Filters

2013 - 20163

More filters

20193
Reset filters

Settings

Author: Dziwnik, Marion ×

Search Results

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Existence of solutions to an anisotropic degenerate Cahn-Hilliard-type equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dziwnik, Marion; Jachalski, Sebastian
    We prove existence of solutions to an anisotropic Cahn-Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate auxiliary results which play an essential part in the present existence proof. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by 1.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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