Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 62
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A moderate velocity of the piston would then imply an enormouspressure, tending to break the connecting rod. Although Thomson did not say why, heprobably reasoned by combining Bemoulli's law and the impossibility of negative pressureat the edge, as he had done earlier for the flow around a globe. In the real world, Thomsonwent on, the connecting rod would either break, or yield slightly, thus allowing the liquidto leave the solid wall before it comes to the edge.
In neither case would there be a slip ofliquid over liquid. 59The argument backfired. In his response (20-21 , 26 Dec. 1 898, ST), Stokes placed thecylinder and pistons vertically, and counterpoised the double piston and liquid by meansof a string, pulley, and weight (see Fig. 5.1 1). Then a housefly perching on the upper pistonwould suffice to break a connecting rod of large, but finite, resistance to traction. Stokes'ssolution to this paradox was the formation of a surface of discontinuity past the edge,despite the lack of a strict angular point. 6057Stokes to Thomson, 27 Oct. 1894, 26 Dec. 1898; Thomson to Stokes, 23 Dec.
1898, ST. See also their lettersofl3, 1 8-20 Dec. 1900, and 4 Jan. 1901, ST.59Thomson to Stokes, 25 Nov. 1898, ST."Stoke to Thomson, 22-23 Oct. 1898, ST.60Stokes to Thomson, 20-21, 26 Dec. 1898 (paradox), 14 Feb. 1899 (solution). Another escape from theparadox would be to note that the fly cannot communicate a finite velocity to the piston, and therefore cannotinduce an infinite pressure of the fluid if the 'mike' solution still applies.WORLDS OF FLOW204Fig.
5 . 1 0.Thomson's diagram for a thought experiment regarding flow around a sharp edge. From Thomsonto Stokes, 25 Nov. 1898, ST.Fig. 5. 1 1 .1898,5.4.4Stokes's device for his housefly paradox, from the description in Stokes to Thomson, 20-21, 26 Dec.ST.Separation and boundary layerFrom the beginning, Stokes believed that surfaces of discontinuity were formed evenbehind smoothly-shaped obstacles. In previous letters, he had only focused on the infinitely-sharp edge because the instability of the 'mike' (minimum kinetic energy) solutionwas the easiest to understand in this case.
Two days after he enunciated the houseflyparadox, he re-expressed his conviction that the 'mike' solution for a uniform flow arounda cylinder was unstable at the rear of the cylinder and challenged Thomson for a proof ofstability in this case. He referred to the turbulent flow behind the pillars of a bridge as aninstance of this instability. 'It is hard to imagine', he reflected, 'that the instability whichthe commonest observation shows to exist is wholly due to viscosity, especially as anincrease of viscosity seems to tend to increased stability, not the reverse.' 6 1A week later, Stokes described how surfaces of discontinuity could be generated evenwithout a sharp edge:62I can see in a general way how it is that it is towards the rear of a solid movingthrough a fluid that a surface of discontinuity is formed.
I find that at the point of asolid which is the birthplace of such a surface . . . the flowing fluid must go off at a61Stokes to Thomson, 22 Dec. 1898, ST.62Stokes to Thomson, 27 Dec. 1898, ST.205INSTABILITYtangent, and the fluid at the other side of the surface of discontinuity must just at thebirthplace be at rest.In a crossing letter, Thomson denied instability in the perfect-liquid case, and proceededto explain the practical instability for a real fluid of small viscosity and negligible compressibility, such as water. He first considered the fluid motion induced by a suddenacceleration (from rest) of an immersed solid body: 63The initial motion of the water will be exceedingly nearly that of an incompressibleinviscid liquid (the motion of minimum kinetic energy).
There will be an exceedinglythin stratum of fluid round the solid through which the velocity of the water variescontinuously from the velocity of the solid to the velocity in the solution for inviscidfluid. It is in this layer that there is instability. The less the viscosity, the thinner is thislayer for a given value ofthe initial acceleration; but the surer the instability. Not verylogical this.Thomson did not say why he thought the thin layer ofvorticity to be unstable.
He onlyalluded to his earlier argument about the practical instability of the plane Poiseuille flow(parallel flow between two fixed parallel plates), to be discussed shortly. 64 He moved on toconsider what would happen to the fluid if the acceleration ceased and the body (now aglobe) was kept moving uniformly:If the velocity is sufficiently great, the motion of the fluid at small distances from itssurface all round will always be very nearly the same as if the fluid were inviscid, andthe difference will be smaller near the front part than near the rear of the globe.Here we have a description of what Ludwig Prandtl later called the boundary layer. Therest is more personal to Thomson:If now the whole fluid suddenly becomes inviscid and the globe be kept movinguniformly, the rotationally moving fluid will be washed off from it, and left movingturbulently in the wake, and mixing up irrotationally moving fluid among it.Thus, Thomson made viscosity responsible for the formation of an unstable state ofmotion, but regarded the instability of this state as unrelated to viscosity and thereforefelt free to 'turn off' viscosity to discuss it.
Although, for a given state of motion at a giveninstant, viscosity could only have a stabilizing effect, it could make a stable state evolveinto an unstable one. 65In his reply to this letter, Stokes expressed his agreement with everything Thomson hadsaid, except for what would happen if the viscosity were suddenly brought to zero. In hisopinion,the streams of right-handedly revolving and left-handedly revolving fluid at the twosides would have the rotationally moving fluid washed away, at least in the side trails,and the streams would give place to streams bounded by surfaces of finite slip,commencing at the solid, and then being paid out from thence.
The subsequent63Thomson to Stokes, 27 Dec. 1898, ST.65Thomson to Stokes, 27 Dec. 1898, ST."'Thomson [1887c]. See later onpp.211-3.206WORLDS OF FLOWmotion would doubtless be of a very complicated character [owing to the HelmholtzKelvin instability].Again, Stokes wanted the inviscid behavior to be a limit of the low-viscosity behavior.
If adiscontinuity surface was formed in the ideal inviscid fluid case, then it had to play a role inthe practical case of a slightly-viscous fluid. 665.4.5EpilogueThe debate continued until Stokes's last letter to Thomson, dated 23 October 1901. In thislate period the two old friends stuck at their positions. They could not even agree on the(in)compatibility of Lagrange's theorem with the formation of discontinuity surfaces.Stokes refmed his picture of the formation of a discontinuity surface behind a moving solidsphere, so as to reach 'continuity in the setting of discontinuity'.
In the new picture thecontact line of the solid and surface began as a tiny circle around the rearward pole of thesphere, and then widened out until the surface took its final, steady shape. Stokes also madethe spiral unrolling of the discontinuity surface the true cause of eddying behind a solidobstacle:67It seems evident that the mere viscosity of water would be utterly insufficient toaccount for [the eddies] when they are formed on a large scale, as in a mill pool orwhirlpool . . . Of course eddies are modified by viscosity, but except on quite a smallscale I hold that viscosity is subordinate.
Of course, it prevents a finite slip, which itconverts into a rapid shear, but viscosity tends to stability, not to instability.Throughout their long, playful disagreement, Stokes and Thomson were driven bydifferent interests. Whereas Stokes wanted to understand the behavior of real liquids,Thomson primarily reasoned on the ultimate perfect liquid of the world. Thus, theyhad opposite prejudices on the stability properties of the flow of a perfect liquid pasta solid obstacle.
As the intrinsic mathematical difficulty of the subject prevented asettling of issues by a rigorous argument, they relied on intuition and past experience.Stokes appealed to the natural world and conjectured that the behavior of perfectliquids should reflect that of real liquids with small viscosity and compressibility.Thomson instead appealed to the energy-based dynamics that founded his natural philosophy. Hence he promoted the minimum-energy flow and an energy-based criterionof stability.The Thomson-Stokes debate is not only instructive for the kind of theoretical prejudicesit reveals, but also as an indication of the powers and limits of intuitive discussions ofhydrodynamic instability.
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