  The problem of melting a crystal dendrite is modelled as a quasi-steady
Stefan problem. By employing the Baiocchi transform, asymptotic results are
derived in the limit that the crystal melts completely, extending previous
results that hold for a special class of initial and boundary conditions. These
new results, together with predictions for whether the crystal pinches off and
breaks into two, are supported by numerical calculations using the level set
method. The effects of surface tension are subsequently considered, leading to
a canonical problem for near-complete-melting which is studied in linear
stability terms and then solved numerically. Our study is motivated in part by
experiments undertaken as part of the Isothermal Dendritic Growth Experiment,
in which dendritic crystals of pivalic acid were melted in a microgravity
environment: these crystals were found to be prolate spheroidal in shape, with
an aspect ratio initially increasing with time then rather abruptly decreasing
to unity. By including a kinetic undercooling-type boundary condition in
addition to surface tension, our model suggests the aspect ratio of a melting
crystal can reproduce the same non-monotonic behaviour as that which was
observed experimentally.
