Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 57
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1 36-7.INSTABILITY187Therein Stokes saw a symptom of instability:When the quantity of fluid carried with the cylinder becomes considerable comparedwith the quantity displaced, it would seem that the motion must become unstable, inthe sense in which the motion of a sphere rolling down the highest generating line ofan inclined cylinder may be said to be unstable.If the cylinder moved long enough in the same direction (as would be the case for thesuspending wire of a very slow pendulum) then 'the quantity of fluid carried by the wirewould be diminished, portions being continually left behind and forming eddies.' Stokesalso mentioned that in such an extreme case the quadratic term of the Navier-Stokesequation might no longer be negligible.
According to a much later study by Car! WilhelmOseen, this is the true key to the cylinder paradox. 75.1.3Ether dragAir and water were not the only imperfect fluid that Stokes had in mind. In 1 846 and 1848,he discussed the motion of the ether in reference to the aberration of stars. In his view theether behaved as a fluid for sufficiently slow motions, since the Earth and celestial bodieswere able to move through it without appreciable resistance. However, its fluidity couldonly be imperfect, since it behaved as a solid for the very rapid vibrations implied in thepropagation of light. Stokes explained the aberration of stars by combining these twoproperties in the following manner.8He first showed that the propagation of light remained rectilinear in a moving medium,the velocity of which derived from a potential.
Hence any motion of the ether that met thiscondition would be compatible with the observed aberration. Stokes then invokedLagrange's theorem, according to which the motion of a perfect liquid always meets thiscondition when it results from the motion of immersed solid bodies (starting from rest).For a nearly-spherical body like the Earth, Stokes believed the Lagrangian motion to beunstable (for it implies a diverging flow at the rear of the body). However, his ether was animperfect fluid, with tangential stresses that quickly dissipated any departure from gradient flow: 'Any nascent irregularity of motion, any nascent deviation from the motion forwhich [v dr] is an exact differential, is carried off into space, with the velocity of light, bytransversal vibrations.
'9In the course of this discussion, Stokes noted that his solution of the (linearized) NavierStokes equation in the case of the uniformly-moving sphere did not depend on the value ofthe viscosity parameter and yet did not meet the gradient condition. Hence an arbitrarilysmall viscous stress seemed sufficient to invalidate the gradient solution. Stokes regardedthis peculiar behavior as a further symptom of the instability of the gradient flow.In summary, in the 1 840s Stokes evoked instability as a way to reconcile the solutions ofEuler's equations with observed or desired properties of real fluids, including the ether.
Heregarded a divergence of the lines of flow (in the jet and sphere cases) and fluid inertia (inthe cylinder case) as a destabilizing factor, and imperfect fluidity (viscosity or jelly-likebehavior) as a stabilizing factor (explicitly in the ether case, and implicitly in the pendulum·7Stokes [1850b]pp.65-7. Cf. Lamb [1932]pp.609-17.8Stokes [1 846c], [1 848b]. Cf., e.g., Wilson [1987]9Stokes [1 848b]p.9.pp.132-45.WORLDS OF FLOW188bulb case). His intuition of unstable behavior derived from common observations of realflows and from the implicit assumption that ideal flow behavior should be the limit of realfluid behavior for vanishing viscosity.Stokes did not attempt a mathematical investigation of the stability of flow. He did offera few formal arguments, which today's physicist would judge faiiacious.
His deduction o fjet formation was based on an unwarranted assumption of uniform pressure i n thereceiving vessel. The steady flow around a cylinder, which he believed to be impossible,is in fact possible when the quadratic terms in the Navier-Stokes equation are no longerneglected. The argument based on the zero-viscosity limit of the flow around a sphere failsfor a similar reason. Stokes's contemporaries did not formulate such criticisms. Rather,they noted his less speculative achievements, namely, new solutions of the hydrodynamicequations that bore on the pendulum problem, and rigorous, elegant proofs of importanthydrodynamic theorems.5.2 Discontinuous flowIn Chapter4,we saw how Helmholtz made discontinuity surfaces a basic element ofperfect-liquid dynamics and derived the spiral growth of any bump on such a surface in1 868.This instability, to which Helmholtz attributed important physical consequencesincluding fluid mixing, wave formation, and meteorological perturbations, is now caiiedthe 'Kelvin-Helmholtz' instability, owing to its similarity with another instability studiedby Wiiiiam Thomson in1 871.Thomson's consideration i s related t o the strange episode recounted in Chapter 2, thatwhile slowly cruising on his personal yacht and fishing with a line, he observed a beautifulwave pattern and explained it by the combined action of gravity and capillarity.
In anatural extension of this theory, he took into account the effect of wind over the watersurface, and showed that the waves grew indefinitely when the wind velocity exceeded acertain, smaii limit that vanished with the surface tension. In other words, the plane watersurface is unstable for such velocities. The calculation proceeds as follows.10A solution of Euler's equation is sought for which the separating surface takes the planemonochromatic waveformy=7](x,t)=aei(kx-wt) ,(5. 1 )the x-axis being in the plane o f the undisturbed water surface, and the y-axis being normalto this plane and directed upwards.
Neglecting the compressibility of the two fluids, and'assuming irrotationality, their motions have harmonic velocity potentials cp and cp . Byanalogy with Poisson's wave problem, Thomson guessed the form<p=Ceky+i(kx-wt)(5.2)for the water, and10Thomson [187Ja], [187Jb], [187lc]. See Chapter 2, pp. 87-8. Some commentators, including Lamb ([1932]p. 449), have Thomson say that the plane surface is stable for lower velocities, which leads to an absurdly highthreshold for the production of waves (about twelve nautical miles per hour). Thomson did not and could not stateas much, since he only considered irrotational perturbations of perfect fluids.INSTABILITY''P1 89= VX + C'e-ky+i(kx-wt)(5.3)for the air, where v is the wind velocity.A first boundary condition at the separating surface is that a particle of water originallybelonging to this surface must retain this property.
Denoting byinates of this particle at timet, this givesx(t) and y(t) the coordy(t) = 'Tl(x(t),t)at any(5.4)t, or, differentiating with respect to time,o<po<p O'T/O'T/oyoxot-= --+OXwheny = 'T/(x,t).(5.5)A similar condition must hold for the air. The third and last boundary condition is therelation between pressure difference, surface tension, and curvature. For simplicity, capillarity is neglected in the following so that the pressure difference vanishes. The waterpressure P is related to the velocity potential 'P by the equationP1o<p+ 2 p(\1'P)2 + pgy + p Eft = constant,(5.6)obtained by the spatial integration of Euler's equation.
A similar relation holds for the air.Substituting the harmonic expressions for <p, <p1, and 'T/ into the boundary conditions andretaining only first-order terms (with respect toa, C, and C') leads to the relationsCk = -iaw, C'k = ia(w - kv)(5.7)p(ga - i Cw) = p'[ga - iC'(w - kv)] .(5.8)andEliminatinga, C, and C' givespw2 + p1(w - kv)2 = gk(p - p').w is negative ifg p2 - p'2V2 > .k pp'(5.9)The discriminant of this quadratic equation in---(5.1 0)Hence there are exponentially-diverging perturbations of the separation surface for anyvalue of the velocity v; and the water surface is unstable under any wind, no matter howsma11.
1 111To every growing mode there corresponds a decaying mode by taking the complex�conjugate solution of eqn(5.9). This seems incompatible with the growth derived in the vortex-sheet consideration of Chapter 4, pp. 1 61-2.In fact. it is not, because Thomsen's harmonic perturbations imply an initially heterogeneous distribution ofvorticity on the separating surface, whereas the vortex-sheet argument assumes an initially homogenous distribution (to first order).
For Thomsen's decaying modes, the initial distribution has an excess of vorticity on the lefthand side of every positive arch of the sine-shaped surface, and a defect on the right-hand side.WORLDS OF FLOW190This conclusion only holds when capillarity is neglected. As Thomson showed, thesurface tension implies a wind-velocity threshold for the exponential growth of shortwave, irrotational perturbations. Thomson did not discuss the limiting case of equaldensities for the two fluids.
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