Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 61
Текст из файла (страница 61)
see Chapter 4, footnote 48, p. 163.WORLDS OF FLOW200contest Thomson's assertion that discontinuity surfaces could not be formed by anynatural process. In Stokes's view, a drop of perfect liquid falling on a calm surface ofthe same liquid led to discontinuous motions. So did the 'goring' of fluid on itself, asdrawn in Fig.
5.6. 50Thomson rejected these suggestions, as well as Helmholtz's idea of bringing into contacttwo parallel plane surfaces bounding two portions of liquid moving with different velocities. In every case, he argued, the contact between the two different fluid portions alwaysbegins at an isolated point, and the boundary of the fluid evolves so that no finite slip everoccurs. The drawings of Fig. 5. 7 illustrate his understanding of the goring and raindropcases. In Hehnholtz's plane-contact process, the imperfect flatness of the surfaces achievesthe desired result. 5 1Seven years later, Thomson published another provocative article in Nature against the'doctrine of discontinuity'.
This time his target was the alleged formation of a surface ofdiscontinuity past a sharp edge. The relief from infinitely-negative pressure at the sharpedge, Thomson declared, never was the formation of a surface of discontinuity, whichcontradicted his minimum-kinetic-energy theorem. The true compensatory factors werefinite viscosity, finite compressibility, or the yielding boundary of the fluid. Thomsonillustrated the compensations using the example of a thin moving disc. When the firstfactor dominates, a layer of abrupt velocity change, or, equivalently, a vortex sheet withsmall thickness, is formed behind the moving solid. When the third factor dominates, asuccession of thin hollow rings is created behind the disk in a manner similar to that whichFig.
5.6.The goring of a liquid on itself according to Stokes.A discontinuity surface is formed when fg meets cd. FromStokes to Thomson, 4-7 Feb. 1887, ST.�(a)Fig.5. 7./t.JL(b)(a) The goring of a liquid on itself and (b) the fall of a drop on a plane water surface, according toThomson. From Thomson to Stokes, 6-9 Feb. 1 887, ST.50Stokes to Thomson, 4, 7 Feb. 1887, ST. For cavitation around ship propellers, as discussed in Reynolds[1 873], see later on p. 246 (in real fluids vapor fills the cavities).5 1Thomson to Stokes, 6, 9 Feb.
1887, ST.INSTABILITY201\Fig. 5.8.6f�����-."••w··------- --------··---- ab'L---------------------------------ee'isformed at the edge of the disc A, the axis of which is ina'Thomson described inThomson's drawing for Rayleigh's 'dead water'theory of fluid resistance. A discontinuity surfacethe plane of the figure. From Thomson [1894] p. 220.1 887 for the moving sphere. Both processes imitate a surface ofdiscontinuity when the fluid is nearly perfect. However, the imitation is alwaysimperfect.
52The strict doctrine of discontinuity, Thomson went on, leads to an absurd theory ofresistance. His target was Rayleigh's 'dead-water' theory of resistance of 1 876, accordingto which the fluid remains at rest (with respect to the solid) in the space limited by a tubularsurface of discontinuity extending from the edges to infinity (see Fig.5.8).
The pressure inthe dead water immediately behind the solid is inferior to the pressure on the front of thebody, so that a finite resistance results. Whereas Rayleigh offered this picture as a solutionto d'Alembert's old paradox, Thomson denounced its gross incompatibility with experiment. The dead water, if any, could not realistically extend indefinitely rearwards. Moreover, the resistance measured by William Dines for a rectangular blade under normalincidence was three times larger than that indicated by Rayleigh's calculation in this case.Truncation of the discontinuity surface, Thomson showed, did not remove this discrepancy.
As a last blow to the dead-water theory, he conceived a special case in which it gavezero resistance (see Fig.5.4.35.9). 53Birth ofdiscontinuity surfacesStokes's reaction was strong and immediate. He had never supported the dead-watertheory, and believed instead that the main cause of resistance was the formation of eddies.52Thomson [1 894]. Thomson had already expressed this opinion in a letter to Hehnholtz of 3 Sept. 1868,quoted in Thompson [1910] p. 527: 'Is it not possible that the real cause of the formation of a vortex-sheet may beviscosity which exists in every real liquid, and that the ideal case of a perfect liquid, perfect edge, and infinitely thinvortex sheet, may be looked upon as a limiting case of more and more perfect fluid, finer and finer edge of solid.and consequently thinner and thinner vortex-sheet?'53Thomson [1894]; Rayleigh [1 876b].
See Chapter 4, p. 165.WORLDS OF FLOW202Fig. 5.9.Case of motion for which the dead-water theory gives zero resistance. The hatched tube EA moves tothe left through a perfect liquid, leaving a dead-water wake in its rear cavity and within the cylindrical surfaceof discontinuity which begins at LL. The longitudinal resultant of pressure on the front part E is very nearlyequal to the pressure at infinity times the transverse section of the tube, because the cylindrical part of thetube is much larger than its curved front part. The same equality holds exactly at the rear of the tube, becausethe pressure is continuous across the discontinuity surface and constant within the dead water.
Therefore, thenet longitudinal pressure force on the tube vanishes. From Thomson [ 1894] p. 228.He nonetheless maintained that the continuous, irrotational, and steady motion of aperfect liquid around a solid body with sharp edges was unstable. After conceding toThomson that this motion was that of minimum energy under the given boundaryconditions, he interjected: 'But what follows from that? There is the rub.' Instability, heexplained, was still possible:54What is meant by the motion being unstable? I should say, the motion is said to bestable when whatever small deviation from the phi motion [the minimum-energymotion, for which there exists a velocity potential [cp] is supposed to be produced, andthe fluid thenceforth not interfered with, the subsequent motion differs only by smallquantities from the phi motion, and unstable when the small initial deviation goes onaccumulating, so that presently it is no longer small.-! have a right to take for mysmall initial deviation one in which the fluid close to the edge shoots past the edge,forming a very minute surface of discontinuity.
The question is, Will this alwaysremain correspondingly minute, or will the deviation accumulate so that ultimately itis no longer small? I have practically satisfied myself that it will so accumulate, andthe mode of subsequent motion presents interesting features.Thomson replied that the would-be surface of discontinuity would 'become instantlyruffled, and rolled up into an 'crvTJpL9fLOV "{EA<Y<TfL<Y' (by the last word I mean laughing atthe doctrine of finite slip)'55 and would be washed away and left in the wake. Stokesdeclared himselfundisturbed by this objection. He knew well the instability of discontinuity surfaces, but their spiral unrolling was not a priori incompatible with their continualformation at the edge of a body. 'The rub' was still Thomson's pretense to derive stabilityfrom his minimum-energy theorem.
The theorem, Stokes explained, did not require thatthe actual motion should be that of minimum energy, because the additional energyneeded to create the discontinuity surface could result from work done by the externalpressures that sustained the flow. 5654Stokes to Thomson, 1 1 Oct. 1 894, ST.55Cf. Aechylus, The Prometheus bound, verses 89-90, 'KUf.L<i:rwv &vf}pt.9j.LOV -yEX.a.O'J.Lct', which literally means'a smile of countless waves'. The whole strophe reads (in George Thomsen's translation, Cambridge, 1932, p. 55):'0 divine Sky, and swiftly-winging Breezes,/0 River-springs, and multitudinous gleam/Of smiling Ocean-to thee,All-Mother Earth,IAnd to the Sun's all-seeing orb I cry:/See what I suffer from the gods, a god!'";.rhomson to Stokes, 14 Oct. 1 894; Stokes to Thomson, 27 Oct. 1 894, ST.
See also Stokes to Thomson, 22-23Nov. 1898, ST.INSTABILITY203Perhaps, Stokes wondered, there was another 'Kelvinian theorem' that truly excludedthe discontinuity. The only one that came out in later letters was the theorem that theangular momentum of every spherical portion of a liquid mass in motion, relative to thecenter of the sphere, is always zero, if it is so at any one instant for every spherical portionof the same mass. The theorem, Stokes judged, no more excluded the formation of asurface of discontinuity than Lagrange's and Cauchy's theorems (regarding fluid motionproduced by moving immersed solids) already did, for its proof required the continuity ofthe fluid motion near the walls.
57After a pause of four years, Stokes resumed the discussion with some considerations onthe growth of a 'baby surface of discontinuity' at a sharp edge. Presumably, Thomson hadobjected that the continuity of pressure across the baby surface was incompatible with thediscontinuity of velocity. Stokes explained that the growth of the surface and the resultingunsteadiness of the flow implied an additional term ocpjot in the pressure equation(Bemoulli's law) that counterbalanced the discontinuity of �pif. He also repeated hisconviction that Thomson's minimum-energy theorem was not incompatible with theformation of discontinuity surfaces.
58Thomson replied with a thought experiment (see Fig. 5.10):To keep as closely as possible to the point (edge!) of your letter of the 22nd, let E be anedge fixed to the interior of a cylinder, with two pistons clamped together by aconnecting-rod as shewn in the diagram, and the space between them filled withincompressible inviscid liquid. Let the radius of curvature of the edge be I0- 12 of acentimeter.The curvature still being finite, Thomson thought that Stokes would agree about theperfectly-determinate and continuous character of the fluid motion induced by pushingthe double piston.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
