The beautiful expression for lift of a two-dimensional airfoil, Eq. (8.69), arose from applying the Joukowski transformation, ζ =z+a2/z, where z=x+iyand ζ = η + iβ. The constant a is a length scale. The theory transforms a certain circle in the z plane into an airfoil in the ζ plane. Taking a = 1 unit for convenience, show that
(a) a circle with center at the origin and radius >1 will become an ellipse in the ζ plane, and
(b) a circle with center at x= −ε 1,y= 0, and radius (1+ε ) will become an airfoil shape in the ζ plane. Hint: Excel is excellent for solving this problem.