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Evolution of Microstructure: an Example

In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems

Author

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  • Felix Otto

    (Universität Bonn, Institut für Angewandte Mathematik)

Abstract

We consider a continuum model for the flow of two phases of different mobility and density in a Hele-Shaw cell or a porous medium. As a consequence of the Saffman-Taylor instability, the phase distribution is thought to develop a microstructure, so that its evolution is effectively unpredictable. We identify the constraints on the macroscopic quantities, like the averaged volume fraction of the phases, and show that these constraints allow to derive some predictions on how the macroscopic quantities change over time. Furthermore, we investigate a class of closure hypothesis, which complement these constraints and thereby determine an evolution of the macroscopic quantities themselves, by analyzing the stability of this evolution. Our analysis uses a combination of tools from nonlinear scalar conservation laws and ideas from the theory of effective moduli.

Suggested Citation

  • Felix Otto, 2001. "Evolution of Microstructure: an Example," Springer Books, in: Bernold Fiedler (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pages 501-522, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-56589-2_22
    DOI: 10.1007/978-3-642-56589-2_22
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