Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. I. Theoretical approach

TL;DR

This paper develops a phase-field model for Hele-Shaw flows with arbitrary viscosity contrast, validating it against classical equations and analyzing the effects of interface thickness on accuracy and convergence.

Contribution

It introduces a theoretical phase-field model for Hele-Shaw flows with arbitrary viscosity contrast and analyzes its asymptotic behavior and accuracy.

Findings

Model reproduces Hele-Shaw equations in the sharp-interface limit.

Corrections to the equations are computed to first order in interface thickness.

Convergence to Hele-Shaw dynamics is slower at high viscosity contrasts.

Abstract

We present a phase-field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. With asymptotic matching techniques we check the model to yield the right Hele-Shaw equations in the sharp-interface limit and compute the corrections to these equations to first order in the interface thickness. We also compute the effect of such corrections on the linear dispersion relation of the planar interface. We discuss in detail the conditions on the interface thickness to control the accuracy and convergence of the phase-field model to the limiting Hele-Shaw dynamics. In particular, the convergence appears to be slower for high viscosity contrasts.