1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11. 10 • IMAGE OF A FLUID FLOW 497

(a) Flow over a step. (b) Flow around a blunt object.

Figure 11.87

Furthermore, the image of horizontal streamlines in the z plane are curves
in the w plane given by the parametric equation

w = f (t + ic) = ~ (t^2 - c^2 -1 + i2ct) t

11'

+~Log [t +ic+ (t^2 -c^2 -1 +i2ct)~),

for -oo < t < oo. The new flow is that of a step in the bed of a deep stream
and is illustrated in Figure 11.87(a). The function w = f (z) is also defined for
values of z in the lower half-plane, and the images of horizontal streamlines that
lie above or below the x-axis are mapped onto streamlines that flow past a long
rectangular obstacle, which is illustrated in Figure 11.87(b).

-------~EXERCISES FOR SECTION 11.10

For Exercises 1-4, use the Schwarz- Christoffel formula to find a conformal map-

ping w = f (z) that will map the flow in the upper half-plane Im (z) > 0 onto

the flows indicated.

1. Use Figure 11.88 to find the flow over the vertica.l segment from 0 to i.

Figure 11.88