  Traditional mathematical models of Hele--Shaw flow consider the injection (or
withdrawal) of an air bubble into (or from) an infinite body of viscous fluid.
The most commonly studied feature of such a model is how the Saffman-Taylor
instability drives viscous fingering patterns at the fluid/air interface. Here
we consider a more realistic model, which assumes the viscous fluid is finite,
covering a doubly connected two-dimensional region bounded by two fluid/air
interfaces. For the case in which the flow is driven by a pressure difference
across the two interfaces, we explore this model numerically, highlighting the
development of viscous fingering patterns on the interface with the higher
pressure. Our numerical scheme is based on the level set method, where each
interface is represented as a separate level set function. We show that the
scheme is able to reproduce the characteristic finger patterns observed
experimentally up to the point at which one of the interfaces bursts through
the other. The simulations are shown to compare well with experimental results.
Further, we consider a model for the problem in which an annular body of fluid
is evolving in a rotating Hele--Shaw cell. In this case, our simulations
explore how either one or both of the interfaces can be unstable and develop
fingering patterns, depending on the rotation rate and the volume of fluid
present.
