1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.7 • Two-DIMENSIONAL FLUID FLOw 471

y

Figure 11.49 A uniform parallel flow.

• EXAMPLE 11.23 Consider the complex potential F (z) = 4z^2 , where A is

a positive real number. The velocity potential and stream function are given by

A

<P (x, y) =

2

(x^2 - y^2 ) and 'I/! (x, y) = Axy,

respectively. The streamlines 'I/! (x, y) = constant form a family of hyperbolas

with asymptotes along the coordinate axes. The velocity vector V = Az indi-

cates that in the upper half-plane Im (z) > 0, the fluid flows down along the
streamlines and spreads out along the x-axis, as against a wall, as depicted in
Figure 11. 50.

• EXAMPLE 11.24 Find the complex potential for an ideal fluid flowing from
  left to right across the complex plane and around the unit circle jz j = 1.

Solution We use the fact that the conformal mapping w = S (z) = z + (~)

maps the domain D = {z: jzj < 1} one-tc:rone and onto thew plane slit along

the segment -2 :$ u :$ 2, v = O. The complex potential for a uniform horizontal

flow parallel to this slit in t he w plane is

F1 (w) =Aw,

Figure 11 .50 T he fluid flow with complex potential F(z) = 4z^2.