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Author: Kitavtsev, Georgy × Author: Wagner, Barbara × End date: 2019 × Start date: 2010 × DDC: 510 × WGL Institute: WIAS ×

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Now showing 1 - 4 of 4
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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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    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.
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    Thin film models for an active gel
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Kitavtsev, Georgy; Münch, Andreas; Wagner, Barbara
    In this study we present a free-boundary problem for an active liquid crystal based on the Beris-Edwards theory that uses a tensorial order parameter and includes active contributions to the stress tensor to analyse the rich defect structure observed in applications such as the Adenosinetriphosphate (ATP) driven motion of a thin film of an actin filament network. The small aspect ratio of the film geometry allows for an asymptotic approximation of the free-boundary problem in the limit of weak elasticity of the network and strong active terms. The new thin film model captures the defect dynamics in the bulk as well as wall defects and thus presents a significant extension of previous models based on the Leslie-Erickson-Parodi theory. Analytic expressions are derived that reveal the interplay of anchoring conditions, film thickness and active terms and their control of transitions of flow structure.
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    Center manifold reduction approach for the lubrication equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Kitavtsev, Georgy; Recke, Lutz; Wagner, Barbara
    The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold. The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness eps undergo a slow-time coarsening dynamics. With this situation in mind, we construct an asymptotic approximation of the corresponding invariant manifold, that is parametrized by a family of droplet pressures and positions, in the limit when $epsto 0$. The approach is inspired by the paper by Mielke and Zelik [Mem. Amer. Math. Soc., Vol. 198, 2009], where the center manifold reduction was carried out for a class of semilinear systems. In this study this approach is considered for quasilinear degenerate parabolic PDE's such as lubrication equations. While it has previously been shown by Glasner and Witelski [Phys. Rev. E, Vol. 67, 2003], that the system of ODEs governing the coarsening dynamics, can be obtained via formal asymptotic methods, the center manifold reduction approach presented here, pursues the rigorous justification of this asymptotic limit.