  A one-sided phase-field model is proposed to study the dynamics of unstable
interfaces of Hele-Shaw flows in the high viscosity contrast regime. The
corresponding macroscopic equations are obtained by means of an asymptotic
expansion from the phase-field model. Numerical integrations of the phase-field
model in a rectangular Hele-Shaw cell reproduce finger competition with the
final evolution to a steady state finger the width of which goes to one half of
the channel width as the velocity increases.
