474 CHAPTER 11 • APPLICATIONS OF H ARMONIC FUNCTIONS-------~EXERCISES FOR SECTION 11.7
- Consider the ideal 8uid Bow for the complex potential F (z) =A ( z +~),where
A is a positive real number.
(a) Show that the velocity vector at the point (1, 11 ) , z = re'^9 on t he unit
circle is given by V (1, 11) =A (1 -cos 211 - isin 29 ).
(b} Show that the velocity vector V (1, 11) is tangent to the unit circle lzl =
1 at all points except - 1 and +l. Hint: Show that V · P = 0, where
P = cosl1 +isinO.
(c) Show that the speed at the point (1, 11) on the unit circle is given by
IVI = 2A lsinl11 and t hat the speed attains the ma..ximum of 2A at the
points ± i and is zero at the points ±1. Where is the pressure the
greatest?- Show that the complex potential F (z) = ze_.., + •:~ determines the ideal 8uid
ftow around the unit circle lz l = 1, where the velocity at points distant from the
origin is given approximately by V:::::: e'°'; that is, the direction of the 8ow for large
values of z is inclined at an angle a with the x-axis, as shown in Figure 11.53.
Figure 11. 53- Consider the ideal 8uid Bow in the channel bounded by the hyperbolas xy = 1 and
xy = 4 in the first quadrant, where the complex potential is given by F(z) = 4z^2
and A is a positive real number.
(a) Find the speed at each point, and find the point on t he boundary at
which the speed attains a minimum value.
(b) Where is t he pressure greatest?4. Show that the stream function is given by 'I/! (r, 11) = Ar^3 sin 311 for an ideal fluid
ftow around the angular region 0 < 9 < i indicated in Figure 11.54. Sketch several
streamli nes of the ftow. Hint: Use the conformal mapping w = z^3.