Showing papers in "Interfaces and Free Boundaries in 2000"

Phenomenological modelling of polymer crystallization using the notion of multiple natural configurations

Abstract: Crystallization and solidification in polymers is a problem of great importance to the polymer processing industry. In these processes, the melt is subjected to deformation while being cooled into the desired shape. The properties of the final product are strongly influenced by the deformation and thermal histories and the final solid is invariably anisotropic. In this work we present a model to capture the effects during solidification and crystallization in polymers within a purely mechanical setting, using the framework of multiple natural configurations that was introduced recently to study a variety of non-linear dissipative responses of materials undergoing phase transitions. Using this framework we present a consistent method to model the transition from a fluid-like behaviour to a solid-like behaviour. We also present a novel way of incorporating the formation of an anisotropic crystalline phase in the melt. The anisotropy of the crystalline phase, and consequently that of the final solid, depends on the deformation in the melt at the instant of crystallization, a fact that has been known for a long time and has been exploited in polymer processing. The proposed model is tested by solving three homogenous deformations.

Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs

Abstract: The efficient numerical simulation of the curvature-driven motion of interfaces is an important tool in several free- boundary problems We treat the case of an interface which is given as a graph The highly non-linear problem is discretized in space by piecewise linear finite elements Although the problem is not in divergence form it can be written in a variational form which allows the use of the modern adaptive techniques of finite elements The time discretization is carried out in a semiimplicit way such that in every time step a linear system with symmetric positive matrix has to be solved Optimal error estimates are proved for the fully discrete problem under the assumption that the time-step size is bounded by the spatial grid size

Simulation of dendritic crystal growth with thermal convection

Abstract: The dendritic growth of crystals under gravity influence shows a strong dependence on convection in the liquid. The situation is modelled by the Stefan problem with a Gibbs–Thomson condition coupled with the Navier–Stokes equations in the liquid phase. A finite element method for the numerical simulation of dendritic crystal growth including convection effects is presented. It consists of a parametric finite element method for the evolution of the interface, coupled with finite element solvers for the heat equation and Navier–Stokes equations in a time dependent domain. Results from numerical simulations in two space dimensions with Dirichlet and transparent boundary conditions are included.

On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model

Abstract: We study the Γ -convergence of a family of vectorial integral functionals, which are the sum of a vanishing anisotropic quadratic form in the gradients and a penalizing double-well potential depending only on a linear combination of the components of their argument. This particular feature arises from the study of the so-called ‘bidomain model’ for the cardiac electric field; one of its consequences is that the L1-norm of a minimizing sequence can be unbounded and therefore a lack of coercivity occurs. We characterize the Γ -limit as a surface integral functional, whose integrand is a convex function of the normal and can be computed by solving a localized minimization problem.

Evolution of compressible and incompressible fluids separated by a closed interface

Abstract: This work solves the problem governing the simultaneous motion of two viscous liquids of different kinds: compressible and incompressible. The boundary between the fluids is considered as an unknown (free) interface where the surface tension is taken into account. Although the fluids occupy the whole space R3, one of them should have a finite volume. Local (in time) unique solvability of this problem is obtained in the Sobolev–Slobodetskii spaces of functions. Estimates of the solution of a model problem for the Stokes equations are considered in detail, the interface between the fluids being a plane. The Schauder method is used to study a linear problem with a compact boundary. The passage to the nonlinear problem is made by successive approximations.

On Maxwellian equilibria of insulated semiconductors

Abstract: A semi-linear elliptic integro-differential equation subject to homogeneous Neumann boundary conditions for the equilibrium potential in an insulated semiconductor device is considered. A variational formulation gives existence and uniqueness. The limit as the scaled Debye length tends to zero is analysed. Two different cases occur. If the number of free electrons and holes is sufficiently high, local charge neutrality prevails throughout the device. Otherwise, depletion regions occur, and the limiting potential is the solution of a free boundary problem.

Error bounds for a difference scheme approximating viscosity solutions of mean curvature flow

Abstract: We analyse a finite difference scheme for the approximation of level set solutions to mean curvature flow. The scheme which was proposed by Crandall & Lions (Numer. Math. 75, (1996) 17–41) is a monotone and consistent discretization of a regularized version of the underlying problem. We derive an L ∞ -error bound between the numerical solution and the viscosity solution to the level set equation provided that the space and time step sizes are appropriately related to the regularization parameter.

On a constrained variational problem with an arbitrary number of free boundaries

Abstract: We study the problem of minimizing the Dirichlet integral among all functions u ∈ H 1 (Ω) whose level sets {u = li } have prescribed Lebesgue measure αi . This problem was introduced in connection with a model for the interface between immiscible fluids. The existence of minimizers is proved with an arbitrary number of level-set constraints, and their regularity is investigated. Our technique consists in enlarging the class of admissible functions to the whole space H 1 (Ω), penalizing those functions whose level sets have measures far from those required; in fact, we study the minimizers of

A free-boundary problem for Stokes equations: classical solutions

Abstract: The problem considered is that of evolution of the free boundary separating two immiscible viscous fluids with different constant densities. The motion is described by the Stokes equations driven by the gravity force. For flows in a bounded domain Ω ⊂ Rn , n 2, we prove existence and uniqueness of classical solutions, and concentrate on the study of properties of the moving boundary separating the two fluids.

Behaviour of interfaces in a diffusion-absorption equation with critical exponents

Abstract: with continuous compactly supported initial data u(x, 0) = u0(x) 0 in the critical case m + p = 2 of the range of parameters m > 1, p 1) and strong absorption ( p < 1).

Thermo-kinetically controlled pattern selection

Abstract: Through a combination of asymptotic and numerical approaches we investigate bifurcation and pattern formation for a free boundary model related to a rapid crystallization of amorphous films and to the self-propagating high-temperature synthesis (solid combustion). The unifying feature of these diverse physical phenomena is the existence of a uniformly propagating wave of phase transition whose stability is controlled by the balance between the energy production at the interface and the energy dissipation into the medium. For the propagation on a two-dimensional strip with thermally insulated edges, we develop a multi-scale weakly-nonlinear analysis that results in a system of ordinary differential equations for the slowly varying amplitudes. We identify a nonlinear parameter which is responsible for the pattern selection, and utilize the amplitude system for predicting the evolving patterns. The pattern selection is confirmed by direct numerical simulations on the free boundary problem. Some numerical results on strongly nonlinear regimes are also presented.

Distribution of vortices in a type-II superconductor as a free boundary problem: existence and regularity via Nash-Moser theory

Abstract: This paper is concerned with a model describing the distribution of vortices in a Type-II superconductor. These vortices are distributed continuously and occupy an unknown region D with ∂ D representing the free boundary. The problem is set as follows: two constants H0 > H1 > 0 are given, to find an open subset D of the smooth bounded open set Ω ⊂ R2 and a function H defined on Ω\D such that:   div(F(|∇H |2)∇H) − H = 0 in Ω\D where the function F is analytic positive increasing H = H0 on ∂Ω H = H1 on ∂ D ∂ H ∂n = 0 on ∂ D.

The `hump' effect in solid propellant combustion

Abstract: An eikonal equation modelling the propagation of combustion fronts in striated media is studied via a level set formulation. Travelling fronts are obtained, and their speeds turn out to be monotone with respect to the angle of the striations. An effective equation for thin meniscus striations is derived from a homogenization process, thus explaining the so-called ‘hump’ effect. Finally, the time-asymptotic stabilization of unsteady fronts propagating in straight striations is proved.

A free-boundary problem in combustion theory

Abstract: This model appears in combustion theory in the analysis of the propagation of curved flames. It is derived in the framework of the theory of equi-diffusional premixed flames analysed in the relevant limit of high- activation energy for Lewis number 1. In this application, v e represents the fraction of

On convergence of solutions of the crystalline Stefan problem with Gibbs-Thomson law and kinetic undercooling

Abstract: †This paper presents a study of the relations between the modified Stefan problem in a plane and its quasi-steady approximation. In both cases the interfacial curve is assumed to be a polygon. It is shown that the weak solutions to the Stefan problem converge to weak solutions of the quasi-steady problem as the bulk specific heat tends to zero. The initial interface has to be convex of sufficiently small perimeter.

Existence and uniqueness of a classical solution for a mathematical model describing the isobaric crystallization of a polymer

Abstract: In this paper a global existence and uniqueness result is presented for the classical solution of a free boundary problem for a system of partial differential equations (p.d.e.s) with non-local boundary conditions describing the crystallization process of a cylindrical sample of polymer under prescribed pressure. The system of equations is discussed in [16] as the model for coupled cooling and shrinking of a sample of molten polymer under a given constant pressure. The velocity field generated by the thermal and chemical contraction enters the model only through its divergence. Such an approximation is discussed on the basis of a qualitative analysis.

Global existence for a non-local mean curvature flow as a limit of a parabolic-elliptic phase transition model

Abstract: We consider a free boundary problem where the velocity depends on the mean curvature and on some non-local term. This problem arises as the singular limit of a reaction–diffusion system which describes the microphase separation of diblock copolymers. The interface may present singularities in finite time. This leads us to consider weak solutions on an arbitrary time interval and to prove the global-in-time convergence of solutions of the reaction–diffusion system.

Flux pinning and boundary nucleation of vorticity in a mean field model of superconducting vortices

Abstract: We study a one-dimensional mean field model of superconducting vortices with a finite London penetration depth, flux pinning and nucleation of vorticity at inflow boundary sections. The existence of a unique weak solution is proved and the long time behaviour is studied. A numerical discretization of the model is derived and it is shown that as the time step and the mesh size tend to zero, the discrete solution converges to the unique weak solution of the continuous model. Some numerical computations are presented which illustrate the effects of flux pinning and the finite penetration depth.

A free boundary problem involving a cusp : breakthrough of salt water

Abstract: In this paper we study a two-phase free boundary problem describing the stationary flow of fresh and salt water in a porous medium, when both fluids are drawn into a well. For given discharges at the well (Qf for fresh water and Qs for salt water) we formulate the problem in terms of the stream function in an axial symmetric flow domain in Rn(n=2,3). We prove the existence of a continuous free boundary which ends up in the well, located on the central axis. Moreover, we show that the free boundary has a tangent at the well and approaches it in a C1 sense. Using the method of separation of variables we also give a result concerning the asymptotic behaviour of the free boundary at the well. For a given total discharge Q:=Qf + Qs we consider the vanishing Qs limit. We show that a free boundary arises with a cusp at the central axis, having a positive distance from the well. This work is a continuation of [5,6].