live forces.Z1Navier found Belidor's treatment ofhydraulic problems most defective, as appears fromhis judgment of the included theory of efflux:The preceding theory, with which the author seems so pleased, now appears to be oneof the most defective of his work. In truth one had not, at the time he was writing,gathered a sufficient amount of experiments so as to establish the exact measure ofphenomena; but this does not justify the totally vicious theory that he gives of it, northe trust with which he presents it.In order to correct Belidor on this subject, Navier only had to return to Daniel Bemoulliand to refer to Venturi's relevant experiments.22Fluid resistance was harder to rectify.
As Navier well knew, numerous experiments byJean-Charles de Borda in the 1760s and by Bossut and Du Buat in the 1780s and 1 790s haddisproved the old impact theory recalled by Belidor. In his notes, Navier could onlydeplore that contemporary hydrodynamics did not permit a definitive solution to thisproblem.
He agreed with Euler that momentum balance applied to the tubes of flowaround the immersed body should yield the value of the resistance. No more than Euler,however, could he justifY the truncation of the tubes that allowed for a nonzero resistanceproportional to the squared velocity. Nor could he account for the negative pressure thatDu Buat had found to exist at the rear of the body.23From Coulomb, Navier also knew that the resistance became proportional to thevelocity for very slow motion. He agreed with Coulomb that in this case the retardingforce resulted from 'the mutual adhesion of the fluid molecules among themselves or at thesurface of the immersed bodies.' In summary, he considered two causes of fluid resistance,namely, a non-balanced distribution of pressure around the immersed body owing to someparticularity in the shape of the lines of flow around the body, and friction occurringbetween the body and the successive layers of fluid owing to 'molecular adhesion'.
Herespected Belidor's omission of pipe flow. 24•3.2.2Laplacian physicsAnother novelty of Navier's edition was the respect he paid to Laplace's new molecularphysics. Imitating Newton's gravitation theory and some of his queries, the French2 1 Cf. McKeon [1974], Prony [1864], Grattan-Guinness [1990] vol. 2, pp. 969-74.22Navier, note to Belidor [1819] p.
285n.23Ibid. pp. 339n-356n. On the fluid-resistance experiments by Borda, Bossut, and Du Buat, cf. Dugas [1950]pp. 297-305, Rouse and !nee [1957] pp. 124, 128, 133-4.2"Navier, note to Belidor [1819] p. 345n. Navier briefly mentioned (ibid. p. 292n) 'friction of the fluid on the[pipe] walls' (but not the internal adhesion in this case).VISCOSITYIllastronomer sought t o explain the properties o f matter by central forces acting betweenmolecules. His first successful attempt in this direction was a theory of capillarity published in 1 805/06. In the third edition of his Systeme du monde, published in 1808, he alsoindicated how optical refraction, elasticity, hardness, and viscosity could all be reduced toshort-range forces between molecules.
In an appendix to the fifth volume of the Mecaniqueceleste, published in 1 821, he gave a detailed molecular theory of sound propagation,based on his and Claude-Louis Berthollet's idea that molecular repulsion depended on thecompression of elastic atmospheres of caloric.Z5In the foreword to his edition of Belidor, Navier approved Laplace's idea of theconstitution of solids:Even though the intimate constitution of bodies is unknown, the phenomena whichthey show allow us to clearly perceive a few features of this constitution .. From thefaculty that solid bodies have to dilate under heating, to contract under cooling, andto change their figure under effort, it cannot be doubted that they are made of partswhich do not touch each other and which are maintained in equilibrium at very smalldistances from each other by the opposite actions of two forces, one of which is anattraction inherent in the nature of matter, and the other a repulsion due to theprinciple of heat.At that time, Navier used this conception of solids only to banish the ideally-hard bodiesof rational mechanics from collision theory.
He referred to Laplace's theory of capillarityin a footnote. The conditions of equilibrium of fluids, he emphasized, could not berigorously established without the molecular viewpoint. A fortiori, fluid motion had todepend on molecular processes, as he argued in his discussion of Coulomb's fluid-frictionexperirnents.Z63.2.3Elastic beams andplatesIn his engineering role, Navier acted mostly as an expert on bridge construction. In the1 8 1 0s, he designed three new bridges on the River Seine, and oversaw an important bridgeand embankment project in Rome.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
