Schwarz-Christoffel Theory of Flow Past an Opening

The Schwarz-Christoffel transformation for a steady two-dimensional plane flow past an eccentric, normal constriction of zero thickness is formulated. The transformation formula contains two unknown complex constants and four unknown parameters (real numbers). It is found that there are five identities for the parameters and two additional equations for two additional physical parameters representing the geometry of the constriction in the channel. Except one of the identities, they are all expressed in the form of elliptic integrals of the first and third kinds; four of them are sufficient for unique solutions of the four parameters. The five identities that do not contain the physical parameters cannot be used in the method of solution alone, but the remaining two equations relating the two physical parameters to the four unknown parameters of the transformation can be reduced to a set of three simple trigonometric relationships, from which the analytical solutions of the parameters are obtained. Once the parameters are determined, the corresponding flow pattern can be readily computed from the transformation.