between 0 and co. To find the form of the free boundaries, let

us consider the portion of the stream-line

= 0 for which >0.

The first term of (59) is then real, the second imaginary, so that if we write

If s denote the arc of this line, measured from the edge of the aperture, we have

аф

ds

if the origin of z (hitherto arbitrary) be taken at the edge of the aperture. The final width of the stream is given by the difference (the velocity being unity), i.e. it is

of the extreme values of equal to π; and since when centre of the stream is 1+π.

The equations (61) and (62) combined give the form of the boundary of the issuing jet. They are obtained in the above manner by Lord Rayleigh*, who has also given a drawing of the curve in question.

ds

Since slog cos 0, the radius of curvature of the boundary

is =tan 0, and therefore vanishes at the edge. Kirchhoff has de

shewn that this is a general property of free boundaries.

97. Example 11. Fluid escapes from a large vessel by a straight canal projecting inwards. This illustrates one of the cases of the tube, spoken of in Art. 31.

An inspection of Fig. 8, giving the forms of the boundaries in the planes of z, w, Y, and of a new variable √, will shew that

this case may be obtained from the preceding by merely writing √ for, so that we now have

Along the free boundary = 0, $> 0, we have in the same way as before

if the origin of x and of s be at the extremity of the fixed wall. Also

y = − } π + e ̄® √1 − e−2+ arc sin e ̃o,

the constant of integration being so chosen as to make y = 0 for 80. When 8∞,y=-7, so that, the final breadth of the stream being as before equal to T, the breadth of the canal is 2π. The coefficient of contraction is therefore.

This example, the first of its class which was solved, is due to Helmholtz (l.c. Art. 92).

π

If in (58) we write a for we obtain the solution of the case where the inclination a of the walls of the canal has any value whatever.

98. Example 12. A steady stream impinges directly on a fixed plane lamina. The region of dead water behind the lamina is bounded on each side by a surface of discontinuity at which Չ is (for the moving fluid) constant (say = 1).

The middle stream-line, after meeting the lamina at right angles, branches off into two parts, which follow the lamina to the edges, and thence the surfaces of discontinuity. Let this be the line for which = 0, and let us further assume that at the point of divergence we have = 0. The forms of the boundaries in the planes of z, 5, w are shewn in Fig. 9. The region occupied by moving fluid corresponds to the whole of the plane of w; but the two sides of the straight line = 0, $> 0 are internal boundaries. The assumption w/w transforms this double line into

the axis of abscissæ, which may of course be regarded as a circle of infinite radius. The rule of Art. 95 then gives

==

The last assumption fixes the breadth of the lamina in terms of the unit of length. Of these conditions (a) gives A=— C, (b) gives B= D, and (c) makes A-B*. - B*. Hence (63) becomes

If y = 0, and lie between ± 1, the right-hand side of (64) is

* That is, we assume √/w= +1 when = +1, and therefore √w= −1 for = − 1. The alternative supposition leads to the same results.

real. To find the breadth of the lamina in terms of the unit of length, we have

To find the excess of pressure on the anterior face of the lamina, we have,

and for the dead water

Now at the edge of the lamina, we have pp', q= 1, so that C = C' + 1, and the required excess is given by

If we multiply this by dx, and integrate between the limits +1,

To make this result intelligible we must get rid of the arbitrary assumptions made for simplicity of calculation. If q, be the general velocity of the stream, 4, the value of ☀ at the edge of the lamina, we have, instead of (64),