Novel applications of complex analysis to effective parameter quantification in transport theory

Abstract

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.