and (66) by

πα Φ......

.......

.(68).

Multiplying (68) by p, and eliminating po, we obtain for the total excess of pressure on the anterior face

If the stream be oblique to the lamina, making an angle a with its plane, the condition (a) is replaced by

which gives

(=e-ia, for &= ± ∞,

A: C=cos a · 1: cos a + 1;

(b) is unaltered, whilst (c) is no longer applicable, there being no longer symmetry as regards the line 0. The former conditions. however reduce (63) to the form

(+)

which shews that the

value of the remaining constant K only affects the scale of w. If we assign to it any real value, we make the cusp = 1 of the lune in the plane of correspond to some definite point of the boundary in the plane of w'. The simplest assumption is K=1, which gives, after some reduction,

For the discussion of this result, and the calculation of the resultant pressure on the lamina we must refer to the paper by Lord Rayleigh, already cited (l. c. Art. 96).

CHAPTER V.

ON THE MOTION OF SOLIDS THROUGH A LIQUID.

99. THE chief subject treated of in this chapter is the motion of a solid through an infinite mass of liquid under the action of any given forces. The same analysis applies with little or no alteration to the case of a liquid occupying a cavity in a moving solid. We shall consider, though less fully, cases where we have more than one moving solid, or where the fluid does not extend in all directions to infinity, being bounded externally by fixed rigid walls.

We shall assume in the first instance that the motion of the fluid is entirely due to that of the solid, and is therefore characterized by the existence of a single-valued velocity-potential & which besides satisfying the equation of continuity

v3$ = 0.......

аф

(1)

fulfils the following conditions: (a) the value of do, dn denoting

dn'

as usual an element of the normal at any point of the surface of the solid drawn towards the fluid, must be equal to the velocity of the surface at that point normal to itself, and (b) the differential аф аф аф coefficients must vanish at an infinite distance, in dx' dy' dz every direction, from the solid. The latter condition is rendered necessary by the consideration that a finite velocity at infinity would imply an infinite kinetic energy, which could not be generated by finite forces acting for a finite time on the solid. It is also the condition to which we are led by supposing the fluid enclosed within a fixed vessel infinitely large and infinitely distant all round from the moving body. For on this supposition

the space occupied by the fluid may be conceived as made up of tubes of flow which begin and end on the surface of the solid, so that the total flux across any area, finite or infinite, drawn in the fluid must be finite, and therefore the velocity at infinity zero.

In the case of a fluid occupying a cavity in a moving solid, the condition (b) does not apply; the surface of the cavity is then the complete boundary of the fluid, and the condition (a) is therefore sufficient.

We have seen in Arts. 49, 52 that under either of the above sets of conditions the motion of the fluid is determinate. Our problem then divides itself into two distinct parts; the first or kinematical part consisting in the determination of the motion of the fluid at any instant in terms of that of the solid, and the second, or dynamical part, in the calculation of the effect of the fluid pressures on the surface of the latter.

Kinematical Investigations.

100. Let us take a system of rectangular axes Ox, Oy, Oz fixed in the body, and let the motion of the latter at any time t be defined by the instantaneous angular velocities p, q, r about, and the translational velocities u, v, w of the origin O parallel to, the instantaneous positions of these axes. We may then write, as Kirchhoff does,

$=u&、+v$2+w$3 + PX1 + 9X2 + rX3··

where, as will immediately appear, 4,, &c., X1, &c., are certain functions of x, y, z depending only on the shape and size of the solid. In fact, if l, m, n denote the direction-cosines of the normal (drawn on the side of the fluid) at any point (x, y, z) of the surface of the solid, the condition (a) of Art. 99 may be written

= (u + qz − ry) 1 + (v + rx − pz) m + (w +py − qx) n ;

аф

=

dn

The functions,, &c. must of course separately satisfy (1), and have their derivatives zero at infinity; the surface-conditions (3) then render them completely determinate.

101. When the motion is in two dimensions (xy) we have only three functions to determine, viz. 1, 2, X. In the last chapter (Arts. 88, 89) general methods for discovering cases in which one of these functions is known were given. In any case of a liquid filling a cavity in a moving solid it is plain that the conditions (3) are satisfied by 41, 42, 43 = x, y, z, respectively, in other words that if the solid have a motion of translation only, the enclosed fluid moves as if it formed a rigid mass. We may therefore regard the kinematical part of our problem as solved for the cases where the cavity is in the form of an elliptic cylinder, or a triangular prism on an equilateral base, for which x, has been found*.

X3

In the more difficult problem of a cylindrical body moving through an infinite mass of liquid, the complete solution has been obtained for the case where the section of the cylinder is elliptic, and for this case only.

102. The number of cases in three dimensions for which the functions 1, &c., X1, &c. have been completely determined is very small. We give here the chief of them.

Example 1. An ellipsoidal cavity whose semiaxes are a, b, c. If the principal axes of the ellipsoid be taken as axes of co-ordinates, and h be the perpendicular from the centre on the tangent plane at (x, y, z), we have then

respectively, so that the last three of conditions (3) give

other forms of cylindrical cavity for which solutions can be obtained.

Example 2. A cavity in the form of a rectangular parallelepiped. If the axes of co-ordinates be taken parallel to the edges of the cavity, it is plain that the conditions (3) are satisfied by making X, a function of y, z only, &c., so that the problem becomes one of two dimensions. For the complete solution, effected by means of Fourier's series, we refer the student to Stokes*, or to Thomson and Tait†.

Example 3. An ellipsoid moves in an infinite mass of liquid which is at rest at infinity.

This problem, the only one of its class which has been completely worked out, was solved by Green‡ in 1833, for the particular case where the motion of the ellipsoid is one of pure translation. The complete solution was published by Clebsch§, in 1856; it is here reproduced much in the form given to it by Kirchhoff.

The principal axes of the ellipsoid being taken as axes of coordinates, let the component attractions which would be exerted at the point (x, y, z) by an ellipsoid of unit density, coincident in shape, size, and position with the given one, be denoted by X, Y, Z. It is known that X is the potential of the ellipsoid when magnetized uniformly with unit intensity parallel to æ negative, and therefore that it is the potential of a distribution of matter, dX of surface-density - 7, over the surface T. Hence is discon

* Camb. Phil. Trans., Vol. v., pp. 131, 409. + Natural Philosophy, Art. 707 (B).

dn

'Researches on the Vibration of Pendulums in Fluid Media,' Trans. R.S. Edin. 1833. Reprinted in Mr Ferrers' edition of Green's works, pp. 315 et seq. § Crelle, tt. 52, 53 (1956-7).

|| Vorlesungen über Math. Physik. Mechanik., c. 18.

¶ See, as to these points, Maxwell, Electricity and Magnetism, Art. 437.