Darrigol O., Frisch U. «From Newton's mechanics to Euler's equations» (1123933), страница 3
Текст из файла (страница 3)
Frisch / Physica D 237 (2008) 1855–18691859rigor of its dynamical method, the depth of physical insight, andthe abundance of long-lasting results.152.3. Johann Bernoulli’s hydraulicaFig. 5. Effects of the velocity-dependence of pressure according to Bernoulli(1738: plate).which means that the pressure exerted by a moving fluid on thewalls is lower than the static pressure, the difference being halfthe squared velocity (times the density). D. Bernoulli illustratedthis effect in two ways (Fig. 5): by connecting a narrow verticaltube to the horizontal tube EFDG, and by letting a vertical jetsurge from a hole on this tube.
Both reach a water level wellbelow AB.The modern reader may here recognize Bernoulli’s law. Infact, D. Bernoulli did not quite write Eq. (5), because he chosethe ratio s/ε rather than the velocity v as the relevant variable.Also, he only reasoned in terms of wall pressure, whereasmodern physicists apply Bernoulli’s law to the internal pressureof a fluid.There were other limitations to D. Bernoulli’s considerations, of which he was largely aware. He knew that in somecases, part of the live force of the water went to eddying motion,and he even tried to estimate this loss in the case of a suddenlyenlarged conduit. He was also aware of the imperfect fluidity ofwater, although he decided to ignore it in his reasoning.
Mostimportantly, he knew that the hypothesis of parallel slices onlyheld for narrow vessels and for gradual variations of their sections. But his method confined him to this case, since it is onlyfor systems with one degree of freedom that the conservation oflive forces suffices to determine the motion.14To summarize, by means of the principle of live forces,Daniel Bernoulli was able to solve many problems of quasionedimensional flow and thereby related wall pressure tofluid velocity.
This unification of hydrostatic and hydraulicconsiderations justified the title Hydrodynamica which he gaveto the treatise he published in 1738 in Strasbourg. Besidesthe treatment of efflux, this work included all the typicalquestions of contemporary hydraulics except fluid resistance(which D. Bernoulli probably judged as being beyond the scopeof his methods), a kinetic theory of gases, and considerations onCartesian vortices. It is rightly regarded as a major turning pointin the history of hydrodynamics, because of the uniformity and14 Bernoulli, 1738: 12 (eddies), 124 (enlarged conduit); 13 (imperfect fluid).In 1742, Daniel’s father Johann Bernoulli published hisHydraulica, with an antedate that made it seem anterior to hisson’s treatise. Although he had been the most ardent supporterof Leibniz’s principle of live forces, he now regarded thisprinciple as an indirect consequence of more fundamental lawsof mechanics.
His asserted aim was to base hydraulics on anincontrovertible, Newtonian expression of these laws. To thisend he adapted a method he had invented in 1714 to solve theparadigmatic problem of the compound pendulum.Consider again the pendulum of Fig. 1. According toJ. Bernoulli, the gravitational force m B g acting on B isequivalent to a force (b/a)m B g acting on A, because accordingto the law of levers two forces that have the same moment havethe same effect. Similarly, the “accelerating force” m B bθ̈ of themass B is equivalent to an accelerating force (b/a)m B bθ̈ =m B (b/a)2 a θ̈ at A.
Consequently, the compound pendulum isequivalent to a simple pendulum with a mass m A + (b/a)2 m Blocated on A and subjected to the effective vertical force m A g +(b/a)m B g. It is also equivalent to a simple pendulum of length(a 2 m A + b2 m B )/(am A + bm B ) oscillating in the gravity g,in conformity with Huygens’ result. In sum, Johann Bernoullireached his equation of motion by applying Newton’s secondlaw to a fictitious system obtained by replacing the forcesand the momentum variations at any point of the system withequivalent forces and momentum variations at one point of thesystem. This replacement, based on the laws of equilibriumof the system, is what J.
Bernoulli called “translation” in theintroduction to his Hydraulica.16Now consider the canonical problem of water flowing byparallel slices through a vertical vessel of varying section(Fig. 2). J. Bernoulli “translates” the weight gSdz of theslice dz of the water to the location z 1 of the frontal sectionof the fluid. This gives the effective weight S1 gdz, becauseaccording to a well-known law of hydrostatics, a pressureapplied at any point of the surface of a confined fluid isuniformly transmitted to any other part of the surface of thefluid. Similarly, J. Bernoulli translates the “accelerating force”(momentum variation) (dv/dt)Sdz of the slice dz to the frontalsection of the fluid, with the result (dv/dt)S1 dz.
He then obtainsthe equation of motion by equating the total translated weightto the total translated accelerating force as:Z z1Z z1dvS1gdz = S1dz.(6)z0z 0 dtFor J. Bernoulli the crucial point was the determination of theacceleration dv/dt. Previous authors, he contended, had failed15 On the Hydrodynamica, cf. Truesdell, 1954: XXIII–XXXI; Calero, 1996:422–459; Mikhailov, 2002.16 Bernoulli, 1714; 1742: 395.
In modern terms, J. Bernoulli’s procedureamounts to equating the sum of moments of the applied forces to the sum ofmoments of the accelerating forces (which is the time derivative of the totalangular momentum). Cf. Vilain, 2000: 448–450.1860O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869to derive correct equations of motion from the general laws ofmechanics because they were only aware of one contributionto the acceleration of the fluid slices: that which correspondsto the instantaneous change of velocity at a given height z,or ∂v/∂t in modern terms. They ignored the acceleration dueto the broadening or to the narrowing of the section of thevessel, which J.
Bernoulli called a gurges (gorge). In modernterms, he identified the convective component v(∂v/∂z) of theacceleration. Note that his use of partial derivatives was onlyimplicit: thanks to the relation v = (S0 /S)v0 , he could split vinto a time dependent factor v0 and a z-dependent factor S0 /Sand thus express the total acceleration as (S0 /S)(dv0 /dt) −(v02 S02 /S 3 )(dS/dz).17Thanks to the gurges, J. Bernoulli successfully applied Eq.(6) to various cases of efflux and retrieved his son’s results.18He also offered a novel approach to the pressure of a movingfluid on the side of its container. This pressure, he asserted,was nothing but the pressure or vis immaterialis that contiguousfluid parts exerted on one another, just as two solids in contactact on each other:19J.
Bernoulli interpreted the relevant pressure as an internalpressure analogous to the tension of a thread or the mutualaction of contiguous solids in connected systems. Yet, he didnot rely on this new concept of pressure to establish theequation of motion (6). He only introduced this concept as ashort-cut to the velocity-dependence of wall-pressure.21To summarize, Johann Bernoulli’s Hydraulica departedfrom his son’s Hydrodynamica through a more direct relianceon Newton’s laws. This approach required the new conceptof a convective derivative. It permitted a generalization ofBernoulli’s law to the pressure in a non-steady flow.
J. Bernoullihad a concept of internal pressure, although he did not use it inhis derivation of his equation of fluid motion. Like his son’s,his dynamical method was essentially confined to systems withone degree of freedom only, so that he could only treat flow byparallel slices.The force that acts on the side of the channel through which the liquid flows. . .
isnothing but the force that originates in the force of compression through whichcontiguous parts of the fluid act on one another.In 1743, the French geometer and philosopher Jean leRond d’Alembert published his influential Traité de dynamique,which subsumed the dynamics of connected systems undera few general principles. The first illustration he gave ofhis approach was Huygens’s compound pendulum. As wesaw, Johann Bernoulli’s solution to this problem leads to theequation of motion:Accordingly, J.
Bernoulli divided the flowing mass of waterinto two parts separated by the section z = ζ . Following thegeneral idea of “translation”, the pressure that the upper partexerts on the lower part is:Z ζP(ζ ) =(g − dv/dt)dz.(7)z0More explicitly, this is:Z ζZ ζZ ζ∂v∂vdzP(ζ ) =gdz −v dz −∂zz 0 ∂tz0z011∂= g(ζ − z 0 ) − v 2 (ζ ) + v 2 (z 0 ) −22∂t3. D’Alembert’s fluid dynamics3.1. The principle of dynamicsm A g sin θ + (b/a)m B g sin θ = m A a θ̈ + (b/a)m B bθ̈ ,which may be rewritten asa(m A g sin θ − m A a θ̈ ) + b(m B g sin θ − m B bθ̈) = 0.Zζvdz.(8)z0In a widely different notation, J. Bernoulli thus obtained ageneralization of his son’s law to non-stationary parallel-sliceflows.2017 Bernoulli, 1742: 432–437.
He misleadingly called the two parts ofthe acceleration the “hydraulic”and the “hydrostatic” components. Truesdell(1954: XXXIII) translates gurges as “eddy” (it does have this meaning inclassical latin), because in the case of sudden (but small) decrease of sectionJ.
Bernoulli imagined a tiny eddy at the corners of the gorge. In his treatiseon the equilibrium and motion of fluids (1744: 157), d’Alembert interpretedJ. Bernoulli’s expression of the acceleration in terms of two partial differentials.18 D’Alembert later explained this agreement: see below, pp. 7–8.19 Bernoulli, 1742: 442.20 Bernoulli, 1742: 444. His notation for the internal pressure was π . In thefirst section of his Hydraulica, which he communicated to Euler in 1739, heonly treated the steady flow in a suddenly enlarged tube. In his enthusiasticreply (5 May 1739, in Euler, 1998: 287–295), Euler treated the acceleratedefflux from a vase of arbitrary shape with the same method of “translation,”not with the later method of balancing gravity with internal pressure gradient,contrary to Truesdell’s claim (1954: XXXIII).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
