Potential flow in a semi-infinite channel with multiple sub-channels using the Schwarz–Christoffel transformation

Abstract

In this paper we consider the potential fluid flow in a semi-infinite channel with multiple semi-infinite sub-channels using the Schwarz–Christoffel transformation and complex potential theory. The Schwarz–Christoffel transformation contains several unknown parameters, which are solely dependent on the dimensions of the region being considered, and an alternative iterative mathematical technique to that found elsewhere in the literature is developed to determine these parameters using a Runge–Kutta–Merson method of integration. Once these parameters have been determined we numerically integrate the Schwarz–Christoffel transformation using a variable-step Adams method. Now the mapping from the region being considered to the upper half of the complex plane is complete. In order to illustrate this mathematical technique we consider a semi-infinite room with an inlet/outlet placed on the ceiling and an outlet attached to the wall. The inlet and outlet channels are normal to the surface to which they are attached and through each of these channels we have uniform flow at infinity. Hence the whole region is modelled by a semi-infinite channel with two sub-channels attached.