Asymptotic analysis of the Guyer–Krumhansl–Stefan model for nanoscale solidification

Matthew G. Hennessy, Marc Calvo Schwarzwälder, Timothy G. Myers*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

11 Citations (Scopus)

Abstract

Nanoscale solidification is becoming increasingly relevant in applications involving ultra-fast freezing processes and nanotechnology. However, thermal transport on the nanoscale is driven by infrequent collisions between thermal energy carriers known as phonons and is not well described by Fourier's law. In this paper, the role of non-Fourier heat conduction in nanoscale solidification is studied by coupling the Stefan condition to the Guyer–Krumhansl (GK) equation, which is an extension of Fourier's law, valid on the nanoscale, that includes memory and non-local effects. A systematic asymptotic analysis reveals that the solidification process can be decomposed into multiple time regimes, each characterised by a non-classical mode of thermal transport and unique solidification kinetics. For sufficiently large times, Fourier's law is recovered. The model is able to capture the change in the effective thermal conductivity of the solid during its growth, consistent with experimental observations. The results from this study provide key quantitative insights that can be used to control nanoscale solidification processes.

Original languageEnglish
Pages (from-to)1-17
Number of pages17
JournalApplied Mathematical Modelling
Volume61
DOIs
Publication statusPublished - Sept 2018

Bibliographical note

Funding Information:
We thank Brian Wetton for his advice regarding the numerical solution of this problem. This project has received funding from the European Union Horizon 2020 research and innovation programme under grant agreement No 707658. MC acknowledges that the research leading to these results has received funding from ‘la Caixa’ Foundation. TGM acknowledges the support of a Ministerio de Ciencia e Innovación grant MTM2014-56218. The authors have been partially funded by the CERCA Programme of the Generalitat de Catalunya.

Funding Information:
We thank Brian Wetton for his advice regarding the numerical solution of this problem. This project has received funding from the European Union Horizon 2020 research and innovation programme under grant agreement No 707658 . MC acknowledges that the research leading to these results has received funding from ‘la Caixa’ Foundation. TGM acknowledges the support of a Ministerio de Ciencia e Innovación grant MTM2014-56218. The authors have been partially funded by the CERCA Programme of the Generalitat de Catalunya.

Publisher Copyright:
© 2018 Elsevier Inc.

Keywords

  • Asymptotic analysis
  • Guyer-Krumhansl equation
  • Nanoscale phase change
  • Stefan problem

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