Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 5
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The modem concept of dimensional quantities emerged atthe turn of the nineteenth century, and found its first systematic formulation in Fourier'stheory of heat. 8 A full history of early hydrodynamics would necessarily take into accountthis important transformation in the writing style of physico-mathematical equations.Modernized notation is nevertheless used in what follows, because the main points to bemade resist this perversion of the original text.In order to appreciate the daringness of Daniel Bemoulli's approach, one must rememberthat until the nineteenth century energy considerations were very rarely used in mechanicsand elsewhere. Gottfried Wilhelm Leibniz's principle of the conservation ofvis viva, whichhad both Huygenian and Cartesian roots, had little impact because the concept of live force(roughly our kinetic energy) was usually interpreted as a metaphysical threat to therNewtonian concept of accelerating force.
The most significant exception to this general attitudewas Daniel Bernoulli's father Johann, who used Leibniz's principle to ease the solution ofvarious mechanical problems. Father and son also agreed with Leibniz that every apparentloss of live force in the universe was a dissimulation oflive force in small-scale motions. Theyeven believed that potential forms of live force should be reducible to invisible motions, as9exemplified in Daniel's kinetic explanation of gas pressure.As the compound pendulum was the implicit paradigm of the Bernoullis' use of theconservation of live forces, some of Huygens' treatment must be recalled.
Consider apendulum made of two point masses A and B, rigidly connected to a massless rod that canoscillate around the suspension point 0 (see Fig.1.1). In modem notation, the equality ofthe potential ascent to the actual descent reads:mA(Ji/2g) + ms(ifs/2g)= zo,mA + mswhere(1.3)m denotes a mass, v a velocity, g the acceleration of gravity, and ZG the descent ofthe gravity center of the two masses measured from the highest elevation of the pendulumduring its oscillation. This equation leads to a first-order differential equation for the angle() that the suspending rod makes with the vertical.
The comparison of this equation with(a2mA + b2ms)/(amA + bm8) for thea = OA and b = OB).10that of a simple pendulum then yields the expressionlength of the equivalent simple pendulum (with1 . 1 .2 EffluxAs Daniel Bernoulli could not fail to observe, there is a close analogy between this problemand the hydraulic problem of efflux, as long as the fluid motion occurs by parallel slices.Under the latter hypothesis, the velocity of the fluid particles that belong to the same7D. Bernoulli [1 738] pp. 1 1 , 30.9Cf. Costabel [1983], Seris [1987].8Cf. Ravetz [1961].10Cf.
Vilain [2000] pp. 32-6.6WORLDS OF FLOWAzFig. 1.1.Compound pendulum.section of the fluid is normal to and uniform through the section. Moreover, if the fluid isincompressible and continuous (no cavitation), then the velocity in one section of thevessel completely determines the velocity in all other sections.
The problem is thus reducedto the fall of a connected system of weights with one degree of freedom only, just as it is forthe case of a compound pendulum.This analogy inspired Daniel Bernoulli's treatment of efflux. Consider, for instance, avertical vessel with a sectionSdepending on the downward vertical coordinatez(see Fig. 1.2).A mass of water falls through this vessel by parallel, horizontal slices. The continuity of theincompressible water implies that the product Sv is a constant through the fluid mass. Theequality of the potential ascent and the actual descent implies that at every instant111 (�)s 1dz = zS dz,(1.4)zozoS(z)�IIIIIIZo0z--------Fig.
1.2.ZtParallel-slice flow of water in a vertical vessel.11D. Bernoulli gave a differential, geometric version of this relation ([1738] pp. 31-5).7THE DYNAMICAL EQUATIONSwherezoandZJdenote the (changing) coordinates of the two extreme sections of the fluidmass, the origin of the z-axis coincides with the position of the gravity center of this massat the beginning of the fall, and the units are chosen so that the density of the fluid isAsvis inversely proportional to the known function S ofbetweenzoandvo = zo,z,one.this equation yields a relationwhich can be integrated to give the motion of the highest fluidslice, and so forth.
Bernoulli's investigation of efflux amounted to a repeated applicationof this procedure to vessels of various shapes.The simplest sub-case of this problem is that of a broad container with a small openingof sections on its bottom (see Fig.I .3). As the height h of the water varies very slowly, theescaping velocity quickly reaches a steady valueu. As the fluid velocity within the vessel isnegligible, the increase of the potential ascent in the time dt is simply given by the potentialascent(ifj2g)sudt of the fluid slice that escapes through theopening at velocity u. Thishsudt.
Therefore, the velocity u of efflux is thethe height h, in conformity with the law formulated byquantity must be equal to the actual descentvelocity..f2ili offree fall fromEvangelista Torricelli in1 644YII¥Fig. 1.3.1.1.3-IJ.UIdealized effiux through small openiog (without vena contracta).Bernoulli's lawBernoulli's most innovative application of this method concerned the pressure exerted by amoving fluid on the walls of its container, a topic of importance for the physician andphysiologist that he was. Previous writers on hydraulics and hydrostatics had onlyconsidered the hydrostatic pressure due to gravity.
In the case of a uniform gravityhg,the pressure per unit area on a wall portion was known to depend only on the depth ofthis portion below the free water surface. According to the law enunciated by Simon Stevinin1 605,it is given by the weightnormal section and the heighth.gh of a water column (of unit density) that has a unitIn the case of a moving fluid, Bernoulli defined andderived the 'hydraulico-static' .wall pressure. as follows.1312D.
Bernoulli [1738] p. 35. This reasoning assumes a parallel motion of the escaping fluid particle. Therefore,it only gives the velocity u beyond the contraction of the escaping fluid vein that occurs near the opening (Newton'svena contracta, cf.
Lagrange [1788] pp. 430- 1, Smith [1998]). On Torricelli's law and early derivations, cf. Blay[1985], [1992] pp. 331-352.1 3D. Bernoulli [1738] pp. 258-60. Mention of physiological applications is found in Bernoulli to Shoepflin, 25Aug. 1734, in D. Bernoulli [2002] p. 89: 'Hydraulico-statics will also be useful to understand animal economy withrespect to the motion of fluids, their pressure on vessels, etc.'WORLDS OF FLOWFig.
1.4.Daniel Bernoulli's figure accompanyinghis derivation of the velocity-dependence of pressure ([1738] plate).The section S of the vertical vessel ABCG of Fig. 1.4 is supposed to be much larger thanthe section s of the appended tube EFDG, which is itself much larger than the section e ofthe hole o.
Consequently, the velocity u of the water escaping through o is ,j2i/i. Owing to14the conservation of the flux, the velocity v within the tube is (e/s)u. Bemoulli goes on:If in truth there were no barrier FD, the final velocity of the water in the same tube[sfs times greater]. Therefore, the water in the tube tends to a greater[nisus] is hindered by the applied barrier FD.
By this pressingand resistance [nisus et renisus] the water is compressed [comprimitur], which compression [compressio] is itself kept in by the walls of the tube, and thence these toosustain a similar pressure [pressio]. Thus it is plain that the pressure [pressio] on thewould bemotion, but its pressingwalls is proportional to the acceleration . .
. that would be taken on by the water ifevery obstacle to its motion should instantaneously vanish, so that it were ejecteddirectly into the air.Based on this intuition, Bemoulli imagined that the tube was 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 actual descent of the water during thetime dt must be equal to the potential ascent it acquires while passing from the largesection S to the smaller section s, plus the increase of the potential ascent of the portionEabG of the fluid.
This gives (again, the fluid density iswherebpressurepressureone)hsvdt= (�)svdt+bsd(�}= Ea. The resulting value of the acceleration dv/dt isgh(gh-!Jl)jb.(1.5)The wallP must be proportional to this quantity, and it must be identical to the staticin the limiting case v = 0. It is therefore given by the equation(1.6)14D. Bernoulli [1738] pp. 258-9. Translated In Truesdell [1954] p.
XXVII. The compressio in this citationperhaps prefigures the internal pressure later introduced by Johann Bernoulli.THE DYNAMICAL EQUATIONS9which means that the pressure exerted by a moving fluid on the walls is lower than thestatic pressure, the difference being half the squared velocity (times the density). Bernoulliillustrated this effect in two ways (see Fig.1.5): by connecting a narrow vertical tube to thehorizontal tube EFDG, and by letting a vertical jet surge from a hole in this tube.The modern reader may here recognize Bernoulli's law.
In fact, Bernoulli did not quite(1.6), because he chose the ratio sfe rather than the velocity v as the relevantwall pressure, whereas modern physicistsapply Bernoulli's Jaw to the internal pressure of a fluid.write eqnvariable. Also, he only reasoned in terms ofThere were other limitations to Bernoulli's hydrodynamics, of which he was largelyaware. He knew that in some cases part of the live force of the water went to eddyingmotion, and he even tried to estimate this loss in the case of a suddenly enlarged conduit.He was also aware of the imperfect fluidity of water, although he decided to ignore it in hisreasoning.
Most importantly, he knew that the hypothesis of parallel slices only held fornarrow vessels and for gradual variations of their section. But his method confined him tothis case, since it is only for systems with one degree of freedom that the conservation oflive forces suffices to determine the motion.151.2 Johann Bernoulli's HydraulicaIn1742, Johann Bernoulli published his Hydraulica, with an antedate (1732) that made itpredate his son's treatise.
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