Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 46
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In other words, vortex linesfollow the motion of the fluid. Equation (4.5) also implies that, during the motion of thefluid, vortex filaments stretch in the same proportion as the rotational velocity varies.Since the fluid is incompressible, this longitudinal stretching implies a sectional shrinkingin inverse proportion.
In other words, the product w · dS of a section of the filament bytwice the amount of rotation in this section remains the same during the motion of thefluid.14Lastly, this product is the same all along a given filament. In order to prove this,Helmholtz integrated the vector w across the closed tubular surface delimiting a piece ofvortex filament. This integral is equal to the difference of the products w dS taken at thetwo extremities of the piece; and it is also equal to the integral of 'l w over the volume ofthe piece, which is zero following the definition of w.15In summary, vortex filaments are stable structures of the fluid. The product of therotation by the section of a filament, which Helmholtz called 'intensity', does not vary intime, and is the same all along the filament.
From the latter property, Helmholtz concludedthat vortex filaments could only be closed on themselves or end at the limits of the liquid. 16A striking feature of Helmholtz's demonstration of these theorems is the intimateassociation of analytical relations with geometrical representations. In nineteenth-centuryphysics, this quality seems more typically British.
In fact, Stokes, Thomson, and James··12Helmholtz [1858] pp. 1 1 0-11. For d'Aiembert's anticipation, see Chapter I, p. 20.13Ibid. pp. 1 1 1-12.14Ibid. pp. 1 02-3, 1 12-13. 'Vortex lines' and 'vortex filaments' are Tait's translations for ' Wirbellinien' and' Wirbelfaden', cf. Tait to Helmho1tz, 22 Apr. 1967, HN.15He1mholtz [1858] pp. 1 1 3-14.16Ibid. p.1 14. The latter conclusion is only true in topologically-simple cases (cf. Epple [1998] pp.
313-14).VORTICES151Clerk Maxwell anticipated some elements of Helmholtz's reasoning. In 1 845, Stokesintroduced the decomposition of the instantaneous motion of an element of fluid intotranslation, dilations, and rotation, and gave the analytical expression for the rotation. In1 849 and for magnetism, Thomson defined 'solenoidal' distributions of the magneticpolarization M for which the relation \7 M = 0 holds; and he decomposed the corresponding magnets into elementary tubes, as Helmholtz later decomposed vortex motioninto vortex filaments. In his memoir 'On Faraday's lines of force', published in 1 855/56,Maxwell defined 'tubes of force' in a manner quite similar to Hehnholtz's definition ofvortex filaments.
As a friend ofThomson's, Helmholtz may have been partly aware of thisBritish field geometry.·4.2.2The electromagnetic analogyThe British outlook on Helmho1tz's paper is also evident in the next section concernedwith the inverse problem of determining the velocity of the fluid when the distribution ofthe vorticity oo is known. Helmholtz sought the solutions of the equations V' x v = oo and\7 v = 0 in the form·V = V' <p + V'The potential<p satisfiesXA.(4.6)the equation !!.cp = 0 in the fluid mass, and the vector A satisfies\7(\7 A) - !!.A = oo.(4.7)·Helmholtz wanted to retrieve the simpler equation !!.A = -oo which makes the components of A the potentials of fictitious masses measured by the components of ooj41T.
This isimmediately possible if all the vortex filaments of the fluid are closed, since the vectorpotential,A(r) =141TJ lr - r' l droo(r'),(4.8)then satisfies \7 A = 0. In the general case, for which some vortex filaments abut on thesurface of the liquid, Hehnholtz prolonged the filaments beyond the real liquid so that theyall became closed, which brought him back to the previous, simpler problem. 17Applying the operation V' x to the expression (4.8) for the potential A, Hehnholtzrecognized the Biot-Savart formula of electromagnetism: the fluid velocity correspondingto a given distribution of vorticity oo is exactly like the magnetic force produced by theelectric-current distribution oo.
Helmholtz abundantly exploited this analogy, which gavehim a direct intuition for the fluid motion around vortices. 1 8Similar reasoning is easily identified in British sources. In his memoir on diffraction of1 849, Stokes introduced the decomposition (4.6) to determine a vector from its curl. In 'OnFaraday's lines of force' (1 855), Maxwell applied this method to the determination of themagnetic field H generated by the current j. His starting-point was the equation\7 x H = j, which he had obtained by studying the geometry of the magnetic field arounda current loop, and which corresponds to Helmholtz's \7 x v = oo.·17Helmholtz [1 858] pp.
l l4-1 17.1 8/bid. pp. JJ7-[8.WORLDS OF FLOW1 52The recourse to an analogy between electromagnetism and continuum mechanics wasalso a British specialty, inaugurated by Thomson. There is an interesting difference,however. Whereas Thomson and Maxwell used such analogies to shed light on electromagnetic phenomena and structure their theories, Helmholtz did the reverse. He usedelectromagnetic action at a distance, which was most familiar to him, for a better understanding of the motions of a mechanical continuum. This inversion explains why he didnot mention that his analogy between electromagnetism and vortex motion led to the fieldequation V' x H = j, which was unknown on the Continent.Lastly, Helmholtz shared the British engagement in energetics, of which Thomson andhimself were the main founders.
For the ideal fluid on which Helmholtz reasoned, thekinetic energy(4.9)is invariable if the walls do not move, because external forces deriving from a potentialcannot perform any work on an incompressible fluid. If, in addition, the vortex motionoccurs very far from the walls, recourse to eqn (4.6) and integration by parts yieldT=�Jpw · A dT.(4. 1 0)Helmholtz exploited the invariance of this integral in his subsequent discussion of theinteractions between two vortices. 194.2.3Vortex sheets, lines, and ringsIn the last section of his memoir, Helmholtz applied his general theorems and analogies tosimple cases of vortex motion in an infinite fluid.
The most trivial case is that of a uniform,plane vortex sheet. The incompressibility of the fluid implies that the normal velocity ofthe fluid should be the same on both sides of the sheet, while the equation V' x v = wimplies a discontinuity ew of the tangential velocity if w represents the average intensity ofthe vorticity within the sheet and e is the infinitesimal thickness of the sheet.
Within thesheet the fluid moves at a velocity intermediate between the velocities on both sides. Sincevortex lines follow the motion of the fluid of which they are made, the sheet must move at avelocity which is the average of the fluid velocities on both sides.20As we will see shortly, this special example of vortex motion played an essential role inHelmholtz's later hydrodynamics, at least because it showed that tangential discontinuitiesof the fluid motion were compatible with Euler's equations. Earlier investigators usuallyassumed the existence of a velocity potential, and thus excluded finite slips in the flow.Helmholtz not only demonstrated the mathematical existence of such solutions, but alsoindicated a way to realize them, namely, by bringing together two masses of liquid movingat different, parallel velocities.
2119Helmholtz [1858] pp. 123-4.21/bid.20/bid. pp. 121-2.p. 122. As will be shown in Chapter 5, pp. 185-6, Stokes repeatedly considered discontinuities ofHelmholtz's type (already in Stokes [1 842]), but never developed their analysis very far.VORTICES!53The next simple case of vortex motion is that of a single, rectilinear vortex filament. Forsymmetry reasons the filament must remain in a constant position.
According to theelectromagnetic analogy, the fluid rotates around the filament at a linear velocity inverselyproportional to the distance from the filament. The next case considered by Helmholtz isthat of two parallel rectilinear filaments with the intensities i1 and i . The corresponding2fluid velocity is the superposition of the velocities due to each filament.
The velocity of thefluid in the first filament is equal to the rotation due to the second, and vice versa. As thefilaments must move with the velocity of the fluid of which they are made, their mutualinfluence results in a uniform rotation around their barycenter for the masses i1 and i •222The case of a vortex ring is more complex, because the velocity imparted on the vortexby the vortex itself no longer vanishes. The rings must have a finite section for this selfinteraction to remain finite. Using elliptic integrals, energy conservation, and barycentricproperties, Helmholtz proved that a single circular ring must move along its axis withoutsensible change of size, in the direction of the flow at the center of the ring.
When thesection of the ring becomes infinitely thin, this velocity diverges logarithmically.23When two rings are present, they are also subjected to the fluid velocity imparted by theother ring. Let us start, for instance, with two identical rings that have the same axis.Initially, they travel along this axis with the same velocity.
The front ring, however, mustwiden under the effect of the other ring, and the back ring must shrink. Since wider ringstravel slower, the back ring catches up with the front ring after a while. Then the sizevariations are inverted, and the relative motion slows down until the two rings are again ofthe same size, which brings us back to the initial configuration. In summary, the rings passalternately through each other (see Fig. 4.2).24Helmholtz indicated how to observe this dance of the vortex rings with a spoon and acalm surface ofwater. Immersing the spoon vertically and withdrawing it quickly creates ahalf vortex ring, whose two ends form small dips on the water surface. As anyone canverify, these rings do behave as Helmholtz says.
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