Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 40
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etc.None of these methods, however, sufficed to solve the outstanding problem of the engineerof wood and iron structures, namely the flexion and torsion of prisms. For some twentyyears, Saint-Venant worked hard to avoid the simplifications used in previous solutions,such as the absence of slides, small deformation, perpendicularity of longitudinal fibersand transverse sections, flatness of transverse sections, etc. 7674Saint-Venant (1837], [!843a] p. 943: 'Je fais entrer dans le calcul les effets de glissement lateral dus a cescomposantes traosversales dont I'omission a ete l'objet principal d'une sorte d'accusation portee par M.
Vicatcontre toute la theorie de la resistaoce des solides.' Cf. Boussinesq and Flamaot (1886] p. 560 (bridge on the RiverCreuze), Todhunter aod Pearson (1886-1893] vol. I, pp. 834--6, 843, vol. 2, pp. 394-5, Benvenuto [1998] pp. 20-4.75Saint-Venaot (1843b], (1834/35]. In their molecular theories, Cauchy and Poisson used a less consistentdefinition ofpressure that makes it the resultant of the forces between all the molecules on one side of the plane ofthe surface element and the molecules belonging to a straight cylinder based on the other side of the element.Cauchy [1845] approved Saint-Venaot's defmition.
Cf. Darrigol (2002a] pp. 122-3.76Saint-Venaot (1834/35]. For the successive steps ofSaint-Venant's work on the flexion aod torsion of prisms,see Saint-Venaot (1864c].VISCOSITY131His most impressive achievement was the 'semi-inverse' method he developed in the1830s. The 'direct' problem of elasticity, which is the determination of impressed forcesknowing the deformation, is easily solved by applying the stress-strain relation. In contrast, the practically important 'inverse' problem, which is the determination of deformations under given impressed forces, leads to differential equations whose integration infmite terms is usually impossible. Saint-Venant's important idea was to replace the inverseproblem with a solvable, mixed problem in which the deformation and the impressed forceswere both partly given.
He then showed that the exact solutions of the latter problem didnot significantly differ from the practically needed solution of the inverse problem. 773.5.4Onfluid motionAlthough Saint-Venant is best known for his work on elasticity, he also had a constantinterest in hydraulics. Early in his career, he reflected on waterwheels and the channels andweirs that fed them.
He also began to think about the scientific control of waters in ruralareas, which he later called hydraulique agricole. In this field, as for elasticity, Saint-Venantavoided narrow empiricism. He wanted to base the determination of channel and pipe flowon fundamental hydrodynamics. Navier's failed attempt in this direction no doubt stimulated him.78In 1834, Saint-Venant submitted to the Academy of Sciences a substantial, though neverpublished, memoir on the dynamics of fluids. To start with, he expressed his approbationof the mecanique physique by citing Poisson: 'It is important for the progress of sciencesthat rational mechanics should no longer be an abstract science, founded on definitionsreferring to an imaginary state of bodies.' He rejected ideal solids, argued for central forcesand point-atoms, and proved the discontinuity of matter in the earlier-mentioned manner.He defined the average 'translatory' motion observed in hydraulic experiments and theinvisible 'non-translatory' motion that molecular interactions necessarily implied.
Then hegave his molecular definition of internal pressures (which he called 'impulsions'), andshowed the existence of transverse pressures in moving fluids by a detailed considerationof the perturbation of the translatory motion by molecular encounters. In harmony withhis kinematics of elastic bodies, he characterized the transverse pressure as being opposedto the sliding of successive layers of the fluid on one another.79This transverse pressure depends on the microscopic non-translatory motion of themolecules, which propagates through the whole fluid mass 'and gets lost to the outside byproducing, in the walls and in the exterior air foreign agitation and other effects foreign tothe translatory motion of the fluid.' The live force of the macroscopic motion thusdiminishes at the price of hidden microscopic motion.
Later, in the 1 840s, Saint-Venantidentified the non-translatory motions with heat. 8077Saint-Venant introduced this method in 1847 and 1853. His fullest study of the torsion and flexion of prismsis Saint-Venant [1855].78Cf. Melucci [1996], Darrigol [2001].79Saint-Venant [1834] sects 1 (molecular mechanics), 2 (no continuous matter), 4 (undulated motion ofmolecules), 5 (definition of impulsions), 6-7 (transverse pressures); Poisson [1831a] p.
130.80Saint-Venant [1834] sect. 7. For the identification with heat, cf. Saint-Venant [1887b] p. 73 n.WORLDS OF FLOW132Deterred by the complexity he saw in the friction-related molecular motions, SaintVenant renounced a purely molecular derivation of the pressure system. Instead, heappealed to a symmetry argument in the spirit of Cauchy's first theory of elasticity. Heassumed that the transverse pressure on a face was parallel to the fluid slide on this face,and (erroneously, he later realized) took the slide itself to be parallel to the projection ofthe fluid velocity on the face. This led him to an equation of motion far more complicatedthan Navier's, with five parameters instead of one, and with variations of these parametersdepending on the internal, microscopic commotions of the fluid.
Saint-Venant applied thisequation to flow in rectangular or semi-circular open channels and described a newmethod of fluid-velocity measurement. He thus wanted to prepare the experimentaldetermination of the unknown functions that entered his equations. 813.5.5A first-class burialThe commissioners Ampere, Navier, and Felix Savary approved Saint-Venant's memoir.Yet Savary, who was supposed to write the report, never did so and instead expresseddisagreements in letters to the author. From Saint-Venant's extant replies, we may inferthat Savary did not know of the contradiction between Du Buat's measurements andNavier's equation and that he condemned the recourse to adjustable parameters infundamental questions of hydrodynamics. In his defense, Saint-Venant clarified thepurpose of his memoir: 'My principal goal is all practical: it is the solution of the openchannel problem for a bed of variable and arbitrary figure.' He then formulated aninteresting plea for a semi-inductive method: 82My equations contain indeterminate quantities and even indeterminate functions;but is it not good to show how far, in fluid dynamics, we may proceed with a theorythat is free of hypotheses (save forcontinuity, at least on average), that brings forththe unknown and prepares its experimental determination? A bolder march maysometimes quickly lead to the truth .
. . . However, you will no doubt admit that insuch an important matter it may be advantageous to consider things from anotherpoint of view, to avoid every supposition and to appeal to experimenters to fix thevalues of indeterminate quantities by means of special experiments prepared so as toisolate the effects that the theory will later try to explain with much more assuranceand to represent by expressions that are as free of empiricism as possible.3.5.6Re-founding Navier's equationThree years later, Saint-Venant discovered his error about the direction, of slides, andceased to request a report from Savary.
Instead, he inserted a more cogent argument in themanuscript deposited at the Academy. He still assumed that the transverse pressure on aface was parallel to the slide on this face, or, equivalently and even more naturally, that thetransverse pressure was zero in the direction of the face for which the slide vanished.However, he now used the correct expression o;vj + OjVi for the slides (per unit time)corresponding to the fluid velocity v and the orthogonal directions i and j. He further81Saint-Venant [1834] sects 1 1 (hypothesis), 1 5 (equation), 1 8-24 (consequences), 25-8 (suggested experiments).82Saint-Venant to Savary, 25 Aug.
1 834, Bibliotheque de l'Institut de France, MS 4226; see also the letters of27 July and 10 Sept. 1 834, ibid.VISCOSITY133noted that ru - Tjj represented twice the transverse pressure along the line bisectingthe angle i}, and 8;v; - OjVj represented the slide along the same line. Granted that thecomponents of slide must be proportional to the components of transverse pressure,the ratios Tif/(o;vj + Gjv;) and (ru - Tjj)/2(8;v; - Ojvj) are all equal for every choice of iand j.
Denoting by s their common value at a given point of the fluid and w anundetermined isotropic pressure, this implies that(3.31)As Saint-Venant noted, this stress system yields the Navier-Cauchy-Poisson equation inthe special case of a constant s, with a gradient term contributing to the normal pressure. 83For a modern reader familiar with tensor calculus, Saint-Venant's reasoning looks likeanother proof of the fact that the expression (3 .31) is the most general symmetrical secondrank tensor that depends linearly and isotropically on the tensor eif.
Yet this is not the case,because Saint-Venant did not assume the linearity. Admittedly, his hypothesis of theparallelism of slides and tangential pressures implies more than mere isotropy; forinstance, it excludes terms proportional to eikekj· However, it allows for an s that variesfrom one particle of the fluid to another, and from one case of motion to another. 84Saint-Venant believed that Du Buat's and others' experiments on pipe and channel flowrequired a variable s, which expressed the effects of local 'irregularities of motion' oninternal friction.
The velocity v in his reasoning referred to the average, smooth, large-scalemotion. Smaller-scale motions only entered the final equation as a contribution to tangential pressures defmed at the larger scale. Whether or not Saint-Venant regardedNavier's equation with constant s as valid at a sufficiently small scale is not clear. Inany case, he believed that the value of s should be determined experimentally withoutprejudging its constancy from place to place or from one case of motion to another.
85In the mid-1840s, the military engineer Pierre Boileau undertook a series of experimentson channel and pipe flow. Unlike most hydraulicians, who were only interested in theglobal discharge, Boileau planned measurements of the velocity profile of the flow. SaintVenant congratulated him for this intention, because such knowledge was necessary toestimate the friction between successive fluid filaments, or the variable s of his equation offluid motion. He advised Boileau on the most suitable channel and pipe shapes and on thetechnique of velocity measurement. As we will see in Chapter 6, this sort of experiment andthe correlative idea of an effective, eddy-related viscosity had a future.
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