This system of curves includes the ellipse whose parameter is ʼn provided

Ce-2n = wc2,

By the same reasoning as in the last article we see that at a great 1 distance from the origin the velocity is of the order

The above formula therefore give the motion of an infinite mass of liquid, otherwise at rest, produced by the rotation of an elliptic cylinder about its axis with angular velocity w.

A drawing of the stream-lines in this case is given in the Quarterly Journal of Mathematics, December, 1875.

90. If w be a function of z, it follows at once from the definition of Art. 74 that z is a function of w. The latter form of the assumption is sometimes more convenient analytically than the former.

The relations (11) are then replaced by

where q is the resultant velocity at (x, y). Hence if the properties

dz dw

of the function (=, say,) be exhibited graphically in the manner

already explained, the vector drawn from the origin to any point in the plane of will agree in direction with, and be in magnitude the reciprocal of, the velocity at the corresponding point of the plane of z.

which may, by (48), be put into the equivalent forms

simply expresses the fact that corresponding elementary areas in the planes of z and w are in the ratio of the square of the modulus

=

y = c.

The curves = const. are, Art. 88 (d), a system of confocal ellipses, and the curves const. a system of confocal hyperbolas; the common foci of the two systems being the points (± c, 0).

Since at the foci we have = (2n + 1) π, ↓ = 0, n being some integer, we see by (49) that the velocity is infinite there. We must therefore exclude these points from the region to which our formulæ apply. If the ellipses be taken as the stream-lines the motion is cyclic; the circulation in any circuit embracing either focus alone is -π, that in a circuit embracing both is -2π.

At an infinite distance from the origin is infinite, and the velocity zero.

When c = 0 this case coincides with that of Example 2. We

leave it as an exercise for the student to deduce the formulæ of that example from those of the present article.

If we take the hyperbolas as the stream-lines, the portions of the axis of which lie beyond the points (± c, 0) may be taken as fixed boundaries. We obtain in this manner the case of a liquid flowing from one side to the other of a rigid plane, through an aperture of breadth 2c made in the plane; but since the velocity at the edges of the aperture is infinite, this kind of motion cannot be realized with actual fluids.

92. Example 8. Let

whence

is

z = A (w+e"),

x = A + Ae cos ¥\
y=A+ Ae sin

Along the stream-lines + Aπ, we have

As

=

x= A (-e), y = ± Аπ.

increases from through zero to +∞, x increases from -∞, reaches a certain maximum value, and then goes back to ∞. The maximum value is readily found to be when = 0, and A. Hence the portions of the straight lines y = + Aπ which lie between and x-A, may be taken as fixed boundaries. Let us next trace the course of a stream-line infinitely near to one of the former; say = A (π-a), where a is infinitesimal. This gives

=

x= A ($+ e cos ¥), y=Aπ-Ax+ Ax eo, approximately. As increases from - ∞, x increases, whilst y remains at first approximately constant and equal to A (π − a); when, however x approaches its maximum value, y increases to the value Aπ. As increases beyond the value zero, x diminishes, whilst the excess of y over Aπ slowly but continuously increases.

The formulæ (51) express then the motion of a liquid flowing from a canal bounded by two parallel planes into open space*. We see, however, from (50), that the velocity at the edges of these planes (where =0, T) is infinite; so that the motion.

=

*The above example is due to Helmholtz, Phil. Mag. Nov. 1868. A drawing of the curves = const., &=const., is given in Maxwell's Electricity and Magnetism, Vol. 1., Plate XII.

cannot be realized, for the reasons explained in Art. 30. If however the motion be very slow, we may take two stream-lines very near toπ as fixed boundaries, and so obtain a possible case.

93. Example 9. Kirchhoff has, à propos of certain problems to be discussed below, given a method by which the determination of the motion in several cases of interest may be readily effected. The method rests upon the property of the function explained in Art. 90. If the boundaries of the fluid be fixed and rectilinear, the corresponding lines in the plane of, which are also straight, are easily laid down. Also, since the fixed boundaries are streamlines, the corresponding lines in the plane of w are straight lines const. It is then in many cases not difficult to frame an assumption of the form

=

(=ƒ (w),

by which the correspondence of these lines in the planes of and w may be established. The relation between z and w is then to be found by integration.

Example 8, above, is very easily treated in this manner. We take however a somewhat less simple case; viz. that in which a current flows from a uniform canal into an open space which is bounded by an infinite plane perpendicular to the length of the canal, and in which the mouth of the latter lies. See Fig. 6.

The middle line of the canal is evidently a stream-line; say that for which 0. Also for the stream-line BAC let =π ;

==

on account of the symmetry we need only consider the motion between these two lines. If the velocity in the canal at a distance from the mouth be taken to be unity, the half-breadth of the canal will be π. The boundaries in the plane of are shewn in the figure, corresponding points in the planes of z and being indicated by corresponding Roman and Greek letters. The point A corresponds to the origin of because the velocity there is infinite. (Art. 80.)

=

=

We have now to connect and w by a relation such that yz8 and B shall correspond to the two straight lines 0, π. If we assume z' = (3, then in the plane of z' the lines y2ß and ß∞ become parts of the same straight line. The assumption z' =1+er then converts these two parts into the straight lines y = 0, y = π. See Art. 78. We have then

The constant of integration is so chosen that the origin of z (hitherto arbitrary) shall be at the intersection of the middle line of the canal with the plane of its mouth.

Discontinuous Motions*.

94. We have had frequent occasion to remark, concerning forms of fluid motion which we have obtained, that they cannot be realized in practice on account of the infinite velocity and consequent negative pressure which they would involve at some point of the boundary. We are led to solutions of this nugatory character whenever a sharp projecting edge forms part of the boundary. Edges of absolute geometrical sharpness do not of course occur in practice; but even if the edge be slightly rounded, (as for instance in Example 8 above, by the substitution of a neighbouring stream-line as the fixed boundary,) the velocity in the immediate neighbourhood will, unless the motion be every

Helmholtz, l.c. Art. 92.