  The classical model for studying one-phase Hele-Shaw flows is based on a
highly nonlinear moving boundary problem with the fluid velocity related to
pressure gradients via a Darcy-type law. In a standard configuration with the
Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic.
Therefore, conformal mapping techniques and boundary integral methods can be
readily applied to study the key interfacial dynamics, including the
Saffman-Taylor instability and viscous fingering patterns. As well as providing
a brief review of these key issues, we present a flexible numerical scheme for
studying both standard and non-standard Hele-Shaw flows. Our method consists of
using a modified finite difference stencil in conjunction with the level set
method to solve the governing equation for pressure on complicated domains and
track the location of the moving boundary. Simulations show that our method is
capable reproducing the distinctive morphological features of the
Saffman-Taylor instability on a uniform computational grid. By making
straightforward adjustments, we show how our scheme can easily be adapted to
solve for a wide variety of configurations, including cases where the gap
between the plates is linearly tapered, the plates are separated in time, and
the entire Hele-Shaw cell is rotated at a given angular velocity.
