Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 43
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For large pipes, Stokes assumed a tangential pressure proportional tothe velocity squared at the walls, jnstified in Du Buat's and Coulomb's manner by surface irregularities.101 Stokes [l850b] pp. 7, 14-15.fig' I.r=----· __,.·--��c;I�< 'uFig. 3.9.11Poiseuille's apparatus to measure fluid discharge through capillary tubes (from Poiseuille (1844]).The reservoir P, originally ftlled with compressed air by the pump AXY, is connected to a barometric device(on the right), and to the flask M, which in turn feeds the elaborate glass part CABEFGD (enlarged above).VISCOSITY143in a previous memoir, he focused on the behavior of capillary vessels and decided toexamine experimentally the effects of pressure, length, diameter, and temperature on themotion of various liquids through capillary glass tubes.
He judged Girard's anteriormeasurements to be irrelevant, because capillary blood vessels were about one hundredtimes narrower than Girard's tubes.108Poiseuille produced the flow-generating pressure with an airpump and reservoir, invague analogy with the hearts of living organisms (see Fig. 3.9). He avoided the irregularities of open-air efflux and controlled temperature by immersing his capillary tubes in athermostatic bath.
He determined the discharge from the lowering of the fluid level in thefeeding flask. The most delicate parts of the measurements were the optical and hydrauliccontrol of the cylindricity of the capillary tubes, and the determination of the pressurehead. Like Girard, Poiseuille overlooked the entrance effect, which is fortunately negligible for very narrow tubes. He properly took into account hydrostatic head, viscousretardation in the larger tube leading to the capillary tube, and the pressure shift in a givenrun.
The description of his protocol was so meticulous as to include prescriptions for thefilters he used to purify his liquids. His results compare excellently with modern theoreticalexpectations. They of course include the Poiseuille law Q = KPR4 IL, P being the fall ofpressure and K a temperature-dependent constant. 109Poiseuille only mentioned Naviers theory to condemn it for leading to the wrong PR3 ILlaw. Unfortunately, Navier did not live long enough to know of Poiseuille's result. TheAcademicians who reviewed the physician's memoir (Arago, Babinet, and Piobert) did notknow that Navier had already obtained the R4 dependence in the case of a square tube ofside R with zero shift at the walls. It was left to Franz Neumann, who had probably knownHagen in Konigsberg, to give the first public derivation of the Hagen-Poiseuille law.Assuming zero velocity at the walls and making the internal friction proportional to thetransverse velocity gradient, Neumann derived the quadratic velocity profile and integrated it to obtain the discharge.
His student Heinrich Jacobson published this proof in1 860. The Base! physicist Eduard Hagenbach published a similar derivation in the sameyear, with an improved discussion of entrance effects and a mention of the Erschutterungswiderstand (agitation resistance) that occurred for larger pipes. Lastly, the Frenchmathematician Emile Mathieu published a third similar proof in 1 863.1103.7.3A slow integrationIt would be wrong to believe that these derivations of Poiseuille's law were meant tovindicate the Navier-Stokes equation.
Neumann and Mathieu did not mention Navier'stheory at all. Hagenbach did, but imitated Poiseuille in globally condemning Navier'sapproach. Newton's old law of the proportionality between friction and transverse velocity gradient was all that these physicists needed. Hermann Helmholtz was probably thefirst physicist to link the Navier-Stokes equation to the Hagen-Poiseuille law.p.108Poiseuille [1 844]; Arago, Babinet, and Piobert [1842]. Cf. Rouse and Ince [1957] pp.
160-1, Schiller [1933]89, Pedersen [1975].109Poiseuille [1844] p. 519. For a modern evaluation, cf. Schiller [1933] pp. 85-9.1 10Poiseuille [1 844] p. 521; Jacobson [1860] Hagenbach [1860] Mathieu [1863].WORLDS OF FLOW140proportional to density, which is at variance with the approximate constancy later proved10by James Clerk Maxwell. 2Stokes also considered the case of uniform translation, still in the linear approximationof the Navier-Stokes equation. For a sphere of radius R moving at the velocity V, heobtained the expression -67Tp,R V for the resistance, now called 'Stokes' formula'.passant,Enhe explained the suspension of clouds: according to his formula, the resistanceexperienced by a falling droplet decreases much more slowly with its radius than its weightdoes.
In the case of a cylinder, he found that no steady solution existed, because thequantity of dragged fluid increased indefinitely. He speculated that this accumulation103implied instability, a trail of eddies, and nonlinear resistance.At that time, Stokes did not discuss other cases of nonlinear resistance, such as theswiftly-moving sphere. In later writings, he adopted the view that the Navier-Stokescondition with the zero-shift boundary condition applied generally, and that the nonlinearity of the resistance observed beyond a certain velocity corresponded to an instabilityof the regular solution of the equation, leading to energy dissipation through a trail of104eddies.
This is essentially the modem viewpoint.3.7 The Hagen-Poiseuille law3.7.1Hagen's pipesStokes's pendulum memoir contains the first successful application of the Navier--Stokesequation with the boundary condition which is now regarded as correct. For narrow-pipeflow, Stokes (and previous discoverers of the Navier--Stokes equation) knew only ofGirard's results, which seemed to confirm the Navier-Poisson boundary condition. Yeta different law of discharge through narrow tubes had been published twice before Stokes'study, in 1839 and in 1 84 1 .The German hydraulic engineer Gotthilf Hagen was the first t o discover this law,without knowledge of Girard's incompatible results.
Hagen had learned precision measurement under Bessel and had traveled through Europe to study hydraulic constructions.As he had doubts about Prony's and Johann Eytelwein's widely-used formulas for piperetardation, he performed his own experiments on this subject in 1839. In order to bestappreciate the effect of friction, he selected pipes of small diameter, between 13mm.mmandAlthough the principle of the experiment was similar to Girard's, Hagen eliminatedimportant sources of error that had escaped Girard's attention. For example, he carefullymeasured the internal diameter of his pipes by weighing their water content.
Also, heavoided the irregularities of open-air efflux by having the pipe end in a small tank with a105constant water level (see Fig. 3.8).(2To his surprise, Hagen observed that, beyond a critical pipe-flow velocity of ordergh) 112, with h being the pressure head, the flowbecame highly irregular.
For better102Stokes [1850b] sects 2-3. On the wrong value of the viscosity coefficient, cf. Stokes, note appended to his[1850b], SMPP 3, pp. 137-41; Stokes to Wolf, undated (c. 1991), in Larmor [1907], vol. 2 pp. 323-4.103Stokes [1850b] 59, pp. 66-7. More on the cylinder case will be said in Chapter 5, pp. 186-7.104Cf. Stokes's letters of the 1870s and 1880s in Larmor [1907].105Hagen [1839]. Cf. Schiller [1933] pp. 83-4, Rouse and Ince [1957] pp. 157--61.VISCOSITYFig. 3.8.141Hagen's apparatus for measuring fluid discharge (from Hagen [1839)). The tank F feeds the cylinderB through the regulating device H.
The water level in the cylinder is determined by reading the scale Cattached to the floating disk D. The discharging tube A ends in the overflowing tank Kexperimental control, he decided to operate below this threshold. His experimental resultsare summarized by the formula(3.33)wherehis the pressure head,Qis the discharge,L is the length,a is a temperaturedependent constant, and {3 is a temperature-independent constant.
In true Konigsbergstyle, Hagen determined the coefficients and exponents by the method of least squares and106provided error estimates.Hagen correctly interpreted the quadratic term as an entrance effect, corresponding tothe Jive force acquired by the water when entering the tube. Assuming a conic velocityprofile, he obtained a good theoretical estimate for the f3 coefficient.
He attributed thelinear term to friction, and justified the 1 /.K+ dependence by combining the conic velocityprofile with an internal friction proportional to the squared relative velocity of successivefluid layers. Perhaps because this concept of friction later appeared to be mistaken, fullcredit for the discovery of theQL/.K+ Jaw has often been given to Poiseuille. Yet Hagen'spriority and the excellence of his experimental method are undeniable.3.7.2107Dr Poiseuille's capillary vesselsJean-Louis Poiseuille, a prominent physician with a Polytechnique education, performedhis experiments on capillary-tube flow around 1 840, soon after Hagen.
He had noparticular interest in hydraulics, but wanted to understand 'the causes for which someorgan received more blood than another.' Having eliminated a few received explanations1 06Hagen [1839] pp. 424, 442.101/bid. pp. 433, 437, 441.144WORLDS OF FLOWHelmholtz's interest in fluid friction derived from his expectation that it would explain aleftover discrepancy between theoretical and measured resonance frequencies in organpipes. In 1 859, he derived a hydrodynamic equation that included internal friction, andasked his friend William Thomson whether it was the same as Stokes's, of which he hadheard but had not seen. The answer was yes.
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