  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.
