  Viscous fingering experiments in Hele-Shaw cells lead to striking pattern
formations which have been the subject of intense focus among the physics and
applied mathematics community for many years. In recent times, much attention
has been devoted to devising strategies for controlling such patterns and
reducing the growth of the interfacial fingers. We continue this research by
reporting on numerical simulations, based on the level set method, of a
generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is
altered. First, we investigate how imposing constant and time-dependent
injection rates in a Hele-Shaw cell that is either standard, tapered or
rotating can be used to reduce the development of viscous fingering when an
inviscid fluid is injected into a viscous fluid over a finite time period. We
perform a series of numerical experiments comparing the effectiveness of each
strategy to determine how these non-standard Hele-Shaw configurations influence
the morphological features of the inviscid-viscous fluid interface. Tapering
plates in either converging or diverging directions leads to reduced metrics of
viscous fingering at the final time when compared to the standard parallel
configuration, especially with carefully chosen injection rates; for the
rotating plate case, the effect is even more dramatic, with sufficiently large
rotation rates completely stabilising the interface. Next, we illustrate how
the number of non-splitting fingers can be controlled by injecting the inviscid
fluid at a time-dependent rate while increasing the gap between the plates.
Simulations compare well with previous experimental results for various
injection rates and geometric configurations. Further, we demonstrate how the
fully nonlinear dynamics of the problem affect the number of fingers that
emerge and how well this number agrees with predictions from linear stability
analysis.
