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End date: 2019 × Start date: 2019 × Has files: Yes × Type: Text × Subject: free boundaries ×

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Now showing 1 - 6 of 6
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    Anisotropy in wavelet based phase field models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Korzec, Maciek; Münch, Andreas; Süli, Endre; Wagner, Barbara
    Anisotropy is an essential feature of phase-field models, in particular when describing the evolution of microstructures in solids. The symmetries of the crystalline phases are reflected in the interfacial energy by introducing corresponding directional dependencies in the gradient energy coefficients, which multiply the highest order derivative in the phase-field model. This paper instead considers an alternative approach, where the anisotropic gradient energy terms are replaced by a wavelet analogue that is intrinsically anisotropic and linear. In our studies we focus on the classical coupled temperature
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    Stability analysis of non-constant base states in thin film equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dziwnik, Marion; Korzec, Maciek D.; Münch, Andreas; Wagner, Barbara
    We address the linear stability of non-constant base states within the class of mass conserving free boundary problems for degenerate and non-degenerate thin film equations. Well-known examples are the finger-instabilities of growing rims that appear in retracting thin solid and liquid films. Since the base states are time dependent and do not have a simple travelling wave or self-similar form, a classical eigenvalue analysis fails to provide the dominant wavelength of the instability. However, the initial fronts evolve on a slower time-scale than the typical perturbations. We exploit this time-scale separation and develop a multiple-scale approach for this class of stability problems. We show that the value of the dominant wavelength is rapidly attained once the base state has entered an approximately self-similar scaling. We note that this value is different from the one obtained by the linear stability analysis with "frozen modes", frequently found in the literature. Furthermore we show that for the present class of stability problems the dispersion relation behaves linear for large wavelengths, which is in contrast to many other instability problems in thin film flows.
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    A phase-field model for solid-state dewetting and its sharp-interface limit
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dziwnik, Marion; Münch, Andreas; Wagner, Barbara
    We propose a phase field model for solid state dewetting in form of a Cahn-Hilliard equation with weakly anisotropic surface energy and a degenerate mobility together with a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. The resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the natural boundary conditions, as indicated from the first variation of the total free energy, imply a contact angle condition for the dewetting front, which, in the isotropic case, is consistent with the well-known Young's equation
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    Stationary solutions for two-layer lubrication equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Jachalski, Sebastian; Huth, Robert; Kitavtsev, Georgy; Peschka, Dirk; Wagner, Barbara
    We investigate stationary solutions of flows of thin liquid bilayers in an energetic formulation which is motivated by the gradient flow structure of its lubrication approximation. The corresponding energy favors the liquid substrate to be only partially covered by the upper liquid. This is expressed by a negative spreading coefficient which arises from an intermolecular potential combining attractive and repulsive forces and leads to an ultra-thin layer of thickness e. For the corresponding lubrication models existence of stationary solutions is proven. In the limit e to 0 matched asymptotic analysis is applied to derive sharp-interface models and the corresponding contact angles, i.e. the Neumann triangle. In addition we use G-convergence and derive the equivalent sharp-interface models rigorously in this limit. For the resulting model existence and uniqueness of energetic minimizers are proven. The minimizers agree with solutions obtained by matched asymptotics.
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    A model for the austenite-ferrite phase transition in steel including misfit stress
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Dreyer, Wolfgang; Hömberg, Dietmar; Petzold, Thomas
    We present a thermodynamically consistent model to describe the austenite-ferrite phase transition in steel. We consider the influence of the mechanical displacement field due to eigenstrains caused by volumetric expansion. The model equations are derived in a systematical framework. They are based on the conservation laws for mass and momentum and the second law of thermodynamics. By means of numerical computations for a simplified interface controlled model, we examine the influence of the mechanical contributions to the transformation kinetics and the equilibrium states.
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    The degenerate and non-degenerate Stefan problem with inhomogeneous and anisotropic Gibbs-Thomson law
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Kraus, Christiane
    The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end approximate solutions are constructed by means of variational functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law in a weak generalized BV-formula