Darrigol O., Frisch U. «From Newton's mechanics to Euler's equations» (1123933), страница 2
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In1644, Evangelista Torricelli gave the law for the velocity of theescaping fluid as a function of the height of the water level; inthe last quarter of the same century, Edme Mariotte, ChristiaanHuygens, and Isaac Newton tried to improve its experimentaland theoretical foundations of this law.6More originally, Newton devoted a large section of hisPrincipia to the problem of fluid resistance, mainly to disprovethe Cartesian theory of planetary motion. One of his results, theproportionality of inertial resistance to the square of the velocityof the moving body, only depended on a similarity argument.His more refined results required some drastically simplifiedmodels of the fluid and its motion. In one model, he treatedthe fluid as a set of isolated particles individually impactingthe head of the moving body; in another, he preserved thecontinuity of the fluid but assumed a discontinuous, cataractlike motion around the immersed body.
In addition, Newton5 Grimberg, Pauls and Frisch, 2008. Truesdell, 1954: XXXVIII–XLI.6 Cf. Truesdell, 1954: IX–XIV; Rouse and Ince, 1957: Chaps. 2–9;Garbrecht, 1987; Blay, 1992, Eckert, 2005: Chap. 1.O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–18691857Fig. 1. Compound pendulum.investigated the production of a (Cartesian) vortex through therotation of a cylinder and thereby assumed shear stresses thattransferred the motion from one coaxial layer of the fluid tothe next. He also explained the propagation of sound throughthe elasticity of the air and thereby introduced the (normal)pressure between successive layers of the air.7To sum up, Newton introduced two basic, long-lasting concepts of fluid mechanics: internal pressure (both longitudinaland transverse), and similarity. However, he had no generalstrategy for subjecting continuous media to the laws of his newmechanics.
While his simplified models became popular, hisconcepts of internal pressure and similarity were long ignored.As we will see in a moment, much of the prehistory of Euler’sequation has to do with the difficult reintroduction of internalpressure as a means to derive the motion of fluid elements. Although we are now accustomed to the idea that a continuumcan be mentally decomposed into mutually pressing portions,this sort of abstraction long remained suspicious to the pioneersof Newtonian mechanics.2.2. Daniel Bernoulli’s hydrodynamicaThe Swiss physician and geometer Daniel Bernoulli wasthe first of these pioneers to develop a uniform dynamicalmethod to solve a large class of problems of fluid motion. Hisreasoning was based on Leibniz’s principle of live forces, andmodeled after Huygens’s influential treatment of the compoundpendulum in his Horologium oscillatorium (1673).8Consider a pendulum made of two point masses A and Brigidly connected to a massless rod that can oscillate aroundthe suspension point O (Fig.
1). Huygens required the equalityof the “potential ascent” and the “actual descent,” whosetranslation in modern terms reads:2 /2g) + m (v 2 /2g)m A (vAB B= zG,mA + mB(2)where m denotes a mass, v a velocity, g the acceleration ofgravity, and z G the descent of the gravity center of the two7 Cf. Smith, 1998. Newton also discussed waves on water and the shape of arotating fluid mass (figure of the Earth).8 Bernoulli, 1738; Huygens, 1673.Fig.
2. Parallel-slice flow in a vertical vessel.masses measured from the highest elevation of the pendulumduring its oscillation. This equation, in which the modern readerrecognizes the conservation of the sum of the kinetic andpotential energies, leads to a first-order differential equation forthe angle θ that the suspending rod makes with the vertical. Thecomparison of this equation with that of a simple pendulumthen yields the expression (a 2 m A + b2 m B )/(am A + bm B ) forthe length of the equivalent simple pendulum (with a = OAand b = OB).9As D. Bernoulli could not fail to observe, there is a closeanalogy between this problem and the hydraulic problem ofefflux, as long as the fluid motion occurs by parallel slices.Under the latter hypothesis, the velocity of the fluid particlesthat belong to the same section of the fluid is normal toand uniform through the section.
If, moreover, the fluid isincompressible and continuous (no cavitation), the velocity inone section of the vessel completely determines the velocity inall other sections. The problem is thus reduced to the fall of aconnected system of weights with one degree of freedom only,just as is the case of a compound pendulum.This analogy inspired D. Bernoulli’s treatment of efflux.Consider, for instance, a vertical vessel with a section Sdepending on the downward vertical coordinate z (Fig.
2). Amass of water falls through this vessel by parallel, horizontalslices. The continuity of the incompressible water implies thatthe product Sv is a constant through the fluid mass. The equalityof the potential ascent and the actual descent implies that atevery instant10Z z1Z z1 2v (z)S(z)dz =zS(z)dz,(3)2gz0z0where z 0 and z 1 denote the (changing) coordinates of the twoextreme sections of the fluid mass, the origin of the z-axis9 Cf.
Vilain, 2000: 32–36.10 Bernoulli, 1738: 31–35 gave a differential, geometric version of thisrelation.1858O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869Fig. 3. Idealized efflux through small opening (without vena contracta).coincides with the position of the center of gravity of this massat the beginning of the fall, and the units are chosen so thatthe density of the fluid is one. As v(z) is inversely proportionalto the known function S of z, this equation yields a relationbetween z 0 and v(z 0 ) = ż 0 , which can be integrated to give themotion of the highest fluid slice, and so forth.
D. Bernoulli’sinvestigation of efflux amounted to a repeated application ofthis procedure to vessels of various shapes.The simplest sub-case of this problem is that of a broadcontainer with a small opening of section s on its bottom(Fig. 3). As the height h of the water varies very slowly, theescaping velocity quickly reaches a steady value u. As thefluid velocity within the vessel is negligible, the increase of thepotential ascent in the time dt is simply given by the potentialascent (u 2 /2g)sudt of the fluid slice that escapes through theopening at the velocity u. This quantity must be equal to theactual descenthsudt.
Therefore, the velocity u of efflux is the√velocity 2gh of free fall from the height h, in conformity withTorricelli’s law.11D. Bernoulli’s most innovative application of this methodconcerned the pressure exerted by a moving fluid on the wallsof its container, a topic of importance for the physician andphysiologist he also was. Previous writers on hydraulics andhydrostatics had only considered the hydrostatic pressure dueto gravity. In the case of a uniform gravity g, the pressure perunit area on a wall portion was known to depend only on thedepth h of this portion below the free water surface. Accordingto the law enunciated by Simon Stevin in 1605, it is given bythe weight gh of a water column (of unit density) that has a unitnormal section and the height h. In the case of a moving fluid,D.
Bernoulli defined and derived the “hydraulico-static” wallpressure as follows.12The section S of the vertical vessel ABCG of Fig. 4 issupposed to be much larger than the section s of the appendedtube EFDG, which is itself much larger than the section ε of11 Bernoulli, 1738: 35. This reasoning assumes a parallel motion of theescaping fluid particle. Therefore, it only gives the velocity u beyond thecontraction of the escaping fluid vein that occurs near the opening (Newton’svena contracta): cf. Lagrange, 1788: 430–431; Smith, 1998.12 Bernoulli, 1738: 258–260. Mention of physiological applications is foundin D.
Bernoulli to Shoepflin, 25 Aug 1734, in Bernoulli, 2002: 89: “Hydraulicostatics will also be useful to understand animal economy with respect to themotion of fluids, their pressure on vessels, etc.”Fig. 4. Daniel Bernoulli’s figure accompanying his derivation of the velocitydependence of pressure (1738: plate).the hole o. Consequently,the velocity u of the water escaping√through o is 2gh. Owing to the conservation of the flux, thevelocity v within the tube is (ε/s)u.
D. Bernoulli goes on tosay:13If in truth there were no barrier FD, the final velocity of the water in the sametube would be [ s/ε times greater]. Therefore, the water in the tube tends to agreater motion, but its pressing [nisus] is hindered by the applied barrier FD.By this pressing and resistance [nisus et renisus] the water is compressed [comprimitur], which compression [compressio] is itself kept in by the walls of thetube, and thence these too sustain a similar pressure [pressio]. Thus it is plainthat the pressure [pressio] on the walls is proportional to the acceleration. .
. thatwould be taken on by the water if every obstacle to its motion should instantaneously vanish, so that it were ejected directly into the air.Based on this intuition, D. Bernoulli imagined that the tubewas suddenly broken at ab, and made the wall pressure Pproportional to the acceleration dv/dt of the water at thisinstant. According to the principle of live forces, the actualdescent of the water during the time dt must be equal to thepotential ascent it acquires while passing from the large sectionS to the smaller section s, plus the increase of the potentialascent of the portion EabG of the fluid. This gives (the fluiddensity is one) 2v2vhsvdt =svdt + bsd,(4)2g2gwhere b = Ea.
The resulting value of the acceleration dv/dtis (gh − v 2 /2)/b. The wall pressure P must be proportional tothis quantity, and it must be identical to the static pressure ghin the limiting case v = 0. It is therefore given by the equation1P = gh − v 2 ,2(5)13 Bernoulli, 1738: 258–259, translated in Truesdell, 1954: XXVII. Thecompressio in this citation perhaps prefigures the internal pressure laterintroduced by Johann Bernoulli.O. Darrigol, U.
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