On harmonic functions satisfying a mixed boundary condition with an application to the flow past a porous wall

Summary

The problem of calculating the incompressible two-dimensional flow of a gas past porous surfaces is shown to involve the determination of an analytic function satisfying a complicated non-linear relation between its real and imaginary parts on these surfaces. In “linearized” flow this boundary condition reduces to a linear relation, and it is possible to determine the analytic function defining the flow. The general problem of determining an analytic function subject to a linear relation between its real and imaginary parts on a single boundary is solved by reducing it to two simultaneous Dirichlet problems. Solutions of this problem have been obtained by Carleman, Gakhov and others (see¹) for references), but the solution given here is more direct, as it does not require the continuation of the function to a “sectional holomorphic” function¹) beyond the original boundary. The flow past an infinite wall, the “porosity” of which varies along its length, is calculated as a simple example of the theory. The theory of an aerofoil in a slotted tunnel is also outlined.