Schwarz Problem for Multiply Connected Domains and its Application to Diffusion Around Fractals

Abstract

Stationary diffusion perpendicular to unidirectional circular cylinders arranged in two-dimensional fractals set is considered. Complex potentials and the method of functional equations are used to determine the concentration field. This result is applied to the calculation of the effective diffusion tensor of fractals randomly diluted on the plane. In particular, we discuss a special fractal derived by a group of Möbius transformations and deduce a generalization of the Clausius-Mossotti approximation.