359 research outputs found
A Multiscale Diffuse-Interface Model for Two-Phase Flow in Porous Media
In this paper we consider a multiscale phase-field model for
capillarity-driven flows in porous media. The presented model constitutes a
reduction of the conventional Navier-Stokes-Cahn-Hilliard phase-field model,
valid in situations where interest is restricted to dynamical and equilibrium
behavior in an aggregated sense, rather than a precise description of
microscale flow phenomena. The model is based on averaging of the equation of
motion, thereby yielding a significant reduction in the complexity of the
underlying Navier-Stokes-Cahn-Hilliard equations, while retaining its
macroscopic dynamical and equilibrium properties. Numerical results are
presented for the representative 2-dimensional capillary-rise problem
pertaining to two closely spaced vertical plates with both identical and
disparate wetting properties. Comparison with analytical solutions for these
test cases corroborates the accuracy of the presented multiscale model. In
addition, we present results for a capillary-rise problem with a non-trivial
geometry corresponding to a porous medium
Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System
In this paper, we present a novel second order in time mixed finite element
scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities.
The scheme combines a standard second order Crank-Nicholson method for the
Navier-Stokes equations and a modification to the Crank-Nicholson method for
the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth
extrapolation and a trapezoidal rule are included to help preserve the energy
stability natural to the Cahn-Hilliard equation. We show that our scheme is
unconditionally energy stable with respect to a modification of the continuous
free energy of the PDE system. Specifically, the discrete phase variable is
shown to be bounded in and the discrete
chemical potential bounded in , for any time
and space step sizes, in two and three dimensions, and for any finite final
time . We subsequently prove that these variables along with the fluid
velocity converge with optimal rates in the appropriate energy norms in both
two and three dimensions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1411.524
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