6 1 8 ANSWERS
11. T(x, y) =^2 ~^0 Re(Arcsin±).
1 • 21 3. Isothermals are T (x, y) = k. The equation 100 -^1 ~^0 arctan -:i: 2 Y-v = k
can be manipulated to yield c =tan 1 ~( 100 - k) = i-x;Y-y• which is better
recognized as the circle x^2 + (y + c)^2 = 1 + c^2.
1 5. T( x, y) = 40 + 20Im( Arcsinz).
Section 11.6. Two-Dimensional Ele ctrostatics: page 463
- tf>(x, y) = 100 + ~~g In lzl.
- ef>(x, y) = 150 - .,i°.2~2.
- tf>(x,y) = 50 +^2 ~Re(Arcsinz).
7. (a) w = S(z) =^2 ,"; 36 , (b} tf>(x, y) = 200 - ~In 12 ; ; 36 1.
Section 11 .7. Two- D im e nsional Fluid Flow: page 474- (a) V(r, 8) = A(l - ek) = A(l -e-^21 11) = A(l-cos28-isin 28), (c) z = 1,
andz=-1.
3a. Speed= A lzl. The minimum speed is A 1 1-ii= Av'2.
3 b. The maximum pressure in the channel occurs at t he point 1 + i.Sa. iIJ (r, 8) =Ar~ sin^3 :.
Section 11 .8. The Joukowski Airfoil: page 484
1. z + ± = w implies that z^2 + 1 = zw. Rewrite as z^2 - zw + 1 = 0 and then
use the quadratic formula.3 ( )^2 + ( )2 ^1 +^2 (b). h · · l- u
(^2) -v (^2) • 2v
. a x y-a - a , use t e mverse x +iy - (l-u)2+v 2 +i(l-u)'+v•
and substitute for x and yin part (a) and obtain the equation ( 14 .£':,j:f:t• = 0,