1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.11 • SOURCES AND SINKS 511

10. Use a Schwarz-Christoffel transformation to find a conformal mapping w = S (z)
    that will map the flow in the upper half-plane onto the flow from a channel back
    into a quadrant, as indicated in Figure 11.108, where wo = 2VZ-2In ( v'2 - I)+ i?r.

Figure 11.108

11. Consider the complex potential F (z) = w given implicitly by z = w + e"'.

(a) Show that F (z) = w determines the ideal fluid flow through an open

channel bounded by the rays

y =tr, -oo < x < - 1 and y = -1r, - oo < x < - 1

into the plane.

(b) Show that the streamline 'l/J (x, y) = c of the flow is given by the para-

metric equations

x=t+ e'cosc and y=c+e'sinc, for - oo < t < oo,

as shown in Figure 11.109, which has been called Borda's mouthpiece.

Figure 11.109