Complex Analysis: Theory and Applications · Problem 10.8.7
Applications of Harmonic Functions — Howell & Matthews
Mathematics — Complex Analysis · joukowski-cardioid-airfoil-trailing-edge
Problem
Let be a cardioid whose cusp is at the origin. The image of under will be a Joukowski airfoil. Show that trailing edge forms an angle of .Intuitive explanation
A Joukowski transformation maps smooth curves to smooth curves, but cusps behave specially—they map to points where the derivative vanishes or the curve has a sharp corner. The trailing edge of an airfoil is where the upper and lower surfaces meet; if it forms a angle, the surfaces meet tangentially (smoothly). You need to find where the cardioid's cusp maps under the given linear fractional transformation, then verify that the image curve approaches this point with matching tangent directions from both sides, creating a cusp-to-smooth-point transition that forces the trailing edge angle to zero.
Intuitive explanation
A Joukowski transformation maps smooth curves to smooth curves, but cusps behave specially—they map to points where the derivative vanishes or the curve has a sharp corner. The trailing edge of an airfoil is where the upper and lower surfaces meet; if it forms a angle, the surfaces meet tangentially (smoothly). You need to find where the cardioid's cusp maps under the given linear fractional transformation, then verify that the image curve approaches this point with matching tangent directions from both sides, creating a cusp-to-smooth-point transition that forces the trailing edge angle to zero.