Oct. 28, 1880] 



NATURE 



619 



I \ 



toixed (like two eggs " whipped " together) that, infinitely near 



to any portion of either, there shall be some of the other. 



[ 2. The case of a single Helmholtz ring, reduced by diminution 



Cif its aperture to an infinitely long tube coiled within the 



Inclosure. 



3. The ca=e of a single vortex column, with two ends on the 

 lljoundary, bent till its middle meets the boundary ; and farther 

 ■bent and extended, till it is broken into two ecjual and opposite 

 ivortex columns ; and then farther dealt with till these two are 

 whipped together to mutual annihilation. 



IV. To avoid for the present the extremely difficult general 

 question illustrated (or suggested) by the consideration of such 

 cases, confine ourselves now to two-dimensional motions in a 

 space bounded by two fixed parallel planes and a closed cylindric 

 surface perpendicular to them, subjected to changes of figure 

 (but always truly cylindric and perpendicular to the planes). It 

 is obvious that, « ith the limitation to two-dimensional motion, 

 the energy cannot be either infinitely small or infinitely great 

 with any given vorticity and given cylindric figure. Hence, 

 under the given conditions, there certainly are at least two stable 

 steady motions. We shall, however, fee further that possibly 

 in every ca^e except cases of a narrow, well-defined character, 

 and certainly in many cases, there is an infinite number of stable 

 steady motions. 



V. In the present case, clearly, though there are an infinite 

 number of unstable steady motions, there are only two stable 

 steady motions — those of absolute maximum and of absolute 

 minimum energy. 



VI. In every steady motion, when the boundary is circular, 

 the stream lines are concentric circles, .and the fluid is distri- 

 buted in co-axial cylindric layers of equal vorticity. In the 

 stable motion of maximum energy, the Viirticity is greatest at 

 the axis of the cylinder, and is less and less outwards to the 

 circumference. In the stable motion of minimum energy the 

 vorticity is smallest at the axis, and greater and greater out- 

 wards to the circumference. To ex ress the conditions symbo- 

 lically, let T be the velocity of the fluid at distance r from the 

 axis (understood that the direction of the motion is perpendicular 

 to the direction of r) ; the vorticity at d'stance r is — 



K^4f)■ 



If the value of thi; expression diminishes from r — oio r — a, 

 the motion is stable, and of maximum energy. If it increases 

 from ?- = o to ?- = a the motion is stable and of minimum energy. 

 If it increases and diminishes, or diminishes and increases, as r 

 increa-es continuously, the motion is unstable. 



VII. As a simplest subcase, let the vorticity be uniform 

 through a given portion of the whole fluid, and zero through 

 the remainder. In the stable motion of greatest energy, the 

 portion of fluid having vorticity will be in the shape of a 

 circular cylinder rotating like a solid round its own axis, 

 coinciding with the axis of the inclosure ; and the remainder of 

 the fluid will revolve irrotationally around it, so as to fulfil the 

 condition of no finite slip at the cylindric interface between 

 the rotational and irrotational portions of the fluid. The 

 expression for this motion in symbols is : — 



T = (r from r = oio r = b ; ', 

 ^ — from r = b to r = a. 



and T ■ 



r 



VIII. In the stable motion of minimum energy the rotational 

 portion of the fluid is in the shape of a cylindric shell, inclosing 

 the irrotational remainder, which in this case is at rest. The 

 symbolical expression for this motion is — 



r = o, when r < VC'^" - ''-) and r = C ('' - ° ^ ''' \ 



when r > V(«" - ^')- 



IX. Let now the liquid be given in the configuration VII. of 

 greatest energy, and let the cylindrical boundary be a sheet of a 

 real elastic solid, such as sheet-metal with the kind of dereliction 

 from perfectness of ela-ticity which real elastic solids present ; 

 that is to say, let its shape when at rest be a function of the 

 stress applied to it, but let there be a resistance to change of 

 shape depending on the velocity of the change. Let the un- 

 stressed shape be truly circular, and let it be capable of slight 

 deformations from the circular figure in cross section, but let it 

 always remain truly cylindrical. Let now the cylmdric boundary 

 be slightly deformed and left to itself, and held so as to prevent 

 it from being carried round by the fluid. The central vortex 



column is set into vibration in such a manner that longer and 

 shorter waves travel round it with less and greater angular 

 velocity.' These waves cause corresponding waves of corruga- 

 tion to travel round the cylindric bounding sheet, by which 

 energy is consumed, and moment of momentum taken out of the 

 fluid. Let this process go on until a certain quantity of moment 

 of momentum has been stopped from the fluid, and now let the 

 canister run round freely in space, and, for simplicity, suppose 

 its material to be devoid of inertia. The whole moment of 

 momentum is initially — 



It is now 



,rCb"-(a--- U'') - M, 

 and continues constantly of this amount as long as the boundary 

 is left free in space. The consumption of energy still goes on, 

 and the way in which it goes on is this : the waves of shorter 

 length are indefinitely multiplied and exalted till their crests run 

 out into fine laminse of liquid, and those of gi-eater length are 

 abated. Thus a certain portion of the irrotationally revolving 

 water becomes mingled \\ith the central vortex column. The 

 process goes on until what may be called a vortex sponge is 

 formed ; a mixture homogeneous on a large scale, but consisting 

 of portions of rotational and irrotational fluid, more and more 

 finely mixed together as time advances. The mixture is, as 

 indicated above, altogether analogous to the mixture of the 

 substances of two eggs whipped together in the well-known 

 culinary operation. Let b' be the radius of the cylindric vortex 

 sponge, /; being as before the radius of the original vortex 

 column — 



hi^ = ib- -^ > .1,. 



X. Once more, hold the cylindric case from going round in 

 space, and continue holding it until some more moment of 

 momentum is stopped from the fluid. Then leave it to itselt 

 again. The vortex sponge will swell by the mingling with it of 

 an additional portion of irrotational liquid. Continue this 

 process until the sponge occupies the whole inclosure. 



After that continue the process further, and the result will be 

 that each time the containing canister is allowed to go round freely 

 in space, the fluid will tend to a condition in which a certain 

 portion of the original vortex core gets filtered into a position next 

 to the boundary, and the fluid within it tends to a more and more 

 nearly uniform mixture of vortex with irrotational fluid. This 

 central vortex-sponge, on repetition of the process of preventing 

 the canister from going round, and again leaving it free to go 

 round, becomes more and more nearly irrotational fluid, and the 

 outer belt of pure vortex becomes thicker and thicker. The 

 final condition towards which the whole tends is a belt consti- 

 tuted of the original vortex core now next the boundai-y ; and 

 the fluid which originally revolved irrotationally round it now 

 placed at rest within it, being the condition (VIII. above) of 

 absolute minimum energy. Begin once more with the condition 

 (VII. above) of absolute maximum energy, and leave the fluid 

 to itself, whether with the canister free to go round sometimes, 

 or always held fixed, provided only it is ultimately held from 

 going round in space ; the ultimate condition is always the 

 same, viz., the condition (VIII.) of absolute minimum energy. 



XI. That there may be an infinite number of configurations 



of stable motions, each of them having the energy of a thorough 

 minimum as said in IV. above, we see, by considering the case, 

 in which the cylindric boundaiy of the containing canister con- 

 sists of two wide portions communicating by a narrow passage, 

 as shown in the sketch. If such a canister be completely filled 

 with irrotationally moving fluid of uniform vorticity, the stream- 

 lines must be something like those indicated in the sketch. 



> See ProcKjiiis^s of ihe Royal Society of Edinburgh for iSSo, or P/n7ir. 

 sophkal Magazine for 1880 ; Vibrations of a Columnar Vortex ; Wm- 

 Thoinson. 



