- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 06:58:26
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=11&iss=1&update_since=2024-03-29
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
11
2009
1
Phase boundary dynamics: transitions between ordered and disordered lipid monolayers
Hans Wilhelm
Alt
Universität Bonn, BONN, GERMANY
Wolfgang
Alt
Rheinische Friedrich-Wilhelms-Universität Bonn, BONN, GERMANY
Based on a general thermodynamical theory of mass and momentum, we propose and investigate a new phase field model for small transition layers between two spatially separated phases with intersecting free energy functions. We use a phase fraction, that only depends on the ratio of the two density components. From the phase field model we derive conditions for the sharp interface velocity and density jumps. The general model is motivated by and applied to the dynamics of lipid monolayers, which appear as surfactant on the strongly expanded and compressed thin water film of lung alveoli. While the \emph{liquid condensed ordered phase} (LC) of a flat lipid monolayer is characterized by high viscosity and limited compressibility, the \emph{liquid expanded disordered phase} (LE) is dominated by diffusion and high compressibility. In order to perform the asymptotic transition layer analysis at moving phase boundaries, a new nonlinear free energy interpolation model is proposed whose excess energy, in comparison to standard linear interpolations, contains an energy hump that has to be surpassed in a permissive transition from one phase to the other. This leads to a unique density jump condition in the case that the ordered phase is extending, whereas in the retracting case the jump densities are not restricted. The transition profiles and the resulting interface speed are numerically determined for a typical example by solving the nonlinear degenerate ODE-system. In a simplified 1-dimensional situation with low Reynolds number, the approximative macroscopic system of differential equations with moving sharp interface is numerically solved and interpreted in application to surfactant monolayers in lung alveoli.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
1
36
10.4171/IFB/202
http://www.ems-ph.org/doi/10.4171/IFB/202
Viscosity solutions for a model of contact line motion
Karl
Glasner
University of Arizona, TUCSON, UNITED STATES
Inwon
Kim
University of California Los Angeles, LOS ANGELES, UNITED STATES
This paper considers a free boundary problem that describes the motion of contact lines of a liquid droplet on a flat surface. The elliptic nature of the equation for droplet shape and the monotonic dependence of contact line velocity on contact angle allows us to introduce a notion of ``viscosity" solutions for this problem. Unlike similar free boundary problems, a comparison principle is only available for a modified short-time approximation because of the constraint that conserves volume. We use this modified problem to construct viscosity solutions to the original problem under a weak geometric restriction on the free boundary shape. We also prove uniqueness provided there is an upper bound on front velocity.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
37
60
10.4171/IFB/203
http://www.ems-ph.org/doi/10.4171/IFB/203
Multiscale analysis of a prototypical model for the interaction between microstructure and surface energy
Andrea
Braides
Università di Roma Tor Vergata, ROMA, ITALY
Caterina Ida
Zeppieri
SISSA, TRIESTE, ITALY
Γ-convergence, Γ-development, phase transitions, homogenization
The combined effect of fine heterogeneities and small gradient perturbations is analyzed by means of an asymptotic development by $\Gamma$-convergence for a family of energies related to (one-dimensional) phase transformations. We show that multi-scale effects add up to the usual sharp-interface limit, due to the homogenization of microscopic interfaces, internal and external boundary layers, optimal arrangements of microscopic oscillations, etc. Several regimes are analyzed depending on the ``size'' of the heterogeneity ({\em small or large perturbations} of a homogeneous situation) and their relative period as compared with the characteristic length of the phase transitions ({\em slow or fast oscillations}).
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
61
118
10.4171/IFB/204
http://www.ems-ph.org/doi/10.4171/IFB/204
A moving boundary problem for periodic Stokesian Hele–Shaw flows
Joachim
Escher
University of Hannover, HANNOVER, GERMANY
Bogdan-Vasile
Matioc
Leibniz University Hannover, HANNOVER, GERMANY
Quasilinear elliptic equation, nonlinear parabolic equation, non-Newtonian fluid, Hele–Shaw flow
This paper is concerned with the motion of an incompressible, viscous fluid in a Hele-Shaw cell. The free surface is moving under the influence of gravity and the fluid is modelled using a modified Darcy law for Stokesian fluids. We combine results from the theory of quasilinear elliptic equations, analytic semigroups and Fourier multipliers to prove existence of a unique classical solution to the corresponding moving boundary problem.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
119
137
10.4171/IFB/205
http://www.ems-ph.org/doi/10.4171/IFB/205
Regularity and nonexistence results for some free-interface problems related to Ginzburg–Landau vortices
Nam
Le
New York University, NEW YORK, UNITED STATES
We study regularity and nonexistence properties for some free-interface problems arising in the study of limiting vorticities associated to the Ginzburg-Landau equations with magnetic field in two dimensions. Our results imply in particular that if these limiting vorticities concentrate on a smooth closed curve then they have a distinguished sign; moreover, if the domain is thin then solutions of the Ginzburg-Landau equations cannot have a number of vortices much larger than the applied magnetic field.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
139
152
10.4171/IFB/206
http://www.ems-ph.org/doi/10.4171/IFB/206
Level set approach for fractional mean curvature flows
Cyril
Imbert
Université Paris-Est Créteil Val de Marne, CRÉTEIL CEDEX, FRANCE
Fractional mean curvature, mean curvature, geometric flows, dislocation dynamics, level set approach, stability results, comparison principles, generalized flows
This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phasefield theory for fractional reaction-diffusion equations. It is defined by using the level set method. The main results of this paper are: on one hand, the proper level set formulation of the geometric flow; on the other hand, stability and comparison results for the geometric equation associated with the flow.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
153
176
10.4171/IFB/207
http://www.ems-ph.org/doi/10.4171/IFB/207