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where

r = {(x − x')2 + a2 + a2 – 2aa' cos (9-9')}}.

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M=-S sin 9, N = S cos ............(47).

If the variable of integration in (44) be changed from A to e, where =′-9, the limits of integration for e are 0 and 2π; and since

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Here F1, E, denote the complete elliptic integrals of the first and second kinds with respect to the modulus

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F1=

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142. The kinds of fluid motion now under consideration come under the class for which a stream-function was shewn, in Art. 103, to exist. By the definition of that article, we have 2πψ = total flux through the circle whose co-ordinates are x, 5,

= ff(lu + mv + nw) dS,

where the integration extends over any surface bounded by that circle. Recalling the expressions (6) for u, v, w, from which Pis now to be omitted, we have by the theorem of Art. 40,

2πψ = [(Ldx + Mdy + Ndz),

the integration here being taken round the circle, or, by (47),

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The formula (28) for the kinetic energy may now be written

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143. Let us take the case of a single circular vortex of strength m. At all points of its infinitely small section the

modulus k of the elliptic integrals in the value of S is nearly

equal to unity. In this case we have*

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approximately, where k' denotes the complementary modulus √(1-k2), so that in our case

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nearly, if & denote the distance between the infinitely near points (x, 5), (x', '). Hence at points within the substance of the vortex the value of S, and therefore by (50) also of 4, is of the order mloge, where e is a small linear magnitude comparable with the dimensions of the section. The velocity at the same point, depending (Art. 103) on the differential coefficient of 4, will be of the order

m

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* See Cayley, Elliptic Functions, Art. 72, and Maxwell, Electricity and Magnetism, Arts. 704, 705.

dx

do

dt

m

We can now make use of (43) to estimate the magnitude of the velocity at of translation of the vortex. By (51) T is of the order m2 log e, and is, as we have seen, of the order. Also x-x is of the order e. Hence the second term on the righthand side of (43) is, in this case, small compared with the first, and the velocity of translation of the ring is of the order m log e, and approximately constant.

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An isolated vortex-ring moves then, without sensible change of size, parallel to its (rectilinear) axis, with nearly constant velocity. This velocity is small compared with that of the fluid in the immediate neighbourhood of its circular axis, but large compared with the velocity of the fluid at the centre of

2πη

,

the ring, with which it agrees in direction.

A drawing of the stream-lines due to a single circular vortex is given by Thomson*.

144. If we have any number of circular vortex-rings, coaxial or not, the motion of any one of these may be conceived as made up of two parts, one due to the ring itself, the other due to the influence of the remaining rings. The preceding considerations shew that the second part is insignificant compared with the first, except when two or more rings approach within a very small distance of one another. Hence each ring will move, without sensible change of shape or size, with nearly uniform velocity in the direction of its (rectilinear) axis, until it passes within a short distance of a second ring. A general notion of the result of the encounter of two rings may, in particular cases, be gathered from the theorem of Art. 130.

:

Thus, let us suppose that we have two circular vortices having the same rectilinear axis. If the sense of the rotation be the same in both, the two rings will advance in the same direction. One effect of their mutual influence will be to increase the radius

*

On Vortex-Motion, Trans. R. S. Edin. 1869. Copied in Maxwell, Electricity and Magnetism, plate xvIII.

of the one in advance, and to contract the radius of the one in the rear. If the radius of the one in front become larger than that of the one in the rear, the motion of the former ring will be retarded, whilst that of the latter is accelerated. Hence if the conditions as to the relative size and strength of the two rings be favourable, it may happen that the second ring will overtake and pass through the first. The parts played by the two rings will then be reversed; the one which is now in the rear will in turn overtake and pass through the other, and so on, the rings alternately passing one through the other. ·

If the rotations in the two rings be opposed, and such that the rings approach one another, the mutual influence will be to enlarge the radius of each ring.

If the two rings be moreover equal in size and strength, the velocity of approach will continually diminish. In this case the motion at all points of the plane which is parallel to the two rings, and half-way between them, is tangential to this plane. We may therefore, if we please, regard this plane as a fixed boundary to the fluid on either side of it, and so obtain the solution of the case where a single vortex-ring moves directly towards a fixed rigid wall.

On the Conditions for Steady Motion.

145. In steady motion, i. e. when

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the equations (2) of Art. 6 may be written

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so that each of the surfaces P=const. contains both stream-lines and vortex-lines. If further dn denote an element of the normal

at any point of such a surface, we have

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where q is the current-velocity, w the angular velocity, and the angle between the stream-line and the vortex-line at that point.

Hence the conditions that a given state of motion of a fluid may be a possible state of steady motion are as follows. It must be possible to draw in the fluid an infinite system of surfaces each of which is covered by a network of stream-lines and vortexlines; and the product of qo sin e dn must be constant over each such surface, dn denoting the length of the normal drawn to a consecutive surface of the system.

These conditions may also be deduced from the considerations that the stream-lines are, in steady motion, the actual paths of the particles, that the product of the angular velocity into the cross-section is the same at all points of a vortex, and that this product is, for the same vortex, constant with regard to the time.

The theorem that the quantity P, defined by (52), is constant over each surface of the above kind is an extension of that of Art. 28, where it was shewn that P is constant along a streamline.

The above conditions are satisfied identically in all cases of irrotational motion.

In the motion of a liquid in two dimensions, the product qdn is constant along a stream-line; the conditions then reduce to this, that w (or ζ, if the axes of co-ordinates be the same as

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