1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.7 • TwO-DI MENSIONAL FLUID FLOW 469

Using the fact that 1/1 11 = ef>x and this equation, we find that the tangent vector

to the curve is

T = ¢>., (x, y) -i1/J., (x, y) = p (x, y) + iq (x, y) = V (x, y).

The main idea of the preceding discussion is the conclusion that, if

F (z) = ¢> (x, y) + i1/I (x, y) (11-35)

is an analytic function, then the family of curves

represents the streamlines of a fluid flow.
The boundary condition for an ideal fluid flow is that V should be parallel
to the boundary curve containing the fluid (the fluid flows parallel to the walls of
a containing vessel). In other words, if Equation (11-35) is the complex potential
for the flow, then the boundary curve must be given by 1fi (x, y) = K for some
constant K; that is, the boundary curve must be a streamline.

We note that the functions

¢>(x, y) = {> (u (x, y), v (x, y)) and 1/1 (x, y) = '11 (u(x, y), v (x, y))