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End date: 2015 × Start date: 2013 × DDC: 510 × Has files: Yes × Subject: asymptotic analysis ×

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Now showing 1 - 8 of 8
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    Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, Jürgen
    This paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates
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    Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, Jürgen
    In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.
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    Rational modeling of electrochemical double-layers and derivation of Butler-Volmer equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dreyer, Wolfgang; Guhlke, Clemens; Müller, Rüdiger
    We derive the boundary conditions for the contact between an electrolyte and a solid electrode. At first we revisit the thermodynamic consistent complete model that resolves the actual electrodeelectrolyte interface and its adjacent boundary layers. The width of these layers is controlled by the Debye length that is typically very small, leading to strongly different length scales in the system. We apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. This approach fully determines the relation of bulk quantities to the boundary conditions of the reduced model. In particular, the Butler-Volmer equations for electrochemical reactions, which are still under discussion in the literature, are rational consequences of our approach. For illustration and to compare with the literature, we consider a simple generic reaction.
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    Rational modeling of electrochemical double layers in thermodynamic non-equilibrium
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dreyer, Wolfgang; Guhlke, Clemens; Müller, Rüdiger
    We consider the contact between an electrolyte and a solid electrode. At first we formulate a thermodynamic consistent model that resolves boundary layers at interfaces. The model includes charge transport, diffusion, chemical reactions, viscosity, elasticity and polarization under isothermal conditions. There is a coupling between these phenomena that particularly involves the local pressure in the electrolyte. Therefore the momentum balance is of major importance for the correct description of the layers. The width of the boundary layers is typically very small compared to the macroscopic dimensions of the system. In a second step we thus apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. For a metal-electrolyte interface, we derive a qualitative description of the double layer capacitance without the need to resolve space charge layers.
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    Modelling compressible electrolytes with phase transition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dreyer, Wolfgang; Giesselmann, Jan; Kraus, Christiane
    A novel thermodynamically consistent diffuse interface model is derived for compressible electrolytes with phase transitions. The fluid mixtures may consist of N constituents with the phases liquid and vapor, where both phases may coexist. In addition, all constituents may consist of polarizable and magnetizable matter. Our introduced thermodynamically consistent diffuse interface model may be regarded as a generalized model of Allen-Cahn/Navier-Stokes/Poisson type for multi-component flows with phase transitions and electrochemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-coupled and a coupled regime, where the coupling takes place between the smallness parameter in the Poisson equation and the width of the interface. We recover in the sharp interface limit a generalized Allen-Cahn/Euler/Poisson system for mixtures with electrochemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satisfy, for instance, a generalized Gibbs-Thomson law and a dynamic Young-Laplace law.
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    A compressible mixture model with phase transition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dreyer, Wolfgang; Giesselmann, Jan; Kraus, Christiane
    We introduce a new thermodynamically consistent diffuse interface model of AllenCahn/NavierStokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a YoungLaplace and a Stefan type law.
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    New insights on the interfacial tension of electrochemical interfaces and the Lippmann equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dreyer, Wolfgang; Guhlke, Clemens; Landstorfer, Manuel; Neumann, Johannes; Müller, Rüdiger
    The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Based on the framework of continuum thermo-electrodynamics we provide some crucial new insights to this relation. In a previous work we have derived a general thermodynamic consistent model for electrochemical interfaces, which showed a remarkable agreement to single crystal experimental data. Here we apply the model to a curved liquid metal electrode. If the electrode radius is large compared to the Debye length, we apply asymptotic analysis methods and obtain the Lippmann equation. We give precise definitions of the involved quantities and show that the interfacial tension of the Lippmann equation is composed of the surface tension of our general model, and contributions arising from the adjacent space charge layers. This finding is confirmed by a comparison of our model to experimental data of several mercury-electrolyte interfaces. We obtain qualitative and quantitative agreement in the 2V potential range for various salt concentrations. We also discuss the validity of our asymptotic model when the electrode radius is comparable to the Debye length.
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    Delayed feedback control of self-mobile cavity solitons
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Pimenov, Alexander; Vladimirov, Andrei G.; Gurevich, Svetlana V.; Panajotov, Krassimir; Huyet, Guillaume; Tlidi, Mustapha
    Control of the motion of cavity solitons is one the central problems in nonlinear optical pattern formation. We report on the impact of the phase of the time-delayed optical feedback and carrier lifetime on the self-mobility of localized structures of light in broad area semiconductor cavities. We show both analytically and numerically that the feedback phase strongly affects the drift instability threshold as well as the velocity of cavity soliton motion above this threshold. In addition we demonstrate that non-instantaneous carrier response in the semiconductor medium is responsible for the increase in critical feedback rate corresponding to the drift instability.