Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 32
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1 1•Venturi intended to prove 'the lateral communication of motion in fluids' and to showits consequences for various kinds of flow. Some of the effects he described, such as theincrease of efflux obtained by adding a divergent conical end to the discharging pipe, werepurely inertial effects already known to Daniel Bernoulli. Others, such as the formation ofeddies, genuinely depended on internal friction. The eddies that Leonardo da Vinci hadbeautifully drawn for the flow past immersed bodies, those evoked by Daniel Bernoulli forsudden pipe enlargement, or those commonly seen in the smoke from chinmeys or in riversbehind bridge pillars, were all due, Venturi explained, to 'motion communicated from themore rapid parts of the stream to less rapidly moving lateral parts' (see Fig.3.1).9Du Buat [1786) vol.
I , pp. 22, 39-41, 58-59, 89-90. Du Buat's notion of fluid viscosity or cohesion was notquite identical with internal friction as we now understand it. Du Buat meant an 'adhesion' of the molecules thatneeded to be overcome to separate them, the resistance to this separation being proportional to its suddenness. Hebelieved (ibid. p. 41) that the microscopic structure of the surface of the pipe or channel had no effect on theretardation, for it was hidden by the adhering layer of fluid.1°Coulomb [1 800] pp. 261 (two kinds of resistance), 287 (quote).
Cf. Gillmor [1971] pp. 1 65-74.11Newton [1687] book 2, prop. 51; Venturi [1797]. Cf. Saint-Venant [1 887b] pp. 41-4 and Dobson [1999](Newton), Rouse and Ince [1957] pp. 134-37 (Venturi).106WORLDS OF FLOW(a)(b)Fig. 3.1.Eddy formation according to (a) da Vinci and (b) Venturi (from Rouse and Ince [1957] p. 46 andVenturi [1797] plate).Accordingly, Venturi made eddy formation one of the principal causes of retardation inrivers, which current wisdom attributed to friction against banks and the bottom.
1 2Venturi prudently avoided deciding whether the lateral communication of motion wasoccasioned 'by the viscidity or mutual adhesion of the parts of the fluids, or their mutualengagement or intermixture, or the divergence of those parts which are in motion.' Nor didhe venture to suggest new equations of fluid motion.
As he explained in his introduction,The wisest philosophers have their doubts with regard to every abstract theoryconcerning the motion of fluids: and even the greatest geometers avow that thosemethods which have afforded them such surprising advancesinthe mechanics ofsolid bodies, do not afford any conclusions with regards to hydraulics, but such as aretoo general and uncertain for the greater number of particular cases.Venturi's memoir enjoyed a favorable review by the French Academicians Bossut, Coulomb, and Prony. Together with Du Buat's and Coulomb's works on fluid friction, itcontributed to revive the old Newtonian notion of friction between two contiguous layersof fluidY3.1.3 Girard's capillary tubesIn 1816, the Paris water commissioner and freshly-elected Academician Pierre-SirnonGirard applied Newton's notion to a six-month-long study of the motion of fluids incapillary tubes.
While his prominent role in the construction of the Canal de l'Ourcq andhis contribution to several hydraulic projects amply justified his interest in flow retardation, Girard had the more philosophical ambition of participating in Laplace's novelmolecular physics. He believed the same molecular cohesion forces to be responsible forthe capillarity phenomena analyzed by Laplace and for retardation in pipe flow. By1 2Venturi [1797] transl. in Tredgold [1826] p.
1 65.13Ibid. pp. 1 32-33, 129; Prony, Bossut, and Coulomb [1 799].VISCOSITY107experimenting on fluid discharge through capillary tubes, he hoped to contribute both tothe theory of molecular forces and to the improvement of hydraulic practice. 14In conformance with Du Buat's observations of reduced flows, Girard assumed that alayer of fluid adhered to the walls of the tube, and that the rest of the fluid moved with aroughly uniform velocity. Flow retardation then resulted from friction between themoving column of fluid and the adherent layer.
Girard favored experiments on capillarytubes, no doubt because measurements were easier in this case, but also because hebelieved (incorrectly by later views) that the uniformity of the velocity of the centralcolumn would apply better to narrower tubes (because of a presumably higher cohesionof the fluid).
He operated with copper tubes of two different diameters2and3mmand lengths (L) varying between20(D)of around2.20 m. The tubes werehorizontal and fed by a large water vessel under a constant height H (see Fig. 3.2). Girardmmcmandtook the pressure gradient in the tube to be equal to pghjL, where g is the acceleration ofgravity and p is the density of water. Following Coulomb and Prony, he assumed the formav +abt}andfor the retarding force on the unit surface of the tube, where v is the flow velocityand b are two tentative constants. The balance of the forces acting on a cylindricalslice of fluid then gives15HpgD4LGirard measured the rate of discharge-- av + bv2 ·7rD2v/4(3. 1)for various lengths and charges, at atemperature varying with the season or controlled artificially.
His first conclusion wasthat the quadratic friction term disappeared for tubes of sufficient length. Consequently, heFig. 3.2.Girard's apparatus for measuring discharge through narrow tubes (from Girard [1816] plate). Thewater from the tank D is maintained at a constant level in the tank A and flows through the horizontal tube(lying on xy) into the bucket T.14Girard [1816]. Cf. Grattan-Guinness [1990] vol.
I, pp. 563-65.15Girard [1816] pp. 257-58, 265.108WORLDS OF FLOWassumed the friction to be fundamentally linear, and the quadratic contribution to be due tovenacontracta and its subsequent oscillations). He then focused on the linear behavior, apparthe lack of (recti)linearity of the flow near the entrance of the tube (involving Newton'sently forgetting the engineer's interest in the quadratic contribution (which dominates inthe case oflarge pipes of any length).
He found that the 'constant'a significantly decreasedwhen the temperature rose, and that it varied with the diameter of the tube.16Girard produced a nice molecular explanation for these effects. A temperature increase,he reasoned, implies a dilation of the fluid and therefore a decrease in the mutual adhesiona. As for the dependence of a on the tube'se of the adherent layer of fluid, which impliesthe substitution of D - 2e for D in eqn (3.1).
For high temperatures the thickness e shouldof the fluid molecules expressed in the constantdiameter, Girard evoked the fmite thicknessbe negligible since there is little adhesion between the fluid and wall molecules. Then theoriginal formula (3.1) (withb = 0) and the proportionality of the discharge to the cube ofthe diameter hold approximately, as Girard's measurements with heated water seemed toconfirm. In a sequel to this memoir, Girard used glass tubes instead of copper and variousliquids instead of water, meaning to confirm his view that the thickness of the adheringlayer depended on molecular forces between the layer and the walL 17As he had little to offer to the hydraulic engineer, Girard wrote something for thephysiologist.
The capillary dimensions of vessels and the wetting of their walls, he noted,was essential to explain blood or sap circulation in animals and plants. Otherwise, bodytemperature could not control the circulation, and friction would wear the vessels. Girardexpressed his amazement at the 'simplicity of the means of Nature and the perfection ofher works' when seen in the light of his own research. His self-confident tone and hisprofessional authority easily convinced his contemporaries, including the Academicianswho welcomed him. Yet his experimental method and his theoretical reasoning falter whencompared with those of the best French experimenters of the day.
18In the absence of contemporary criticism, we may only imagine what flaws a morecareful contemporary could have detected in Girard's work. While estimating the chargeH of the tube, Girard did not include the loss of head due to the entrance in the tube, eventhough Du Buat had noted the importance of this correction for short pipes. In considering the variation of the discharge rate with the diameter of the tube, he used only twodifferent diameters and did not indicate how he had measured them. Judging fromGotthilf Hagen's later measurements, the numbers provided by the manufacturer or asimple external measurement could not be tmsted.These circumstances may in part explain why Girard did not obtain the D4 law for thedischarge, which we know to be quite accurate, why he found glass to provide a strongerdischarge than copper, and why he believed that retardation would be linear for anydiameter and velocity if the tube were long enough.
On the theoretical side, he conflated16Girard [1816] p. 285. Girard insisted (ibid. p. 287) that, contrary to Coulomb's case, the velocity did not needto be small for the quadratic term to disappear. Girard borrowed the expression for the accelerating force and theexpression 'linear motion' from Euler (ibid. p.
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