496 CHAPTER. 11 • APPLICATI ONS OF HARMONIC FUNCTIONS
dz- Show that w = f (z) = J ~ maps the upper half-plane onto a square.
(z -zS)2
1
i 1 - (z+l)•
15. Show that w = f (z) = 2 {z + l}' - Log 1 maps t he upper hal f-plane
l +{z+l)•
Im(z) > 0 onto the domain indicated in Figure 11.86. Hint: Set x 1 = - 1, x2 =0, xs = 1, w 1 = 0, w 2 = d, and W3 = 2v'2 - 2 In ( v'2 - 1) + i1r and let d--> oo.
vFigure 11 .8611.10 Image of a Fluid Flow
We have already examined severa l two-dimensional fluid flows and have shown
that the image of a flow under a conformal transformation is a flow. The con-
formal mapping w = f (z) = u (x, y) +iv (x, y), which we obtained by using
the Schwarz-Christoffel formula, allows us to find the streamlines for flows in
domains in thew plane that are bounded by straight-line segments.
The first technique involves finding the image of a fluid flowing horizontally
from left to right across the upper half-plane Im (z) > O. The image of the
streamline -oo < t < oo, y = c is a streamline given by the parametric equationsu =u(t,c) and v =v(t,c), for - oo < t < oo,
and is oriented in the positive sense (counterclockwise). The streamline u =
u(t,O), v = (t,0) is considered to be a boundary wall for a containing vessel for
the fluid flow.- EXAMPLE 11. 29 Consider the conformal mapping
which we obtained by using the Schwarz-Christoffel formula. It maps the upper
half-plane Im ( z) > 0 onto the domain in the w plane that lies above the boundary
curve consisting of the rays u $ 0, v = 1 and u :e::: 0, v = 0 and the segment
u =O,- l :::;v:::;O.