1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

470 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

y

D

w= S(z)

ljl(x, y) = K

(a) Fluid flow in a plane. (b) Fluid flow in the w plane.

Figure 11. 48 The image of a fluid flow under conformal mapping.

are the new velocity potential and stream function, respectively, for the flow in

D. A streamline or natural boundary curve

1/J(x, y) = K

in the z plane is mapped onto a streamline or natural boundary curve

w(u, v) = K

in thew plane by the transformation w = S (z). One method for finding a flow
inside a domain D in the z plane is to conformally map D onto a domain G in
the w plane in which the flow is known.
For an ideal fluid with uniform density p, the fluid pressure P (x, y) and
speed IV (x, y)I are related by the following special case of Bernoulli's equation:

P(:, y) + ~ IV (x, y)I = constant.

Note that the pressure is greatest when the speed is least.

• EXAMPLE 11.22 The complex potential F (z) = (a+ ib) z bas the velocity
  potential and stream function of

<fo(x, y) = ax - by and 1/J (x, y) =bx+ ay,

respectively, and gives rise to the fluid flow defined in the entire complex plane

that has a uniform parallel velocity of

V (x, y) = F'(z) =a-ib.

The streamlines are parallel lines given by the eq uation bx+ ay = constant and

are inclined at an angle a = -Arctan~, as indicated in Figure 11 .49.