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Periodic structures in a van der Waals fluid

Published online by Cambridge University Press:  14 November 2011

Paul C. Fife  and
Xiao-Ping Wang
Paul C. Fife
Affiliation:
Mathematics Department, University of Utah, Salt Lake City, Utah 84112, U.S.A.
Xiao-Ping Wang
Affiliation:
Mathematics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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Abstract

A system of partial differential equations modelling a van der Waals fluid or an elastic medium with nonmonotone pressure-density relation is studied. As the system changes type, regularisations are considered. The existence of one-dimensional periodic travelling waves, with prescribed average density in a certain range, average velocity and wavelength, is proved. They exhibit layer structure when the regularisation parameter is small. Similarities with the Cahn–Hilliard equation are explored.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 2 , 1998 , pp. 235 - 250
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Cahn, J. W.. Free energy of a nonuniform system. II. Thermodynamic basis. J. Chem. Phys. 30 (1959), 1121–4.Google Scholar
2Carr, J., Gurtin, M. E. and Slemrod, M.. Structured phase transitions on a finite interval. Arch. Rational Mech. Anal. 86 (1984), 317–51.CrossRefGoogle Scholar
3Grant, C.. Spinodal decomposition for the Cahn-Hilliard equation. Comm. Partial Differential Equations 18 (1993), 453–90.CrossRefGoogle Scholar
4Grinfeld, M.. Dynamic phase transition: Existence of cavitation waves. Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), 153–63.CrossRefGoogle Scholar
5Grinfeld, M. and Novick-Cohen, A.. Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments. Proc. Roy. Soc. Edinburgh Sect. A 125 (1993), 351–70.Google Scholar
6Hagan, R. and Slemrod, M.. The viscosity-capillarity criterion for shocks and phase transitions. Arch. Rational Mech. Anal. 83 (1984), 333–61.CrossRefGoogle Scholar
7Hsieh, D.-Y. and Wang, X.-P.. Phase transition in Van der Waals fluid. SIAM J. Appl. Math. 57 (1997), 871–92.Google Scholar
8Korteweg, D. J.. Sur la forme que prennent les equations du mouvement des fluides si I'on tient compte des forces capillaires par des variations de densité. Arch. Neerlandaises Sci. Exactes Naturelles Ser. II 6 (1901), 1–24.Google Scholar
9Maier, S. and Wanner, T.. Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate, Comm. Math. Phys., to appear. Part II: Nonlinear dynamics, preprint.Google Scholar
10Novick-Cohen, A. and Peletier, L. A.. Steady states of the one-dimensional Cahn-Hilliard equation. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 1071–98.Google Scholar
11Serrin, J.. Phase transitions and interfacial layers for van der Waals fluids. In Proceedings of SAFA IV Conference on Recent Methods in Nonlinear Analysis and Applications, eds. Camfora, A., Rionero, S., Sbordone, C. and Trombetti, C. (Naples, 1980).Google Scholar
12Slemrod, M.. Admissability criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81 (1983), 301–15.Google Scholar
13Zheng, S.. Asymptotic behaviour of solutions to the Cahn-Hilliard equation. Appl. Anal. 23 (1986), 165–84.Google Scholar