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Evolution of microstructure: an example. (English) Zbl 1136.76349
Fiedler, Bernold (ed.), Ergodic theory, analysis, and efficient simulation of dynamical systems. Berlin: Springer (ISBN 3-540-41290-5/hbk). 501-522 (2001).
Summary: We consider a continuum model for the ow 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 themselves, by analysing 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.
For the entire collection see [Zbl 0968.00013].

76D27 Other free boundary flows; Hele-Shaw flows
35B35 Stability in context of PDEs
35Q35 PDEs in connection with fluid mechanics
76E99 Hydrodynamic stability
76S05 Flows in porous media; filtration; seepage