ing in consequence of this motion are easily seen to be


and sin x, respectively, so that we must write for

d/ 3a

-b at



v+ u cos x + 15 uv sin x cos x + &c.,

v cos X,

where terms which obviously contribute nothing to the integral (50) have been omitted. Again

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similar omissions being made. Now

[sin x cos xdx=3, sin'x cos'x dx = rs.

so that we have finally for the resultant fluid pressure on B in the direction AB,

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This result is correct to the order inclusive. Since

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We proceed to examine some particular cases, keeping only the most important terms in each.

(a) Let b=a, v=u, so that the motion is symmetrical with respect to the plane bisecting AB at right angles, and is the same as if this plane formed a rigid boundary to the fluid on either side of it. We have thus the solution of the case where a sphere moves directly towards or away from a fixed plane wall. The force repelling the sphere from the wall is*

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where M' is the mass of fluid displaced by the sphere. Hence the principal effect of the plane boundary is to increase the inertia


of the sphere in the ratio 1+ 1, c denoting double the

distance of the centre of the sphere from the plane.

(b) Let us suppose each sphere constrained to move with constant velocity. The force which must be applied to B in order to maintain this motion is u2 approximately, and is in the


direction BA. The spheres therefore appear to repel one another. The forces to be applied to the two spheres are not equal and opposite except when v=u.

(c) Let us suppose that each sphere makes small periodic oscillations about a mean position, the period being the same for each. The average value of the first term of (52) is then zero, and the mutual action of the two spheres is equivalent to a force


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uv, urging them together, where uv denotes the mean value of uv. If u, v differ in phase by less than a quarterperiod this force is one of attraction, if by more than a quarterperiod it is one of repulsion.


(d) Let A perform small periodic oscillations while B is held at rest. The mean force on B is now zero to our order of approximation. To carry the approximation further, we remark do that the mean value of at the surface of B is necessarily zero, dt

and that the next important term in the value (51) of the semi

aεb 4 c2

square of the velocity is, when v=0, 45 u2 sin2

X cos X, and

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the resulting term in (50) is found on integration to be


where u2 denotes the average value of the square of the velocity of A.

This result comes under a general principle enunciated by Thomson. If we have two bodies immersed in a fluid, one of which A performs small vibrations while the other B is held

at rest, the fluid velocity at the surface of B will on the whole

be greater on the side nearer A than on that which is more remote. Hence by (18) the average* pressure on the former side will be less than that on the latter, so that B will experience on the whole an attraction towards A. As practical illustrations of this principle we may cite the apparent attraction of a delicately-suspended card by a vibrating tuning-fork, and other similar phenomena studied experimentally by Guthrie† and explained in the above manner by Thomson §.

The same principle accounts for the indraught of a light powder, strewn on a vibrating plate, towards the ventral segments.

аф dt

Since is by hypothesis periodic function of t, the term in (18) con

tributes nothing to the average effect.

+ Proc. R. S.

§ Reprint, Art. XLI.



126. So far our investigations have been confined for the most part to the case of irrotational motion. We now proceed. to the study of rotational or 'vortex' motion. This subject was first investigated by Helmholtz, in Crelle's Journal, 1858; other and simpler proofs of some of his theorems were afterwards given by Thomson in the paper on vortex motion already cited in Chapter III.

A line drawn from point to point so that its direction is everywhere that of the instantaneous axis of rotation of the fluid is called a 'vortex-line.' The differential equations of the system of vortex-lines are

dx dy



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where §, n, have, as throughout this chapter, the meanings assigned in Art. 38.

If through every point of a small closed curve we draw the corresponding vortex-line, we obtain a tube, which we call a ' vortex-tube.' The fluid contained within such a tube constitutes what is called a 'vortex-filament,' or simply a 'vortex.'

Kinematical Theorems.

127. Let ABC, A'B'C' be any two circuits drawn on the surface of a vortex-tube and embracing it, and let AA' be a

connecting line also drawn on the surface. Let us apply the theorem of Art. 40 to the circuit ABCAA'C'B'A'A and the part

Fig. 10.


of the surface of the tube bounded by it. Since l+my+ng is zero at every point of this surface, the line-integral

f(udx + vdy+wdz),

taken round the circuit, must vanish; i.e. in the notation of Art. 39


which reduces to


Hence the circulation is the same in all circuits embracing the same vortex-tube.

Again, it appears from Art. 39 that the circulation round the boundary of any cross-section of the tube, made normal to its length, is 2wo, where w = (§2+n2 + y2)3 is the angular velocity of the fluid at the section, and σ the (infinitely small) area of the section.

Combining these results we see that the product of the angular velocity into the cross-section is the same at all points of a vortex. This product is conveniently termed the 'strength' of the vortex.

The foregoing proof is due to Thomson; the theorem itself was first given by Helmholtz, who deduced it from the relation

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which follows at once from the values of E, n,

In fact, writing in Art. 64, Cor. 1, §, n, ( for u, we find

SS(l§ + mn + ng) dS = 0



given in Art. 38. v, w, respectively,



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