Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Supercooling and superheating effects in heterogeneous systems

Author: A. Visintin
Journal: Quart. Appl. Math. 45 (1987), 239-263
MSC: Primary 80A20
DOI: https://doi.org/10.1090/qam/895096
MathSciNet review: 895096
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Abstract: In the model for phase transitions in binary systems based on Fourier's and Fick's laws, the interface equilibrium condition $ \theta = w$, relating the temperature $ \theta $ and the chemical activity $ w$, is here replaced by a relaxation dynamics for the liquid concentration $ \chi $:

$\displaystyle \frac{{\partial \chi }}{{\partial t}} + {\tilde H^{ - 1}}\left( \chi \right) = \beta \left( {\theta ,w} \right)$

( $ {\tilde H^{ - 1}}$: inverse of the Heaviside graph); here $ \beta \in {C^0}\left( {{R^2}} \right)$ and $ sign\beta \left( {\theta ,w} \right) = sign\left( {\theta - w} \right)$. In the case of a single dimension of space, with an interface $ x = s\left( t \right)$, a different dynamics can be considered:

$\displaystyle s'\left( t \right) = \beta \left( {\theta \left( {s\left( t \right),t} \right),w\left( {s\left( t \right),t} \right)} \right)$


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DOI: https://doi.org/10.1090/qam/895096
Article copyright: © Copyright 1987 American Mathematical Society

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