Existence of Minimal Surfaces with a Simple Pole at Infinity and Condition of Transversality on the Surface of a Cylinder
Yu Why Chen · Transactions of the American Mathematical Society · 1949
Introduction.Let C be an infinite cylinder in the x, y, z space, which is generated by a smooth curve C in the x, y plane through parallel displacement along the z-axis.Consider a steady, irrotational flow around C, which is independent of z, with the velocity potential ¡p(x, y).Instead of the general pressure-density law of the flow, we shall consider as an approximation that the pressure is linearly dependent on the reciprocal density^)-The differential equation for the velocity potential /3, dz/dy->0 at infinity and the condition of transversality on the cylinder C.This problem is of importance both for practical engineering and for the theory of compressible fluids.A study of the velocity and pressure distribution around a cylindrical body has been made by von Kármán and Tsien(3), Presented to the Society, December 29, 1946; received by the editors January 22, 1948.(') This is the method suggested by A. Chaplygin to treat the subsonic flow of compressible fluid.Using this assumption Chaplygin obtained for the velocity potential and the stream-function two linear partial differential equations with flow speed and flow direction as independent variables.See for reference: von Kármán, The engineer grapples with the nonlinear problems.