79. The hydrodynamical interpretation of cases where w is, in the foregoing sense, a many-valued function of z is obvious from the preceding chapter. It is possible however for w to be ambiguous in another way. Let us take for instance the function

w=√2= r2+(cos + i sin 0)

(26).

If we start from the point_r= 1, 00, with the value w = 1, the value of the function at any other point is

w=r3 (cos ¿0 + i sin § 0),

where must of course be supposed to vary continuously. Hence if the point P (r, e) describe a closed curve not embracing the origin, w will return to its former value; but if the path of P encompass the origin this will not be the case; if the motion of P be in the positive direction will have increased by 2π, and we shall have w=-1. A second circuit round the origin will restore to w its original value. Hence to every point P in the plane of z correspond two values of w, which however pass into one another continuously as P describes a closed curve about the origin, where the two values coincide.

Again, if

w=

(27),

and we start from any point A with the value w。, the value of w at A after one circuit of P round the origin will be aw, after a second circuit aw,, and after a third aw,, where a is a cube root of unity. Hence to every point P of the plane of z correspond three values of w, forming a cycle which recurs at every third circuit of P round the origin.

A point, such as the origin in the above examples, at which two or more values of a function coincide, is called a 'branchpoint'. The similarity in their infinitely small parts of the planes of w and z must obviously break down at a branch-point, so that we must have at such points (Art. 76)

dw

dz

The branch-points of a function w are also branch-points of

and vice versa, as is easily seen from the meaning of the latter function.

Hence if an assumption of the form (26) or (27) be hydrodynamically intelligible, the portion of the plane xy occupied by the fluid must not include any branch-points; otherwise the component velocities would be ambiguous at every point of the fluid. Branch-points may however occur on the boundary of this portion, in which case circulation round them is impossible.

80. We can now proceed to some applications of the foregoing theory.

Example 1. Assume

w = Az",

and suppose A to be real*. Introducing polar co-ordinates r, 0, we have

= Ar" cos no,

y= Ar" sin no.

(a) If n = 1, the stream-lines are a system of straight lines parallel to x, and the equipotential curves are a similar system parallel to y. In this case any corresponding figures in the planes of w and z are similar, whether they be finite or infinitesimal.

=

(b) If n = 2, the curves const, are a system of rectangular hyperbolas having the axes of co-ordinates as their principal axes, and the curves const. are a similar system having the co-ordinate axes as asymptotes. The lines 0=0,0 = 1π are parts of the same stream-line ↓ = 0, so that we may take the positive parts of the axes of x, y as fixed boundaries, and thus obtain the case of a fluid in motion in the angle between two perpendicular walls.

(c) If n=-1, we get two systems of circles touching the axes of co-ordinates at the origin. Since now

A cos 0

=

the

* If A be complex, the curves const., const. are not altered in form but only in position, being turned round the origin through an angle arc tan ß, if

A = a + iß.

velocity at the origin is infinite; to avoid a physical absurdity we must suppose the region to which our formulæ apply to be limited internally by a closed curve surrounding the origin.

=

(d) If n = 2, each system of curves is composed of a double system of lemniscates. The axes of the system const. coincide with x or y; those of the system const. bisect the angles between these axes.

=

(e) By properly choosing the value of n we get a case of irrotational motion in which the boundary is composed of two rigid walls inclined at any angle a. The equation of the stream-lines being

j sin ne = const.,

The component velocities along and perpendicular to î, are

and are therefore zero, finite, or infinite at the origin, according as a is less than, equal to, or greater than π.

or

81. Example 2. The assumption

gives

w = μ log z,

+ if=μ log reio,

φ=μlogr, ψ= μθ.

The equipotential lines are concentric circles about the origin; the stream-lines are straight lines radiating from the origin. Or, we may take the circles r = const. as the stream-lines, and the radii = const. as the equipotential lines. In both cases the

velocity at a distance r from the origin is

suppose the origin excluded (e.g. by drawing it) from the region occupied by the fluid.

μ

r

; ; we must therefore

a small circle round In the second case,

already discussed in Art. 34, the motion is cyclic; the circulation circuit surrounding the origin being 2πμ.

in any

2

If r1, r, denote the distances of any point in the plane xy from the points (±a, 0), and 0,, 0, the angles which these distances make with the positive direction of x, we have

2

=

= 2

=

The curves const., i.e. 0-0, const., are circles passing through the points (± a, 0); the curves const. are the system of circles orthogonal to these. Either of these systems of circles may be taken as the equipotential curves, and the other system will then compose the stream-lines. In either case the velocity at the points (±a, 0) will be infinite; so that we must exclude these points (e.g. by small closed curves drawn round them) from the region to which the formulæ apply, which thus becomes triply-connected. If the curves

r

2

const. be taken as the stream

lines, the circulation in any circuit embracing the first only of the above points is 27μ; that in one embracing the second point only is - 2πμ; whilst that in a circuit embracing both is zero.

83. Example 4. Assume

r

2'

2

If r1, "2, 01, 0, have the same meanings as in the last example, this gives

The curves r11 = const. are a system of lemniscates whose poles are at the points (a, 0). The curves 0, +0,= const., i.e.

are a system of rectangular hyperbolas, orthogonal to the above system of lemniscates, and passing through their poles. A drawing of both systems of curves is given by Lamé*.

The formulæ (28) make the velocity infinite at the poles, which must therefore be excluded from the region to which the formulæ apply. If the lemniscates be taken as the stream-lines, the velocity-potential μ (0, + 0) is a cyclic function; the circulation in any circuit embracing one pole only is 2πμ, that in a circuit embracing both poles is 4πμ.

1

1

2

1

2

If we suppose the region occupied by the fluid to have the axis of x as a boundary, there will be no ambiguity in these values of u, v. Moreover, u, v will be everywhere finite and continuous except on the axis of x. When y =0, and x > a, then 0,= 0,=0; and when y=0, x<-a, then 0,=0,=π. In each case v = = 0. When y = 0, and a> x>—a, we have 0 ̧1 = π, 0,= 0, and therefore v=-μπ. We have thus the solution of the following problem : An infinite mass of liquid bounded by an infinite rigid plane but otherwise unlimited is initially at rest, and a strip of this plane of breadth 2a is supposed detached from the remainder and suddenly pushed inwards with velocity μπ; to find the motion produced in the fluid. In the above formula the rigid plane corresponds to the axis of x, and the fluid lies to the negative side of the latter. It appears from (30) that at the edges of the strip u is infinite.