In this thesis, we present a new transform approach for solving biharmonic boundary value
problems in two-dimensional polygonal and circular domains. Our approach provides a
unified general approach to finding quasi-analytical solutions to a wide range of problems
in Stokes flows and plane elasticity.
We have chosen to analyze various Stokes flow problems in different geometries which
have been solved using other techniques and present our transform approach to solve them.
Our approach adapts mathematical ideas underlying the Unified transform method, also
known as the Fokas method, due to Fokas and collaborators in recent years.
We first consider Stokes flow problems in polygonal domains whose boundaries consist of
straight line edges. We show how to solve problems in the half-plane subject to different
boundary conditions along the real axis and we are able to retrieve analytical results found
using other techniques. Next, we present our transform approach to solve for a flow past
a periodic array of semi-infinite plates and for a periodic array of point singularities in a
channel, followed by a brief discussion on how to systematically solve problems in more
complex channel geometries.
Next, we show how to solve problems in circular domains whose boundaries consist of a
combination of straight line and circular edges. We analyze the problems of a flow past a
semicircular ridge in the half-plane, a translating and rotating cylinder above a wall and a
translating and rotating cylinder in a channel.