This thesis proposes the application of complex analysis to the calculation of effective
parameters of transport problems in multiply connected domains. This can be done by
using special functions called Schottky-Klein prime functions. The effective parameters
focused on in this thesis are electrical resistivity, electrical capacity, and slip lengths of
channels. The prime function is a powerful mathematical function invented by Crowdy for
solving problems in multiply connected domains including transport problems governed
by Laplace’s equation and Poisson’s equation in domains with multiple boundaries. The
functional properties of the prime function make it possible to analyse effective parameters
in multiply connected domains.
First, a new method for solving a new class of boundary value problems in multiply
connected domains is explained. An explicit solution can be derived by multiplying of the
boundary data with a radial slit map written in terms of the prime functions.
We then focus on two electrical transport problems called “the van der Pauw method”
and “electrical capacity”. For the van der Pauw method, the prime function allows us to
derive new formulas for calculating the resistivity of holey samples. A new method for
the electrical capacity of multiply connected domains is formulated by coupling the prime
function with asymptotic matching.
We next construct explicit solutions for flows through superhydrophobic surfaces in
periodic channels and calculate the slip length of these channels. We end the thesis by
mentioning that the new methodology gives accurate estimates for so-called “accessory
parameter problems” associated with conformal maps of multiply connected domains.