Abstracts
Structure development in shear flow using a diffuse interface
model
(H.E.H. Meijer, B.J. Keestra, P.D. Anderson, F.N. van de Vosse)
Patrick D. Anderson
A numerical model, based on a Galerkin type spectral element
method
field is used to study morphology development and the kinetics of phase
separation in a homogeneous shear flow.
The theory of Cahn and Hilliard, describing the free energy of a
non-uniform system with local and non-local terms, is used in the
framework of the non-classical thermodynamics described by de Groot
and
Mazur. A complete set of equations governs both phase separation and
hydrodynamic effects such as coalescence. This model is compared with
results in which the velocity field is inserted directly into the
convective term of the time-dependent Ginzburg-Landau equation. The
results of a coupled solution with high capillary number closely
resemble those of a directly imposed velocity field. However, once a
smaller capillary number is chosen the interfacial forces become more
dominant over the viscous forces, which result in different
morphologies.
Furthermore, a characteristic problem in diffuse interface models is
that the length scale of the interface, represented by the Cahn number,
is much smaller than the global length scale. Computationally it is
difficult to resolve both length scales at the same time. Proper scaling
of the Cahn number and the Peclet number, which shows up in the
composition equation, can solve this problem.
To validate the simulations of the structure development as a result of
phase separation in shear-flow, the results of the diffuse interface
modelling are compared with results from light scattering experiments
(SALS) on a well-known polymer blend system of PMMA and SAN.
P.D.Anderson@tue.nl
Study of diphasic flows through
Cahn-Hilliard type models
Franck Boyer
In this talk, I will address the study of incompressible and
viscous
diphasic flows through order parameter formulation. The chosen model is
a coupling between a non-homogeneous Navier-Stokes equation and a
Cahn-Hilliard equation.
One of the possible derivations of this model will be presented. Then, I
will discuss some of the theoretical results we can prove concerning
this system of equations: existence and uniqueness results but also some
qualitative results of stability.
Finally, I will present numerical results in various physical situations
such as the spinodal decomposition under shear.
fboyer@cmi.univ-mrs.fr
Finite volume methods for van der Waals
fluids and fully
discrete cell-entropy inequalities
Frédéric Coquel
The present talk is devoted to the numerical approximation of the
solutions
of mixed (hyperbolic-elliptic) systems of conservation laws modeling the
dynamics of compressible fluids with phase change. Here the non isothermal
framework is addressed when considering the capillarity coefficient as a
general (positive) function of the specific volume. Starting from a given
capillarity law, we first introduce
an extended version of the system still in conservation form
and which by construction encodes all the
smooth solutions of the original PDEs. A given solution is recovered
provided that the initial data of the extended version is suitably
prescribed. Taking advantage of such a result, the structure of the
extended
system allows us to derive a natural numerical procedure where the
approximation of the interstitial energy plays a central role. The
numerical
procedure we propose results in fully discrete
Finite Volume Methods that obey cell-entropy inequalities under a natural
CFL
condition.
coquel@ann.jussieu.fr
Motion of compressible bubbles in a perfect
fluid
Sergey L. Gavrilyuk
Collective behavior of compressible gas bubbles moving in an
inviscid
incompressible fluid is studied. A kinetic approach is employed, based on
an approximate calculation of the fluid flow potential. Kinetic equations
governing the evolution of a distribution function of bubbles are derived.
These equations are similar to Vlasov equations. Conservation laws which
are direct consequences of the kinetic system are found. For a narrowly
peaked distribution function they form a closed system of hydrodynamical
equations for the mean flow parameters. A variational principle for
the hydrodynamical system is established and the linear stability analysis
is performed. In particular, it is shown that the sound speed velocity
behavior in such a system is analogous to that in liquid - vapor mixtures.
Sergey.Gavrilyuk@univ.u-3mrs.fr
The dynamics of lines between fluids and
solid
surfaces
Henry Gouin
Today, the motion of the contact line formed at the intersection
of
two fluids and a solid is still subject to dispute.
A new picture of the dynamics of wetting is offered through an example
of non-Newtonian liquid motions. The kinematics of liquids at the contact
line and the equations of motions are revisited.
Adherence conditions are required except at the contact line.
Consequently, the velocity field is
discontinuous at the contact line and generates a concept of line
viscosity vector but stresses
and viscous dissipation remain bounded. A Young-Dupré equation
for the dynamic contact angle
between the interface and solid surface is proposed.
Henri.Gouin@univ.u-3mrs.fr
Diffuse interface models of
nucleation
(László Gránásy, Tamás Pusztai, and
Tamás Börzsönyi)
László Gránásy
The superiority of diffuse
interface approaches when addressing
nucleation is demonstrated in three cases. (i)
Nucleation rate is predicted for vapor-liquid nucleation of
non-polar/weakly
polar fluids using the perturbative
density functional approach, which uses
the hard-sphere fluid as a reference and the attractive part of the
interaction
potential is assumed to be of Yukawa form [1]. It will be shown that
for various substances this approach removes the
several orders of magnitude difference seen between the classical sharp
interface theory and experiment. (ii) A
simple density functional theory of crystal nucleation
is presented for the hard-sphere system [2], which relies on a
Ginzburg-Landau expanded free energy Shih et al. proposed [3],
and the square-gradient approximation. The two
model parameters are fixed using the interface thickness
and interface free energy known from molecular dynamics
simulations. Without adjustable parameters a fair
agreement is achieved with nucleation barrier heights from
computer simulations. Finally, (iii) we perform a
quantitative test of the phase-field theory for unary and binary
systems [4], and find a reasonable agreement with
computer simulations and experiment. We demonstrate, that
introducing an orientational order parameter,
complex solidification morphologies can be modeled (Fig. 1).
Figure Fig. 1: Composition, phase-field,
and orientation maps for
nucleation and dendritic solidification in a binary alloy as
predicted by the phase field theory.
References.
[1] D. W. Oxtoby and R. Evans, J. Chem. Phys. 89, 7521 (1988);
L. Gránásy, Z. Jurek, and D. W. Oxtoby, Phys. Rev.
E 62, 7486 (2000).
[2] L. Gránásy and T. Pusztai, unpublished.
[3] W. H. Shih, Z. Q. Wang, X. C. Zeng, and D. Stroud,
Phys. Rev. A 35, 2611 (1987).
[4] L. Gránásy, T. Börzsönyi, and T. Pusztai,
J. Cryst. Growth, in print.
grana@mail.szfki.hu
Application of phase-field modeling to the
calculation of two-phase
Navier-Stokes flows
David Jacqmin
Various difficulties can occur when applying phase-field models
to the
calculation of two-phase flow. These can include problems with
diffuse-interface thickness control, excessive damping of waves, and
unwanted coarsening and coalescences. There is also the general problem
of finding optimal procedures for obtaining convergence. I will discuss
some of my attempts to think through and deal with these issues.
Aspects of the ëxtra physics" of the phase-field Navier-Stokes
equations will be briefly analyzed. These include especially the
behavior of the chemical potential field. Special techniques such as
the use of compact high order stencils for the phase-field equation and
the application of double-barrier chemical potentials will be
described. Example calculations will include contact-line flows,
propagation of waves in unstable Rayleigh-Taylor flows, and sloshing
and rotating flows.
fsdavid@lerc.nasa.gov
Travelling waves in plasma sustained by a
laser beam
Bogdan Kazmierczak
We use the Conley connection index theory
or the implicit function technique to prove
existence of travelling wave solutions to
general reaction diffusion systems satisfying
local monotonicity conditions. The results may be
applied e.g. to laser sustained plasma.
bkazmier@ippt.gov.pl
Thermocapillary convection and vorticity
singularity
Gérard Labrosse
A vorticity singularity occurs with the usual
modelling of the capillary stress in the liquid
bridge configuration. It will be shown that
hydrodynamics is sensitive to the presence of a
small length scale introduced to remove it
explicitly. A discussion will be proposed about
the physics which is behind this singularity.
labrosse@limsi.fr
Phase-field modelling of phase transitions
in multi-component
multi-phase systems
Britta Nestler
A phase-field model for a general class of multi-phase metallic
alloys is proposed which
describes both, multi-phase solidification phenomena as well as
polycrystalline grain structures.
The model serves as a computational method to simulate the motion and
kinetics of multiple phase
boundaries and enables the visualization of the diffusion processes and of
the phase transitions
in multi-phase systems. Numerical simulations are presented which
illustrate the
capability of the phase-field model to recover a variety of complex
experimental growth structures.
In particular, the phase-field model can be used to simulate
microstructure evolutions in eutectic,
peritectic and monotectic alloys. In addition, polycrystalline grain
structures with effects such as
wetting, grain growth, symmetry properties of adjacent triple junctions in
thin film samples and
stability criteria at multiple junctions are described by phase-field
simulations.
britta.nestler@fh-karlsruhe.de
Modeling and dynamics of fluids near
the liquid-vapor critical point
Robert L. Pego
Near the liquid-vapor critical point, many fluid properties
(like compressibility) diverge in anomalous ways
that have greatly interested physicists for over 30 years.
This regime is rich with unusual hydrodynamic phenomena,
involving large density gradients, thermoacoustic coupling,
and arbitrarily small wave-damping rates.
I will describe the modeling of low-speed flows
with strong gravitational stratification and
the main mechanisms of relaxation to equilibrium,
including a possible role for capillary stress.
rpego@ipst.umd.edu
An attempt of kinetic-theoric approach to
van
der Waals fluids
Kazimierz Piechór
A discrete four velocity model of the
Enskog-Vlasov equation and its hydrodynamic
approximation will be presented. The travelling
wave solutions to both models are considered
looking for differences between both descriptions.
kpiechor@ippt.gov.pl
Line tension phenomenon described by
Cahn-Hilliard model
Pierre Seppecher
Line tension is an interesting extension of the classical model of
capillarity. In addition to the classical surface tensions, which
correspond to a concentration of energy on the interfaces dividing the
different phases and on the wall of the container, one considers the
possibility of a concentration of energy along the contact line. We
first study the model of capillarity with line tension and we show that
it must be used carefully. A naive formulation leads to ill-posed
problems since the corresponding energy is not lower semi continuous. A
relaxed energy has to be considered instead and the modification of
Young's law induced by the presence of line tension is not trivial. The
relaxed formulation involves boundary phases as well as volume phases.
From a physical point of view these boundary phases can be interpreted
as precursor films. Then we study the equilibrium of a Cahn-Hilliard
fluid in a container assuming that the potential of the interaction
between the fluid and the wall is a double-well potential. A suitable
assumption on the order of magnitude of this interaction potential leads
to the desired concentration of energy.
seppecher@univ-tln.fr
Dynamic phase boundaries:
regularization through discretization
Lev Truskinovsky
We study the simplest prototypical discrete model of a moving
phase
boundary allowing explicit computation of the functional relation
between the macroscopic driving force and the phase boundary velocity.
The adopted model is purely conservative and contains information only
about elasticities of the constitutive elements. The apparent
dissipation is due to the micro-instabilities and induced radiation of
the high frequency waves. These waves carry the energy away from from
the moving phase boundary but remain ïnvisible" at the macro level.
trusk@aem.umn.edu
Phase relaxation
Augusto Visintin
In the framework of the classic Stefan model of phase
transitions,
diffuse and sharp
interfaces are compared. The former are here represented via
so-called phase
relaxation, [1]. The condition
which represents thermal equilibrium at the moving interface S,
is here replaced by a nonequilibrium condition of the form
|
e |
c
t
|
= j(q,c) in W×]0,T[. (2) |
|
Here q is the relative temperature, c is the phase
function (c = 0 in
the solid, c = 1 in the liquid, 0 < c < 1 in the mushy region),
j is a
continuous function such that j(q,c) = 0 iff c ‘ H(q) (H: =
Heaviside graph), and e is a positive relaxation parameter.
Equation (2) is coupled with the energy balance (which is here written with
normalized coefficients)
|
|
t
|
(q+c) -Dq = f in W×]0,T[. (3) |
|
As the relaxation parameter vanishes, convergence to a weak formulation of
the
Stefan problem is proved by means of L1-type techniques [2].
An analogous approach is then applied to the relaxation of a
forward-backward
quasilinear parabolic equation. In the limit a quasilinear parabolic
equation with
hysteresis is obtained, of the following form:
|
|
t
|
|
È
Î
|
u+F(u) |
˜
š
|
-Du = f in W×]0,T[; (4) |
|
here F:C0([0,T])Æ C0([0,T]) is a hysteresis operator [3].
Open problems include the study of a vector Stefan-like problem issued from
electromagnetic evolution of a metal exhibiting a discontinuous [B\vec]
vs. [H\vec] constitutive relation. [4]
References.
[1] A.V. : Models of phase transitions. Birkhäuser, Boston
(1996)
[2] A.V. : Models of phase-relaxation. Differential and
Integral Equations, 14
(2001), 1469-1486
[3] A.V. : Forward-backward parabolic equations and
hysteresis. Calc. Var.
(in press)
[4] A.V. : On some models of ferromagnetism. In: Free
boundary problems,
theory and applications, I (N. Kenmochi, ed.), Gakkotosho (2000), 411-428.
visintin@science.unitn.it
The nonlinear dynamics of a nematic liquid
crystal in the presence of a
shear flow
Adam Wheeler
In this talk I will describe a the Landau-de Gennes theory of a
nematic
liquid crystal in which the orientational order is represented by a tensor
order parameter. The effect of flow of the material can be incorporated
into
the model and I will describe how a combination of numerical and analytical
techniques can be used to understand the rich variety of
ordered states. In particular time-dependent states exist which involve
so-called wagging and tumbling behaviour which results from the presence of
a Takens-Bogdanov point in the underlying bifurcation structure. I will
show
how wagging and tumbling can be understood in the context of bifurcation
structure.
aaw@maths.soton.ac.uk
File translated from TEX
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On 23 May 2002,
16:07.