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We examine the effect of a kinetic undercooling condition on the evolution ofa free boundary in Hele--Shaw flow, in both bubble and channel geometries. Wepresent analytical and numerical evidence that the bubble boundary is unstableand may develop one or more corners in finite time, for both expansion andcontraction cases. This loss of regularity is interesting because it occursregardless of whether the less viscous fluid is displacing the more viscousfluid, or vice versa. We show that small contracting bubbles are described toleading order by a well-studied geometric flow rule. Exact solutions to thisasymptotic problem continue past the corner formation until the bubblecontracts to a point as a slit in the limit. Lastly, we consider the evolvingboundary with kinetic undercooling in a Saffman--Taylor channel geometry. Theboundary may either form corners in finite time, or evolve to a single longfinger travelling at constant speed, depending on the strength of kineticundercooling. We demonstrate these two different behaviours numerically. Forthe travelling finger, we present results of a numerical solution methodsimilar to that used to demonstrate the selection of discrete fingers bysurface tension. With kinetic undercooling, a continuum of corner-freetravelling fingers exists for any finger width above a critical value, whichgoes to zero as the kinetic undercooling vanishes. We have not been able tocompute the discrete family of analytic solutions, predicted by previousasymptotic analysis, because the numerical scheme cannot distinguish betweensolutions characterised by analytic fingers and those which are corner-free butnon-analytic.

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