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

TL;DR

This paper introduces a phase-field model to simulate Hele-Shaw flows with high viscosity contrast, capturing interface dynamics and finger competition, and reproduces steady-state finger widths that depend on velocity.

Contribution

It presents a novel one-sided phase-field model for high viscosity contrast Hele-Shaw flows, deriving macroscopic equations and demonstrating accurate numerical simulations.

Findings

Reproduces finger competition and steady-state finger widths.

Shows finger width approaches half the channel width at high velocities.

Validates the phase-field model against expected physical behaviors.

Abstract

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.