Potential Flow Past a Circular Cylinder of a Homogeneous Inviscid Fluid

The steady-state two-dimensional potential flow of an inviscid fluid past a circular cylinder is considered. The resulting Laplace’s equation for the velocity potential is converted to cylindrical coordinates by the Calculus of Variations method of transformation of coordinates introduced in a pastoral interlude and that equation solved by a separation of variables technique. Then integration of the pressure determined by Bernoulli’s relation about the cylinder yields D’Alembert’s paradox for a two-dimensional situation that the drag on the cylinder is zero which is a consequence of the assumption that the small viscosity coefficient can be neglected. The problems fill in some details involving vector identities employing the alternating tensor introduced in Chap. 9 , examine the properties of the orthogonal Hermite polynomials similar in behavior to the Legendre polynomials discussed below, and consider the corresponding companion situation of three-dimensional potential flow past a sphere. This requires that the resulting Laplace’s equation for the velocity potential be converted to spherical coordinates by the Calculus of Variations transformation method. Then the separation of variables technique of solution gives rise to Legendre polynomials the properties of which have been deduced by means of the two pastoral interludes that conclude the chapter. Integration of the pressure about the sphere yields D’Alembert’s paradox for a three-dimensional situation that the force on the sphere is zero.