so that the polygon representing Ω conformally has a boundary given by straight lines parallel to the coordinate axes; and then to determine Ω 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 Ω 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 β = ½π/n.
The theory of conformal representation shows that the motion is given by
| ζ = [ | √ (b − a′·u − a) + √(b − a·u − a′) | ] | 1/n | , u = ae−πw/m; |
| √ (a − a′·u − b) |
(5)
where u = a, a′ at the edge A, A′; u = b at a corner B; u = 0 across xx′ where φ = ∞; and u = ∞, φ = ∞ 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 ψ = 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, c = m/Q.
If there are more B corners than one, either on xA or x′A′, the expression for ζ is the product of corresponding factors, such as in (5).
Restricting the attention to a single corner B,