271 research outputs found
Sharp Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility
In this work, the sharp interface limit of the degenerate Cahn-Hilliard
equation (in two space dimensions) with a polynomial double well free energy
and a quadratic mobility is derived via a matched asymptotic analysis involving
exponentially large and small terms and multiple inner layers. In contrast to
some results found in the literature, our analysis reveals that the interface
motion is driven by a combination of surface diffusion flux proportional to the
surface Laplacian of the interface curvature and an additional contribution
from nonlinear, porous-medium type bulk diffusion, For higher degenerate
mobilities, bulk diffusion is subdominant. The sharp interface models are
corroborated by comparing relaxation rates of perturbations to a radially
symmetric stationary state with those obtained by the phase field model.Comment: 27 pages, 2 figure
On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility
We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian
fluids in a bounded domain by allowing for a degenerate mobility. The model has been developed by Abels, Garcke and Grün for fluids with different densities and leads to a solenoidal velocity field. It is given by a nonhomogeneous Navier-Stokes system with a modifed convective term coupled to a Cahn-Hilliard system, such that an energy estimate is fulfilled which follows from the fact that the model is thermodynamically consistent
Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction
In this paper we study the long-time behavior of a nonlocal Cahn-Hilliard
system with singular potential, degenerate mobility, and a reaction term. In
particular, we prove the existence of a global attractor with finite fractal
dimension, the existence of an exponential attractor, and convergence to
equilibria for two physically relevant classes of reaction terms
A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation
We propose a novel second order in time numerical scheme for
Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme
is based on second order convex-splitting for the Cahn-Hilliard equation and
pressure-projection for the Navier-Stokes equation. We show that the scheme is
mass-conservative, satisfies a modified energy law and is therefore
unconditionally stable. Moreover, we prove that the scheme is uncondition- ally
uniquely solvable at each time step by exploring the monotonicity associated
with the scheme. Thanks to the weak coupling of the scheme, we design an
efficient Picard iteration procedure to further decouple the computation of
Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by
the mixed finite element method. Ample numerical experiments are performed to
validate the accuracy and efficiency of the numerical scheme
On the stable discretization of strongly anisotropic phase field models with applications to crystal growth
We introduce unconditionally stable finite element approximations for
anisotropic Allen--Cahn and Cahn--Hilliard equations. These equations
frequently feature in phase field models that appear in materials science. On
introducing the novel fully practical finite element approximations we prove
their stability and demonstrate their applicability with some numerical
results.
We dedicate this article to the memory of our colleague and friend Christof
Eck (1968--2011) in recognition of his fundamental contributions to phase field
models.Comment: 20 pages, 8 figure
Sharp-interface limits of Cahn--Hilliard models and mechanics with moving contact lines
We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces
Thermodynamically Consistent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities
A new diffuse interface model for a two-phase flow of two incompressible
fluids with different densities is introduced using methods from rational
continuum mechanics. The model fulfills local and global dissipation
inequalities and is also generalized to situations with a soluble species.
Using the method of matched asymptotic expansions we derive various sharp
interface models in the limit when the interfacial thickness tends to zero.
Depending on the scaling of the mobility in the diffusion equation we either
derive classical sharp interface models or models where bulk or surface
diffusion is possible in the limit. In the two latter cases the classical
Gibbs-Thomson equation has to be modified to include kinetic terms. Finally, we
show that all sharp interface models fulfill natural energy inequalities.Comment: 34 page
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