Allen-Cahn and Cahn-Hilliard-like equations for dissipative dynamics of saturated porous media
arXiv:1110.5457 · doi:10.1016/j.jmps.2012.08.014
Abstract
We consider a saturated porous medium in the regime of solid-fluid segregation under an applied pressure on the solid constituent. We prove that, depending on the dissipation mechanism, the dynamics is described either by a Cahn-Hilliard or by an Allen-Cahn-like equation. More precisely, when the dissipation is modeled via the Darcy law we find that, for small deformation of the solid and small variations of the fluid density, the evolution equation is very similar to the Cahn-Hilliard equation. On the other hand, when only the Stokes dissipation term is considered, we find that the evolution is governed by an Allen-Cahn-like equation. We use this theory to describe the formation of interfaces inside porous media. We consider a recently developed model proposed to study the solid-liquid segregation in consolidation and we are able to fully describe the formation of an interface between the fluid-rich and the fluid-poor phase.
References in corpus (3)
Cited by in corpus (8)
- Phase field modeling of partially saturated deformable porous media
- Kink Localization under Asymmetric Double-Well Potential
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- Temperature-driven volume transition in hydrogels: phase--coexistence and interface localization
- Long-time behavior of solutions to the generalized Allen-Cahn model with degenerate diffusivity
- Effects of an impermeable wall in dissipative dynamics of saturated porous media
- Stability of the stationary solutions of the Allen-Cahn equation with non-constant stiffness