Corner and finger formation in Hele--Shaw flow with kinetic undercooling regularisation

TL;DR

This paper investigates how kinetic undercooling affects free boundary evolution in Hele-Shaw flow, revealing boundary instabilities, corner formation, and different traveling finger behaviors through analytical and numerical methods.

Contribution

It provides new insights into boundary regularity loss, corner formation, and finger selection mechanisms under kinetic undercooling in Hele-Shaw flow, supported by analytical and numerical evidence.

Findings

Boundaries can develop corners in finite time regardless of fluid displacement direction.

Small contracting bubbles follow a geometric flow leading to a slit as the bubble contracts to a point.

A continuum of corner-free traveling fingers exists above a critical width, depending on kinetic undercooling.

Abstract

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele--Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman--Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.