Compacton formation under Allen--Cahn dynamics

We study the solutions of a generalized Allen-Cahn equation deduced from a Landau energy functional, endowed with a non-constant higher order stiffness. We analytically solve the stationary problem and deduce the existence of so-called compactons, namely, connections on a finite interval between the two phases. The dynamics problem is numerically solved and compacton formation is described

Kink Localization under Asymmetric Double-Well Potential

We study diffuse phase interfaces under asymmetric double-well potential energies with degenerate minima and demonstrate that the limiting sharp profile, for small interface energy cost, on a finite space interval is in general not symmetric and its position depends exclusively on the second derivatives of the potential energy at the two minima (phases). We discuss an application of the general result to porous media in the regime of solid-fluid segregation under an applied pressure and describe the interface between a fluid-rich and a fluid-poor phase. Asymmetric double-well potential energies are also relevant in a very different field of physics as that of Brownian motors. An intriguing analogy between our result and the direction of the dc soliton current in asymmetric substrate driven Brownian motors is pointed out

Phase coexistence in consolidating porous media

The appearence of the fluid-rich phase in saturated porous media under the effect of an external pressure is investigated. For this purpose we introduce a two field second gradient model allowing the complete description of the phenomenon. We study the coexistence profile between poor and rich fluid phases and we show that for a suitable choice of the parameters non-monotonic interfaces show-up at coexistence

Allen-Cahn and Cahn-Hilliard-like equations for dissipative dynamics of saturated porous media

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

A kinetic approach to the plane Poiseuille flow over a porous matrix

The Poiseuille-Couette gas flow in a channel and the gas flow through an adjacent porous medium are considered when the governing equations are obtained via a molecular kinetic approach based on the Boltzmann equation. The mass continuity, momentum balance and energy conservation are written for the gas in the contiguous regions, whereas the behavior of the solid matrix obeys to the heat diffusion equation. Two different space scalings lead to different forms of the equations for the steady flow through the fully saturated matrix. The boundary conditions at the interface between the two domains are investigated via a matching procedure