11.7 • TwO-DrMENSI ONAL FLUID FLOW 475
Figure 11.54
- Consider the ideal fluid flow, where the complex potential is
F(z) = Az~ =Ar! (cos
3
: + isin
3
:), for 0 $ () $ 2ir.
(a) Find the stream function !/J (r, 8).
(b) Sketch several streamlines of the flow in the angular region 0 < () <^4 ;
indicated in Figure 11.55.
Figure 11.55
- Consider the complex potential F (z) = A ( z^2 + z^12 ).
(a) Let A > O. Show that F (z) determines an ideal fluid flow around the
domain r > 1,0 < () < ~ indicated in Figure 11.56, which shows the
flow around a circle in the first quadrant. Hint: Use the conformal
mapping w = z^2 •
(b) Show that t he speed at the point z = e^18 on the quarter-circle
r = 1,0 <()<~is given by V = 4 A lsin281.
(c) Determine the stream function for the flow and sketch several stream-
lines.
