Moving boundary problems for quasi-steady conduction limited melting

Morrow, Liam C; King, John R; Moroney, Timothy J; Mccue, Scott

Abstract

The problem of melting a crystal dendrite is modelled as a quasi-steady Stefan 5 problem. By employing the Baiocchi transform, asymptotic results are derived in the limit that 6 the crystal melts completely, extending previous results that hold for a special class of initial and 7 boundary conditions. These new results, together with predictions for whether the crystal pinches off 8 and breaks into two, are supported by numerical calculations using the level set method. The effects of 9 surface tension are subsequently considered, leading to a canonical problem for near-complete-melting 10 which is studied in linear stability terms and then solved numerically. Our study is motivated in 11 part by experiments undertaken as part of the Isothermal Dendritic Growth Experiment, in which 12 dendritic crystals of pivalic acid were melted in a microgravity environment: these crystals were 13 found to be prolate spheroidal in shape, with an aspect ratio initially increasing with time then 14 rather abruptly decreasing to unity. By including a kinetic undercooling-type boundary condition in 15 addition to surface tension, our model suggests the aspect ratio of a melting crystal can reproduce 16 the same non-monotonic behaviour as that which was observed experimentally. 17

Citation

Morrow, L. C., King, J. R., Moroney, T. J., & Mccue, S. (2019). Moving boundary problems for quasi-steady conduction limited melting. SIAM Journal on Applied Mathematics, 79(5), 2107-2131. https://doi.org/10.1137/18M123445X

• Journal Article Type: Article
• Acceptance Date: Aug 13, 2019
• Online Publication Date: Oct 29, 2019
• Publication Date: Oct 29, 2019
• Deposit Date: Nov 12, 2019
• Publicly Available Date: Nov 13, 2019
• Journal: SIAM Journal on Applied Mathematics
• Print ISSN: 0036-1399
• Electronic ISSN: 1095-712X
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 79
• Issue: 5
• Pages: 2107-2131
• DOI: https://doi.org/10.1137/18M123445X
• Keywords: conduction-limited melting; melting in microgravity; moving-boundary problem; 18 surface tension; extinction; formal asymptotics; level set method 19 AMS subject classifications 35R37; 80A22; 65M99 20
• Public URL: https://nottingham-repository.worktribe.com/output/3236241
• Publisher URL: https://epubs.siam.org/doi/10.1137/18M123445X