- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 16:27:29
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=15&iss=3&update_since=2024-03-28
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
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
15
2013
3
Numerical approximation of capillary surfaces in a negative gravitational field
Hanne
Hardering
Freie Universität Berlin, BERLIN, GERMANY
Capillary surface, obstacle problem, finite elements, discretization error bound
A capillary surface in a negative gravitational field describes the shape of the surface of a hanging drop in a capillary tube with wetting material on the bottom. Mathematical modeling leads to the volume- and obstacle-constrained minimization of a nonconvex nonlinear energy functional of mean curvature type which is unbounded from below. In 1984 Huisken proved the existence and regularity of local minimizers of this energy under the condition on gravitation being sufficiently weak. We prove convergence of a first order finite element approximation of these minimizers. Numerical results demonstrating the theoretic convergence order are given.
Numerical analysis
Differential geometry
263
280
10.4171/IFB/303
http://www.ems-ph.org/doi/10.4171/IFB/303
Regularity of expanding front and its application to solidification/melting in undercooled liquid/superheated solid
Xinfu
Chen
University of Pittsburgh, PITTSBURGH, UNITED STATES
Huiqiang
Jiang
University of Pittsburgh, PITTSBURGH, UNITED STATES
Moving front, regularity, solidification, melting, undercooling, superheating
This article proves that fronts of expanding domains with Hölder continuous speeds are contained in finite unions of Lipschitz graphs. As an application, the global in time existence of a solution to a free boundary problem modelling solidification in undercooled liquid or liquidation in superheated solid is established; here the propagation speed of the liquid/solid interface is assumed to be a known positive smooth function of the temperature, known as a kinetic undercooling/superheating effect.
Partial differential equations
Classical thermodynamics, heat transfer
281
322
10.4171/IFB/304
http://www.ems-ph.org/doi/10.4171/IFB/304
Well-posedness of the linearized plasma-vacuum interface problem
Paolo
Secchi
Facoltà di Ingegneria, Università di Brescia, BRESCIA, ITALY
Yuri
Trakhinin
Siberian Branch of the Russian Academy of Sciences, NOVOSIBIRSK, RUSSIAN FEDERATION
Ideal compressible magneto-hydrodynamics, plasma-vacuum interface, characteristic free boundary, elliptic-hyperbolic system
We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface we assume that the total pressure is continuous and that the magnetic field is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. Under a suitable stability condition satisfied at each point of the plasma-vacuum interface, we prove the well-posedness of the linearized problem in conormal Sobolev spaces.
Fluid mechanics
Partial differential equations
323
357
10.4171/IFB/305
http://www.ems-ph.org/doi/10.4171/IFB/305
Traveling waves from the arclength parameterization: Vortex sheets with surface tension
Benjamin
Akers
Air Force Institute of Technology, WRIGHT-PATTERSON AFB, UNITED STATES
David
Ambrose
Drexel University, PHILADELPHIA, UNITED STATES
J. Douglas
Wright
Drexel University, PHILADELPHIA, UNITED STATES
Arclength, vortex sheet, traveling wave, surface tension
We study traveling waves for the vortex sheet with surface tension. We use the angle-arclength description of the interface rather than Cartesian coordinates, and we utilize an arclength parameterization as well. In this setting, we make a new formulation of the traveling wave ansatz. For this problem, it should be possible for traveling waves to overturn, and notably, our formulation does allow for waves with multi-valued height. We prove that there exist traveling vortex sheets with surface tension bifurcating from equilibrium. We compute these waves by means of a quasi-Newton iteration in Fourier space; we find continua of traveling waves bifurcating from equilibrium and extending to include overturning waves, for a variety of values of the mean vortex sheet strength.
Partial differential equations
Fluid mechanics
359
380
10.4171/IFB/306
http://www.ems-ph.org/doi/10.4171/IFB/306
On the interactions between a solid body and a compressible inviscid fluid
Raul
Borsche
Technische Universität Kaiserslautern, KAISERSLAUTERN, GERMANY
Rinaldo
Colombo
Università degli Studi di Brescia, BRESCIA, ITALY
Mauro
Garavello
Università di Milano-Bicocca, MILANO, ITALY
Mixed PDE–ODE problems, balance laws, ordinary differential equations
This paper presents a model describing the interaction between a solid body and a compressible inviscid fluid in a pipe. The resulting system consists in a 1D hyperbolic balance law coupled with an ordinary differential equation and is proved to be well posed. Simple explicit solutions and numerical integrations show qualitative features of this model. In particular, we consider a ball falling in a vertical tube closed at the bottom. This system results in the ball bouncing on the shocks reflected between the ball and the bottom.
Partial differential equations
Ordinary differential equations
381
403
10.4171/IFB/307
http://www.ems-ph.org/doi/10.4171/IFB/307