Page:EB1911 - Volume 14.djvu/140

Rh direction along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.

It is convenient to introduce the function

= log = log (Q/q) + i (4)

so that the polygon representing conformally has a boundary given by straight lines parallel to the coordinate axes; and then to determine and w as functions of a variable u (not to be confused with the velocity component of q), such that in the conformal representation the boundary of the and w polygon is made to coincide with the real axis of u.

It will be sufficient to give a few illustrations.

Consider the motion where the liquid is coming from an infinite distance between two parallel walls at a distance xx′ (fig. 4), and issues in a jet between two edges A and A′; the wall xA being bent at a corner B, with the external angle = /n.

The theory of conformal representation shows that the motion is given by

(5)

where u = a, a′ at the edge A, A′; u = b at a corner B; u = 0 across xx′ where = ∞; and u = ∞,  = ∞ across the end JJ′ of the jet, bounded by the curved lines APJ, A′P′J′, over which the skin velocity is Q. The stream lines xBAJ, xA′J′ are given by = 0, m; so that if c denotes the ultimate breadth JJ′ of the jet, where the velocity may be supposed uniform and equal to the skin velocity Q,

m = Qc, &emsp; c = m/Q.

If there are more B corners than one, either on xA or x′A′, the expression for is the product of corresponding factors, such as in (5).

Restricting the attention to a single corner B,

(6)

(7)

(8)

∞ &gt; a &gt; b &gt; 0 &gt; a′ &gt; −∞ (9)

and then

(10)

the formulas by which the conformal representation is obtained.

For the polygon has a right angle at u = a, a′, and a zero angle at u = b, where  changes from 0 to /n and  increases by i/n; so that

(11)

And the w polygon has a zero angle at u = 0, ∞, where changes from 0 to m and back again, so that w changes by im, and

(12)

Along the stream line xBAPJ,

= 0, &emsp; u = ae−/m; (13)

and over the jet surface JPA, where the skin velocity is Q,

(14)

denoting the arc AP by s, starting at u = a;

(15)

(16)

∞ &gt; u = aes/c > a, (17)

and this gives the intrinsic equation of the jet, and then the radius of curvature

(18)

not requiring the integration of (11) and (12)

If =  across the end JJ′ of the jet, where u = ∞, q = Q,

(19)

Then

(20)

(21)

Along the wall AB, cos n = 0, sin n = 1,

a > u > b, (22)

(23)

(24)

(25)

(26)

Along the wall Bx, cos n = 1, sin n = 0,

b > u > 0 (27)

(28)

(29)

At x where = ∞, u = 0, and q = q0,

(30)

In crossing to the line of flow x′A′P′J′, changes from 0 to m, so that with q = Q across JJ′, while across xx′ the velocity is q0, so that

m = q0·xx′ = Q·JJ′ (31)

(32)

giving the contraction of the jet compared with the initial breadth of the stream.

Along the line of flow x′A′P′J′, = m, u = a′e−/m, and from x′ to A′, cos n = 1, sin n = 0,

(33)

(34)

0 &gt; u &gt; a′. (35)

Along the jet surface A′J′, q = Q,

(36)

(37)

a′ &gt; u = a′e /sc > −∞, (38)

giving the intrinsic equation.

41. The first problem of this kind, worked out by H. v. Helmholtz, of the efflux of a jet between two edges A and A1 in an infinite wall, is obtained by the symmetrical duplication of the above, with n = 1, b = 0, a′ = −∞, as in fig. 5,

(1)

and along the jet APJ, ∞ > u = aes/c > a,

(2)

(3)