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End date: 2012 × Start date: 2010 × DDC: 510 × Has files: Yes × Subject: asymptotic analysis ×

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Now showing 1 - 7 of 7
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    A diffuse interface model for quasi-incrompressible flows : sharp interface limits and numerics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Aki, Gonca; Daube, Johannes; Dreyer, Wolfgang; Giesselmann, Jan; Kränkel, Mirko; Kraus, Christiane
    In this contribution, we investigate a diffuse interface model for quasi–incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn–Hilliard equation numerically by a Local Discontinuous Galerkin scheme.
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    Hybrid mode-locking in edge-emitting semiconductor lasers: Simulations, analysis and experiments
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Arkhipov, Rostislav; Pimenov, Alexander; Radziunas, Mindaugas; Vladimirov, Andrei G.; Arsenjevi´c, Dejan; Rachinskii, Dmitrii; Schmeckebier, Holger; Bimberg, Dieter
    Hybrid mode-locking in a two section edge-emitting semiconductor laser is studied numerically and analytically using a set of three delay differential equations. In this set the external RF signal applied to the saturable absorber section is modeled by modulation of the carrier relaxation rate in this section. Estimation of the locking range where the pulse repetition frequency is synchronized with the frequency of the external modulation is performed numerically and the effect of the modulation shape and amplitude on this range is investigated. Asymptotic analysis of the dependence of the locking range width on the laser parameters is carried out in the limit of small signal modulation. Our numerical simulations indicate that hybrid mode-locking can be also achieved in the cases when the frequency of the external modulation is approximately twice larger and twice smaller than the pulse repetition frequency of the free running passively mode-locked laser fP . Finally, we provide an experimental demonstration of hybrid mode-locking in a 20 GHz quantum-dot laser with the modulation frequency of the reverse bias applied to the absorber section close to fP / 2.
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    Asymptotic analysis for Korteweg models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Dreyer, Wolfgang; Giesselmann, Jan; Kraus, Christiane; Rohde, Christiane
    This paper deals with a sharp interface limit of the isothermal Navier-Stokes-Korteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models.
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    Spin coating of an evaporating polymer solution
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Münch, Andreas; Please, Colin P.; Wagner, Barbara
    We consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of the thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and due to evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent volume fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system. The main practical interest is in controlling the appearance and development of a ``skin'' on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. The critical parameters controlling this behaviour are found to be eps the ratio of the diffusion to advection time scales, delta the ratio of the evaporation to advection time scales and exp(-gamma), the ratio of the diffusivity of the initial mixture and the pure polymer. In particular, our analysis shows that for very small evaporation with delta ll exp(-3/(4gamma)) eps^3/4 skin formation can be prevented
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    Overcoming the shortcomings of the Nernst-Planck model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Dreyer, Wolfgang; Guhlke, Clemens; Müller, Rüdiger
    This is a study on electrolytes that takes a thermodynamically consistent coupling between mechanics and diffusion into account. It removes some inherent deficiencies of the popular Nernst-Planck model. A boundary problem for equilibrium processes is used to illustrate the new features of our model.
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    A quasi-incompressible diffuse interface model with phase transition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Aki, Gonca; Dreyer, Wolfgang; Giesselmann, Jan; Kraus, Christine
    This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier-Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier-Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.
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    Asymptotics for the spectrum of a thin film equation in a singular limit
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Kitavtsev, Georgy; Recke, Lutz; Wagner, Barbara
    In this paper the linear stability properties of the steady states of a no-slip lubrication equation are studied. The steady states are configurations of droplets and arise during the late-phase dewetting process under the influence of both destabilizing van der Waals and stabilizing Born intermolecular forces, which in turn give rise to the minimum thickness eps of the remaining film connecting the droplets. The goal of this paper is to give an asymptotic description of the eigenvalues and eigenfunctions of the problem, linearized about the one-droplet solutions, as epsto 0. For this purpose, corresponding asymptotic eigenvalue problems with piecewise constant coefficients are constructed, such that their eigenvalue asymptotics can be determined analytically. A comparison with numerically computed eigenvalues and eigenfunctions shows good agreement with the asymptotic results and the existence of a spectrum gap to a single exponentially small eigenvalue for sufficiently small eps.