Math. Model. Nat. Phenom.
Volume 10, Number 6, 2015Nonlocal reaction-diffusion equations
|77 - 89
|02 October 2015
A Mathematical Model for Flash Sintering
1 Mathematical Institute, University of
2 Maxwell Institute for Mathematical Sciences and Department of Mathematics School of Mathematical and Computer Sciences, Heriot-Watt University Riccarton, Edinburgh, EH14 4AS, UK
3 Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, UK
⋆ Corresponding author. E-mail: A.A.Lacey@hw.ac.uk
A mathematical model is presented for the Joule heating that occurs in a ceramic powder compact during the process of flash sintering. The ceramic is assumed to have an electrical conductivity that increases with temperature, and this leads to the possibility of runaway heating that could facilitate and explain the rapid sintering seen in experiments. We consider reduced models that are sufficiently simple to enable concrete conclusions to be drawn about the mathematical nature of their solutions. In particular we discuss how different local and non-local reaction terms, which arise from specified experimental conditions of fixed voltage and current, lead to thermal runaway or to stable conditions. We identify incipient thermal runaway as a necessary condition for the flash event, and hence identify the conditions under which this is likely to occur.
Mathematics Subject Classification: 35K58 / 35B44 / 35Q79 / 35Q60 / 35M30 / 41A60
Key words: flash sintering / non-local problems / non-linear heat equations / blow-up
© EDP Sciences, 2015
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