652 research outputs found
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
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
A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects
In this paper, we develop a phase-field model for binary incompressible
(quasi-incompressible) fluid with thermocapillary effects, which allows for the
different properties (densities, viscosities and heat conductivities) of each
component while maintaining thermodynamic consistency. The governing equations
of the model including the Navier-Stokes equations with additional stress term,
Cahn-Hilliard equations and energy balance equation are derived within a
thermodynamic framework based on entropy generation, which guarantees
thermodynamic consistency. A sharp-interface limit analysis is carried out to
show that the interfacial conditions of the classical sharp-interface models
can be recovered from our phase-field model. Moreover, some numerical examples
including thermocapillary convections in a two-layer fluid system and
thermocapillary migration of a drop are computed using a continuous finite
element method. The results are compared to the corresponding analytical
solutions and the existing numerical results as validations for our model
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