Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 56

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Section 5.2 recounts howWilliam Thomson, in 1 8 7 1 , discussed the instability of a water surface under wind,independently of Helmholtz and with a different method.In this and Helmholtz's case, instability was derived from the hydrodynamic equations.In Stokes's case, it was only a conjecture. Yet the purpose was the same, namely, to explainobserved departures from exact solutions of Euler's equations. In contrast, Thomson'svortex theory of matter required stability for the motions he imagined in the primitive1Thomson and Tail [1 867] par. 346.WORLDS OF FLOW184perfect liquid of the world. These considerations, which began inSection5.3.1867,are discussed inAs Thomson could only prove the stability of motions simpler than those heneeded, for many years he contented himself with an analogy with the observed stability ofsmoke rings.

At last, in the late1 880s,he became convinced that vortex rings wereunstable.Owing to their different interests, Stokes and Thomson had opposite biases abouthydrodynamic (in)stability. This is illustrated in Section5.4 through an account of theirlong, witty exchange on the possibility of discontinuity surfaces in a perfect liquid. Fromhis first paper (1842) to his last letter to Thomson (1901), Stokes argued that the formationof surfaces of discontinuity provided a basic mechanism of instability for the flow of aperfect liquid past a solid obstacle. Thomson repeatedly countered that such a processwould violate fundamental hydrodynamic theorems and that viscosity played an essentialrole in Stokes's alleged instabilities.

The two protagonists never came toeven though they shared many cultural values within and outside physics.Section5.5anagreement,deals with the (in)stability of parallel flow. The most definite nineteenth­century result on this topic was Lord Rayleigh's criterion of 1 880 for the stability of two­dimensional parallel motion in a perfect liquid. The context was John Tyndall's amusingexperiments on the sound-triggered instability of smoke jets. In1883, Osbome Reynolds'sp�ecise experimental account of the transition between laminar and turbulent flow incircular pipes motivated further theoretical inquiries into parallel-flow stability.

Cambridgeauthorities, including Stokes and Rayleigh, selected this question for the Adams prize of1 889.This prompted Thomson to publish proofs of instability for two cases of parallel,two-dimensional viscous flow. Rayleigh soon challenged these proofs. William Orr provedtheir incompleteness in1907, thus showing the daunting difficulty of the simplest questionsof hydrodynamic stability.5.1. Divergent flows5.

1 . 1FluidjetsPioneering considerations of hydrodynamic stability are found in Stokes's first paper,published in1 842and devoted to two-dimensional and cylindrically-symmetric steadymotions of a perfect liquid obeying Euler's equation. From an analytical point of view,most of Stokes's results could already be found in Lagrange's or J. M. C. Duhamel'swritings. Stokes's discussion of their physical significance was, nonetheless, penetratingand innovative. Struck by the difference between computed and real flows, he suggestedthat the possibility of a given motion did not imply its necessity; there could be othermotions compatible with the same boundary conditions, some of which could be stableand some others unstable.

'There may even be no stable steady mode of motion possible,in which case the fluid would continue perpetually eddying.'2As a first example of instability, Stokes cited the two-dimensional flow between twosimilar hyperbolas. An experiment of his own showed that the theoretical hyperbolic flowonly held in the narrowing case. He compared this result with the fact that a fluid passingthrough a hole from a higher pressure vessel to a lower pressure vessel forms a jet, instead2Stokes [1842]pp.10-1 1.INSTABILITY185of streaming along the walls as the most obvious analytical solution would have it (seeFig.5.1).Although Mariotte, Bemoulli, and Borda already knew of such effects, Stokeswas the first to relate them to a fundamental instability of fluid motion and to enunciate ageneral tendency of a fluid 'to keep a canal of its own instead of spreading out'.3In the case represented in Fig.5.1, Stokes argued that, according to Bemoulli's theorem,the velocity of the fluid coming from the first vessel was completely determined by thepressure difference between the two vessels.

This velocity was therefore homogeneous, andthe moving fluid had to form a cylindrical jet in order to comply with flux conservation.Dubious though it may be (for it presupposes a uniform pressure in the second vessel), thisreasoning documents Stokes's early conviction that nature sometimes preferred solutionsof Euler's equation that involved surfaces of discontinuity for the tangential component ofthe velocity.This conviction reappears in a mathematical paper that Stokes published four yearslater. There he considered the motion of an incompressible fluid enclosed in a rotatingcylindrical container, a sector of which has been removed (see Fig.5.2).For an acutesector, the computed velocity is infinite on the axis of the cylinder.

Stokes judged that inthis case the fluid particles running toward the axis along one side of the sector would 'takeFig. 5.1.The formation of a jet as a liquid is forced through a hole in a vessel A into another vessel B. FromStokes to Kelvin, 13 Feb. 1858, ST.cFig. 5.2.The formation of a surface of discontinuity (Oe) during the rotation of a cylindrical container (sectionOABC). After sliding along OA, the fluid particle a shoots off at the edge 0. From Stokes [I 847b] p.

310.3Stokes [1 842] p. 1 1.1 86WORLDS OF FLOWoff to form a surface of discontinuity. For the rest of his life, Stokes remained convincedof the importance of such surfaces for perfect-fluid motion. Yet he never offered amathematical theory of their development.45.1.2The pendulumAs we saw in Chapter 3, much of Stokes's early work was motivated by a more concreteproblem of fluid motion, namely, the effect of the ambient air on the oscillations of apendulum. When applied to the spherical bulb of the pendulum, Euler's hydrodynamicsgave no damping at all.

In 1 843, Stokes considered two kinds of instabilities that couldexplain the observed resistance. Firstly, he imagined that the fluid particles along thesurface of the sphere would come off tangentially at some point, forming a surface ofdiscontinuity. Secondly, he evoked his earlier conviction that divergent flow was unstable:It appears to me very probable that thespreading out motionof the fluid, which issupposed to take place behind the middle of the sphere or cylinder, though dynam­ically possible, nay, theonly motion dynamically possible when the conditions whichhave been supposed are accurately satisfied, is unstable; so that the slightest causeproduces a disturbance in the fluid, which accumulates as the solid moves on, till themotion is quite changed. Common observation seems to show that, when a solidmoves rapidly through a fluid at some distance below the surface, it leaves behind it asuccession of eddies in the fluid.Stokes went on to ascribe fluid resistance to the vis viva of the tail of eddies, as Ponceletand Saint-Venant had already done in France.

To make this more concrete, he recalledthat a ship had the least resistance when it left the least wake. 5In the following years, Stokes realized that these instabilities did not occur in thependulum case. The true cause of damping was the air's internal friction. In 1 845, Stokessolved the linearized Navier-Stokes equation for an oscillating sphere and cylinder,representing the bulb and thread, respectively, of a pendulum. The excellent agreementwith experiments left no doubt about the correctness and stability of his solutions.

6For the sake of completeness, Stokes also examined the case of a uniform translation ofthe sphere and cylinder, which corresponds to the zero-frequency limit of the pendulumproblem. In the case of the sphere, he derived the resistance law that bears his name. In thecase of the cylinder, he encountered the paradox that the resulting equation does not havea steady solution (in a reference system bound to the cylinder) that satisfies the boundaryconditions. Stokes explained:The pressure of the cylinder on the fluid continually tends to increase the quantity offluid which it carries with it, while the friction of the fluid at a distance from thesphere continually tends to diminish it. In the case of the sphere, these two causeseventually counteract each other, and the motion becomes uniform.

But in the case ofa cylinder, the increase in the quantity of fluid carried continually gains on thedecrease due to the friction of the surrounding fluid, and the quantity carriedincreases indefinitely as the cylinder moves on.4Stokes [1 847b] pp. 305-13.6See Chapter 3, pp. 1 39-40.'Stokes [1843] pp. 53-4. See Chapter 3, pp.

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