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A one-sided phase-field model is proposed to study the dynamics of unstableinterfaces of Hele-Shaw flows in the high viscosity contrast regime. Thecorresponding macroscopic equations are obtained by means of an asymptoticexpansion from the phase-field model. Numerical integrations of the phase-fieldmodel in a rectangular Hele-Shaw cell reproduce finger competition with thefinal evolution to a steady state finger the width of which goes to one half ofthe channel width as the velocity increases

Phase-field model of Hele-Shaw flows in the high-viscosity contrast regime