Abstract
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.
Highlights
While there is a variety of simple models to approximate the shape of a melting particle [33, 38], the traditional approach from a mathematical perspective is to employ a Stefan problem, which involves the linear heat equation subject to appropriate boundary conditions on the solid-melt interface
We show that kinetic undercooling acts as a de-stabilising term, and is effectively in competition with surface tension
When these two terms are considered simultaneously, we find that the aspect ratio of a prolate spheroid can initially increase before decreasing suddenly to unity in the extinction limit, which is the same behaviour as observed in the Isothermal Dendritic Growth Experiment (IDGE)
Summary
While there is a variety of simple models to approximate the shape of a melting particle [33, 38], the traditional approach from a mathematical perspective is to employ a Stefan problem, which involves the linear heat equation subject to appropriate boundary conditions on the solid-melt interface. Glicksman et al [21] derive an exact solution to the infinite-Stefan-number problem in an infinite domain in prolate spheroidal coordinates, which applies under the further assumption that the aspect ratio of the dendrite remains constant.
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