Temporal and spatial chaos in a van der Waals fluid due to periodic thermal fluctuations
type: Item
instance of: scholarly article
title: Temporal and spatial chaos in a van der Waals fluid due to periodic thermal fluctuations
zbMATH Open document ID: 0605.76078
DOI: 10.1016/0196-8858(85)90008-9
author: Marshall Slemrod
author: J. E. Marsden
published in: Advances in Applied Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The \textit{V. K. Mel'nikov} method [Tr. Mosk. Mat. O.-va 12, 3-52 (1963; Zbl 0135.310)] to prove the existence of deterministic chaos in periodically forced evolution equations is used to analyze two problems arising in the van der Waal's theory of phase transitions. The first problem is concerned with the dynamics of spinoidal decompositions in a van der Waals type fluid. The effect of a small time- periodic fluctuation of temperature is considered. Once the fluid is initially quenched in the unstable spinoidal region, temporal chaos results as a consequence of the fluctuations about a subcritical temperature. This is proved by application of the relevant Holmes-Marsden Mel'nikov theory [e.g.: \textit{P. Holmes} and \textit{J. E. Marsden}, J. Math. Phys. 23, 669-675 (1982; Zbl 0497.70025)]. The second problem is concerned with the effect of a small thermal perturbation on the equilibrium configuration of an infinite tube of van der Waals type fluid under a given applied load. Once the ambient pressure is appropriately chosen, spatial chaotic solutions form as a result of small spatially periodic perturbations. The van der Waals fluid theory is also applied to the elastic bar model of \textit{J. L. Ericksen} [J. Elasticity 5, 191-201 (1975; Zbl 0324.73067)] with the aid of a non-monotone stress-strain constitutive relation.
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 76E99
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 3978670
zbMATH Keywords: \textit{V. K. Mel'nikov} method
zbMATH Keywords: deterministic chaos
zbMATH Keywords: periodically forced evolution equations
zbMATH Keywords: van der Waal's theory of phase transitions
zbMATH Keywords: spinoidal decompositions
zbMATH Keywords: van der Waals type fluid
zbMATH Keywords: small time-periodic fluctuation of temperature
zbMATH Keywords: unstable spinoidal region
zbMATH Keywords: temporal chaos
zbMATH Keywords: subcritical temperature
zbMATH Keywords: small thermal perturbation
zbMATH Keywords: equilibrium configuration
zbMATH Keywords: infinite tube
zbMATH Keywords: spatial chaotic solutions
zbMATH Keywords: spatially periodic perturbations
MaRDI profile type: Publication
full work available at URL: 0196-8858(85)90008-9
OpenAlex ID: W2012274601
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description: scientific article; zbMATH DE number 3978670