Moving Boundary Problems for Quasi-Steady Conduction Limited Melting

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.

A Unified Analytical Approach to Fixed and Moving Boundary Problems for the Heat Equation

“…In summary, there are very limited known analytical solutions to Stefan problems and existing closed-form solutions strongly depend on prescribed initial and boundary conditions. As such, numerical simulations are mainly used for the study of moving boundary problems, both for linear and nonlinear equations [4][5][6][7][8][9][34][35][36]. However, a recent study adopted a deep neural network approach to solve Stefan problems [37].…”

“…Moving boundary problems occur in numerous important areas of science and engineering [1][2][3]. Recent applications of such problems include modelling biological and tumour invasions [4][5][6][7], drug delivery [8] and melting of crystal dendrite [9]. The classical one-dimensional Stefan problem is the canonical moving boundary problem which was introduced by J. Stefan, a Slovenian physicist, in a series of four papers to model the melting of ice and to calculate the moving boundary where the phase transition from water to ice occurs; see Vuik [10] for some historical notes.…”

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

“…In that study it was shown that our numerical scheme is capable of accurately describing the behaviour of interfaces that contract to either one or multiple points. In this section, we give a brief description of the scheme, and refer the reader to [42] for further details.…”

A Review of One-Phase Hele-Shaw Flows and a Level-Set Method for Nonstandard Configurations

“…The level-set method is a popular tool for studying moving boundary problems in fluid dynamics, and has been used to investigate interfacial instabilities that occur in Stefan problems [25,54] and Hele-Shaw flow [66,84]. Also, we have applied this method to these applications, in particular to conduction-limited melting of crystal dendrites [98], bubbles shrinking and breaking up in a porous medium [97], and bubbles expanding in various Hele-Shaw configurations [99]. We refer to [55,103,121] for more information about the level-set method, including details regarding implementation and applications.…”